[urban_cegrell]_capacities_in_complex_analysis_(as(book4you.org).pdf

Urban Cegrell

Capacities in Complex Analysis

Friedr. Vieweg & Sohn Braunschweig/Wiesbaden

CIP-Titelaufnahme der Deutschen Bibliothek

Cegrell, Urban: Capacities in complex analysis/Urban Cegrell. -

Braunschweig; Wiesbaden: Vieweg, 1988 (Aspects of mathematics: E; Vol. 14) ISBN 3-528-06335-1

N E: Aspects of mathematics / E

Prof. Dr. Urban Cegrell Department of Mathematics, University of Umea , Sweden

AMS S ubject Classification: 32 F 05, 31 B 15,30 C 85, 32 H 10,35 J 60

Vieweg is a subsidiary company of the Bertelsmann Publishing Group.

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© Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig 1988

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Produced by Lengericher Handelsdruckerei, Lengerich

Printed in Germany

ISSN 0179-2156

ISBN 3-528-06335-1

Contents

Introduction

List of notations

I. Capacities

II. Capacitability

III . a Outer regularity

II I .b Outer regularity (cont.)

IV. Subharmonic functions in JRn

V. Plurisubharmonic functions in �n _ the Monge-Ampere capacity

VI. Further properties of the Monge-Ampere

operator

VII. Green's function

VIII. The global extremal function

IX . Gamma capacity

X . Capacities on the boundary

XI . Szego kernels

XII . Complex homomorphisms

VII XI

4

1 1

2 2

30

3 2

56

66

73

81

99

1 1 6

148

Introduction

The purpose of this book is to study plurisubharmonic and

analytic functions in [n

using capacity theory. The case n=1

has been studied for a long time and is very well understood.

The theory has been generalized to mn

and the results are in

many cases similar to the situation in [. However, these

results are not so well adapted to complex analysis in several

variables - they are more related to harmonic than plurihar

monic functions.

Capacities can be thought of as a non-linear generali

zation of measures; capacities are set functions and many of

the capacities considered here can be obtained as envelopes of

measures.

In the mn

theory, the link between functions and capa

cities is often the Laplace operator - the corresponding link

in the ITn theory is the complex Monge-Ampere operator.

This operator is non-linear (it is n-linear) while the Laplace

operator is linear. This explains why the theories in mn

and

[n

differ considerably. For example, the sum of two harmonic

functions is harmonic, but it can happen that the sum of two

plurisubharmonic functions has positive Monge-Ampere mass while

each of the two functions has vanishing Monge-Ampere mass. To

give an example of similarities and differences, consider the

following statements. Assume first that � is an open subset

VIII

of ffin and that K i s a c l os ed subs et of � . C on s i d er t h e

f ol l owi ng p r op ert i es that K may or may n ot have.

(i ) F or every z O EK t h er e i s a s ubharmon i c funct i on �

on � such that

l im � ( z ) < �(z O ) , wh er e z EK, z 1z 0. z-+ z 0

(i i ) Th er e i s a subharmon ic funct i on � on � , �t-oo ,

s uch that

Kc{z E � i �(z ) =-oo} .

(i i i ) Ther e i s a l oca l ly upp er b ounded family ( �i ) i E I of

subharmon i c function s on � s uch that Kc { z E � ; � ( z ) < �*(z ) } ,

wh er e �(z ) =sup �.(z) a nd �* ( z ) =li m �(z ' ) . i E I 1 z'-+z

(i v) If � i s s ubharmon i c out s i d e K and i f

l i m �(z ' ) < +00 , wh er e z '�K, �z E � , z '-+ z

th en � ext ends t o a un i qu el y d et erm i n ed s ubharmon i c funct i on

on � .

In c l a s s i ca l p ot ent i a l t h eory , i t i s a th eor em that t h es e

pr op er t i es a r e equ i va l en t a nd t he c ompa ct s et s that have th e

pr op ert i es a re exact l y t hos e w i t h van i sh i ng Newt on capa c i t y .

To study the c or r espond i ng p r op ert i es i n �n , ffin ha s t o

b e r ep l a c ed b y �n a nd th e s ubharmon i c fun ct i on s b y the p l u r i -

subharmon i c f unction s . C ond i t i on s (i ) - ( i v ) a r e then t ra n s f ormed

IX

into conditions (i')-(iv') and they are no longer equivalent

but:

(i') -;; (iii) <:=;;> (iii') -:; (iii);; (iv').

(cf. the last reference in Section X).

Section I and II are concerne,d wi th general capaci ty

theory, Section III capacities related to function classes.

In Section IV and V we specialize to subharmonic and pluri

subharmonic functions, respectively.

In Section V and VI we also study the complex Monge-Ampere

operator. In Section VII, VIII and IV we use the results

obtained to study certain plurisubharmonic functions, while

Sections IX, X and XI are devoted to capacities and analytic

functions. Finally, Section XII is concerned with the capacity

generated by representing measures on the spectrum of the

algebra of bounded analytic functions.

This book contains the notes I prepared in November 1 98 3

for a couple of seminars at the University of Uppsala. These

notes were made into a more complete form during a series of

lectures at the University of Umea in the fall 1 98 5 and at

Universite Paul Sabatier, Toulouse in December 1 98 6 .

x

General references

Capacity theory

Gustave Choquet, Lectures on analysis, 1-3. W.A. Benjamin,

1969.

0.0. Kellog, Foundations of potential theory. Springer

Verlag, 19 29.

N.S. Landkof, Foundations of modern potential theory.

Springer-Verlag, 197 2.

Complex analysis in several variables

L. Hormander, An introduction to complex analysis in

several varibles. North Holland, 19 73.

Steven G. Krantz, Function theory of several complex

variables. John Wiley & Sons, 198 2.

P. Lelong, Plurisubharmonic functions and positive

differential forms. Gordon and Breach, 1969.

List of Notations

Notat i o n

IN JR

P ( U)

a

au

LP(]J ,U)

Meaning

t he n a tural n umb e r s

the real numbers t he c omplex number s

a ll the subsets of U

the p r oduct space Ux . . . x U

the cha racte r i st i c f u nc t i on of U

the c l a s s of r e a l or c ompl ex val ued

f u nc t i on s on n w i t h c on t i nu ou s deri va t i ve s or

or order �P

the d i f f e r e nt i a l operat or d r -d- d z . Zj J

the d if fe r e nt i a l operat or r -d -dzj dZ. J

the b ou ndary of the set U

the exte r i or p r oduct

]J is a mea s u r e on U and LP (]J, U ) i s the

c l a s s of ]J-mea s u rable funct i on s on U w i th

I Capacities

De f i n i t i on . Let U be a a-compact Hausdorff-space. A capacity

c on U is a set function defined on P (U) , the subsets of

U , with the following properties:

i ) P ( U ) 3 E � c ( E ) E JR+ c(¢) = 0 ,

i i ) P ( U ) 3 E -E, s-++oo, � sup c(E ) = c(E) s

sEN s

iii) If K , s E ill is a decreasing sequence of compact subsets s

of U, K = n K , then s'

s=1

inf c(K ) = c(K). sEN

s

De f in i t i on . A set function satisfying property i) and ii)

above is called a precapacity.

Example 1:1. If W is a positive Radon measure then the outer

measure w* is a capacity.

Capacities are thus a non-linear generalization of measures.

Observe that no linearity is assumed e.g. if f: JR+-+JR+ is a

continuous and increasing function vanishing at the origin,

then f 0 c is a capacity for every capacity c.

Def i n i t i on .

tends to

Let w , s E ill and w be measures. We say that s

w weakly and write if

limf�dWS = f�dw , �� E C

O(U).

s-+oo

- 2 -

Lemma I: 1. If is a sequence of positive measures

such that and if is an upper semicontinuous func-

tion with compact support then

By monotone convergence,

Theorem l�'. Let M be a weak*-compact set of positive measur-

es. Then c ( E ) = sup w*(E) is a capacity. wEM

Proof. Everything but iii) is clear. Given E > O. For every

K., choose w· E M with w.(K.) > c(K.) - E. J J J J - J

compact, there is an accumulation point W E M

There is a continuous function � > XK so that

and since

Since M is 00 for ( lJ . ) . 1.

J J= J(CP-XK)dlJ < E

is a weak*-neighborhood of W, there is a j E ill so that

XK. < � and J

Therefore, c(K) � C(Kj) � f�d�j + ( < fCPdW + 2E < W(K) + 3s

which proves the theorem.

- 3 -

Corollary 1:1. I f O < � i s a l ower s em i cont i nuous f un c t i on

n o n UCR , U c ompact , and i f M= { IJ� O ; O .s. � * d lJ� 1 } t h e n

s up IJ* ( E ) IJEM

i s a capac i ty . ( � * dlJ s tand s f o r convolut i on . )

Proof. It i s c l ea r f r om L emma 1:1 t hat M i s weak * - compact .

II Capacitability

Let c be a capacity.

De f i n i ti o n . A set E is said to be c-capacitable if

c(E) sup{c(K); E�K, compact}.

Def i n i t i on . A set E is said to be universally capacitable

if E is c-capacitable for every capacity c on U .

Before proceeding to the main theorem of this section we

need some topological concepts.

U is as always a a-compact Hausdorff space.

Go consists of the sets that can be wrjtten as a denumerable

intersection of open sets.

Fa consists of the sets that can be written as a denumerable

union of compact sets.

Koo consists of the sets that can be written as a denumerabl e

intersection of sets from Fa.

A set E in a Hausdorff space is called K-analytic if

there is a Koo-set F in a compact space W and a continuous

map f: F�E such that E = f( F) .

metric space is called POlIS h.

A complete, separable and

Finally, a continuous image of a Polish space is called an

analytic set.

- 5 -

Proposition 11:1. If E is a K-analytic set in a compact

space U, then there is a Koo-set F in a compact set W

and a continuous map f: W�U with f(F) = E .

Proof. Put r = {(x,f(x)), x E F} where f: F�E and W�F

is as in the definition of K-analytic. Then E � proj r and

r = rn(FxU) (closure in WxU) since r is closed in FxU by

the continuity of f. Since WxU is compact so is r and

since FxU is a Koo-set so is r (�rnFxU). Thus proj: r�E

is the required function.

Theorem 11:1.

a) Every Hausdorff space that is a continuous image of

K-analytic set is K-analytic.

b) Denumerable unions and intersections of K-analytic sets

are K-anal ytic.

Theorem 11:2

a) Every Polish space is homeomorphic to a Go-set contained

in a compact metric space.

b) Every Borel set in a Polish space is K-analytic.

Proof (of Theorem 11:1).

a) is clear because of the transitivity of continuity.

b) f

Let F �B � E where B is a Ko � -set in the compact

n n n u

space Fn

and f (B ) = X cE n n n ·

Consider Fa = L Fn

and

let F be a compact space containing Fa (E F denotes n

for a moment the topological sum E F = uF x{n}). n n

Let

B = � B and define n

- 6 -

f: B-+E by f(x) = f (x) n

Then f is continuous with feB) = uX n

more B = nB . n . n, l l

where B . is a Ko

of n, l

if

Further-

= u B x{n} n

= u n B .x{n} = n(uB .x{n}). Therefore n n i

n,l i n

n,l

B is a Koo of F so UXn

is K-analytic. n

It remains to prove that nX n

is K-analytic. Let F = ITB n

n the product space and let C = ITB

n. The set C is the inter-

section of the cylinders b n

of F where bn

B x IT F n prfn

p

where each bn is K

oo hence so is C. Furthermore, we

denote by f the canonical extension of n

Every

f is then defined and continuous on C. Then for every pair n

i, j

{z E C: f.(z) = f. (z)} J l

is a closed subset of C. Since C is Koo it follows that

B = n {z E C: f.(z) = fJ.(z)} . . l

l, J

is a Koo-set in F.

Define now f to be the restriction to B of fn

. This

restriction is continuous and since f (B)cX , �n E ill we have n n

f ( B) c nX . n

On the other hand, nX cf(B). n

For let y E

for every n there is an xn

E Bn such that f

n(x

n)

n X , n

y. The

point (xn

) E B and therefore f(xn

) = y. Therefore A = f(B)

which proves the theorem.

- 7 -

Proof (of Theorem 11:2). a) Let 1 =[0,1] and ID I = X [ O , l ] . ID

Then rID is a compact, separable and metrizable space with

metric ro

D(a,b) = L j =1

space with metric d.

consider the map h,

I a .-b·1 J J Let now x be a Polish

(We can assume that 0 < d < 1). We

where is dense in X. We define Y = h(X) and claim

h 1 ) X � Y is a homeomorphism.

2) Y is a Go-set.

For if in X, then d(y.,x )�d(y,x ), j�+oo, tin E ID J n n

so therefore D(h(y.))�D(h(y)), j�+ro which means that h is J continuous. Since is dense in X, h is one-to-one.

If h(Yj)�h(y) E Y,

This means that

then d(y.,x )�d{y,x ), j�+oo, tin E ID. J n n

dry ,y)�O, j�+oo J

for given E: > 0, choose n E:

so that d{y,xnE:

< E:. Then o < d (y . , y) < d (y . , x ) + d (y , xn ) ] - J n

so lim d{y. ,y) < E:. j�+ro ]

E: E:

It remains to show that Y is a Go-subset of lID.

Let Qn be the set of points z in rID such that there

is an r > 0 such that n D(h(x1),z) < rn; D(h(x2),z) < rn � d(x1,x2) < n Qn is open.

Then each

Assume that CD

Zo E Yn n Q and take xP E X so that n=l n

D ( h ( xP ) , z 0 ) -+- 0 , p-+-+oo.

corresponding r and

p > p - n �

n

D ( h ( xP) , z 0 ) <

d(xP,xq) < -n

- 8 -

Given € > O . Take

then Pn so that

r . Hence, if p,q n

< € •

Therefore (xP)oo p=1

is a Cauchy sequence 1n

lim xP = p-+-+oo

00

and therefore

Y = n u {y E rlN; D (y , z) < J..-}. j= 1 zEY J

h > (", choose

> Pn

X so

Finally,

b ) Since rlN is compact every open subset of rlN is a

the

F -set. Therefore, by a ) every Polish space is K-analytic. a Suppose now U is an open subset of a Polish space; we claim

that U is Polish. It is clear that �E is Polish and if

d is a metric on E,

v = {(t,x ) E mxE; t'd(x,E \ U) = 1}

is closed. Therefore V is a Polish space and V 3 (t,x)-+-x E U is a homeomorphism so U is a Polish space.

Let now IT be the family of subset E of X so that E

and its complement are K-analytic. We have just shown that IT contains all open sets and by a) IT contains X. By Theorem

11:1 b) IT is a a-algebra and therefore IT contains all

Borel sets.

Corollary 1 1 : 1 . Every analytic set is K-analytic.

Proof. Let P be a Polish space. By Theorem 11:2a), there is

a G�set n o. contained in a compact metric space so that U j ElN J

- 9 -

f(nO.)=p for a cont i nuous f . J I n a compact met r i c space , every open set i s a Ka-set . There fore P is K-analyt i c and so i s any conti nuous image o f P .

Theorem 1 1 : 3 . Every K-anal yt i c set i n U i s un iversally ca-pac i tabl e . ( Remember , U i s a ssumed to be F ) . For the proo f , a we need two l emmas .

Lemma 1 1 : 1 . Every Koo i s un iversa l l y capac i table .

Proo f . As sume that A=nA where A =uK n , p ' K n n n , p n p and increas i ng i n p . Let c be a g iven capac ity g i ven number < c ( A ) . S i nce ACA 1 there i s a Pl

<Xl c ( AnK 1 »A . Put a 1 =AnK 1 and de f i ne ( an ) n=l , p 1 ' P 1 t i vely : I f a n-' lS chosen , take where a =a ,nK . n n- n , p n

S i nce a CK 1 n . . . nK n , Pl n'Pn

we have that

C ( K 1 n . . . nK »A . , Pl n ' Pn 00

Pn so b ig that

compact and A so that i nduc-

c ( a »A n

The set K= n K i s compact and conta i ned i n A s i nce n=l n'Pn K CA . n , p n n

a

Furthermore , s i nce c ( K » \ by Ax iom i i i ) , the lemma i s proved .

Lemma 1 1 : 2 . I f f: Uo�U is a cont i nuous funct ion between two Hausdor f f spaces and if c i s a capacity on U then cof i s a capac i ty on UO'

- 1 0 -

Proof of Theorem 11 :3 . Let A be a K- a na l yt i c set i n a com-

pac t Hausdor f f space U . By Propos i t i o n 11: 1 , there i s a c om-

pact space B O conta i ni ng a Ko o - s et B a nd a con t i nuous

f u nc t i on f: B O�U s o that A= f( B ) . I f c i s a g i ven capa c i ty

o n U , then cof i s a capac i ty o n B O by Lemma I I : 2 . S ince

B is a Ko o - s et it fo ll ows f rom Lemma 11: 1 that

c ( f( B ) ) =s up { c ( f ( K ) ) : K compact s u b s e t of B } , wh ich completes

the proof s i nce f is con t inuous o n B O '

Capa c i ta b il it y conc e r n s i nner regu l a r i ty; approx i mat i on

f rom t he i n s i de w i t h compact s et s .

Outer regu la r i ty ; approx imat i on from t he out s ide w i t h ope n

set s , i s the top i c i n t he next sect ion. I n c o nt r a s t t o t h e c a s e

o f me asu r e s , t h e r e a r e

Notes a n d r e f erences

F - s e t s that a r e not outer regula r . (J

Theo rem 11:3 i s d ue t o Choque t .

Choquet , G . , Theory o f capa c i t i e s . A nn . In s t . Four ie r 5 ( 1 9 5 3 - 54 ) .

Choquet , G. , Lectures on a na l y s i s . W . A . Ben jami n. Inc .

1 9 6 9 . See a ls o chapter two i n : Federer , H . , Geomet r i c mea s u r e

theory . S pr i nger-Ve r lag , Ber l i n-He i de lberg-New Yor k . 1 9 6 9 . And the append i x i n: Treves , F. , Topo l og ical vector space s ,

d i s t r ibut i on s a nd kerne l s . Academic Pre s s I nc . 1 9 6 7 . There are a na ly t ic s e t s wi th comp lemen t s that a r e not u ni

ver s a l l y capac i ta b le , see: Del lacher i e , D . , En semb l e s a n a l yt ique s ,

capa c it es , me s u r e s de Hausdo r ff. Spr i n ge r LNM. 2 9 5 , 1 9 7 2 , pg 28.

III a Outer Regularity

I n th i s sect ion , we assume S to be a compact and metric space.

Def i n i t i on . Let c be a capac i ty on S. We say that c i s outer regular i f c * ( E ) = i n f{c ( O ) ; EeO, 0 open } i s a capac ity .

To veri fy that a g iven capac i ty i s outer regular , i t i s enough t o check property i i ) . Observe a l so that i f c i s outer regular , then i t fo l l ows from Theorem 1 1:3 that c = c * o n a l l K-analyt i c sets , s i nce they agree o n a l l compacts .

I t i s c lear that every pos i tive measure de f i ne s an outer regul ar capac i ty . The fol l owi ng example shows that there exi st s a compact denumerable set M o f probabi l i ty-measures such that i ) there i s a F -set F w ith c ( F ) < c * ( F ) a

i i ) c ( E ) = 0 � c * ( E ) = 0 , where c ( E ) = sup � * ( E ) . �EM

Exampl e I11 : 1 . Let and 0 , n Elli , be D irac measures n wi th mas s at zero and respect ive ly . Let m be the Lebesgue measure on the un i t i nterva l [ 0 ,1] and denote by M the denumerable compact set o f measures { o i®m }7=o and l et E be the F - set a

00 { ( O , y ) E R2; -21 < y < 1 } u {(� , y ) i 0 < y < -2'}.

, 1 1 1=

If we , a s usua l , de f i ne c ( E ) but c * ( E ) = 1 .

sup � ( E ) � EM

we have c ( E ) 1 == "2

- 1 2 -

Def i ni t ion . Let c be a capac i ty. De f i ne c by

c ( E ) = i n f {c ( F ) I ; ECF , F a G o - s et } .

It i s c lear that c ( E ) < c( E ) < c * ( E ) and we now use Example 111 : 1 to construct a capac ity ( o f the type cons idered i n Theorem 1 : 1 ) such that c � c .

Example III:2. Let M be the compact set o f probab i l i ty measures , M = {ox®m}xE [ O , l ] where 0 i s the Di rac measure x with mass at x and where m i s the Lebesgue measure on [0 , 1 ]. We de f i ne the capacity c by c ( E ) = sup �* ( E ) .

� EM We f i rs t de f i ne En ' n Ern , by i nduct i on . Let E 1 be the

set in Exampl e 111 : 1 . To construct E , n >l , n d i v i de [ 0 , 1 ] in n equal i nterva l s and do the construct i on o f Example 111: 1 i n each i nterval s o that the set so obta i ned does not i ntersect E l ' n- Put

Then

00 E = u E n ' n=l

c ( E ) = -2- hut i f � F we cla i m that c ( F ) = 1 . For i f

i s

open sets then the l-d imens ional

any 00 n 0

s=l sets

G o-set conta i ning E

= F , ECO s' s ElN are s Ss = { x E [ 0 , 1 ] ,

{x } x [ O , l ] C OS } ' s E IN are open and dense i n [ 0 , 1 ] by the con-00

struct ion o f E . Hence , n S i s dense and i n par t i cular s= 1 s 00

non-empty whi ch means that c ( n O ) = c ( F ) = 1 . s=l s

Exampl e 111:3. Let c be the capacity de f ined i n Exampl e I I I : 2 . Then there ex i s t s a Go - set A conta i ned i n [O , l ] x[O , l ] such that i ) c ( A ) = 0

- 1 3 -

i i ) c ( O)::;; f or every open set 0 contai n i ng A .

Th i s fol l ows directly from the ex i stence o f a Go-set A conta i ned i n [ O , l ] x [ O,l ] such that 1) A i s the graph o f a l ower semicont i nuous funct i on . 2) I f K i s a compact set i n [0 , 1 ]x [O ,l 1 with

projlK = [ 0 , 1 ] then AnK � ¢.

Def in i t i on . A set function c i s ca l led s trongly subadd i tive i f

for a l l compacts K 1 , K2•

Theorem 111 : 1 . Every strongly subaddit i ve capac i ty on S i s outer regular .

Lemma 1 11 : 1 . As sume that c is s trong l y subadd i t ive and that U . �V . , 1 1 <

c (

1 < h n U U.)

i = 1 1

are n - E

i =l

open subset o f V . Then n n

c ( U . ) < c ( U V . ) - E c ( V . ) . 1 . , 1 i::;;' 1 1::;;

Proo f o f the l emma . The proof i s by i nduction . As sume that U . �V . , , _< j < n , are open sets . We want to prove that 1 1

n n n n c ( U U . ) + E c ( V . ) < c ( U V . ) + E c ( U . ) .

J'=l J . 1 1 . 1 1 . , 1 J= J = J= Th i s i s true i f n = 1 . If n = 2 we put U = U 1 and V = v,UU2. Then c ( U1UU2 ) + c ( V 1 ) < c ( UuV ) + c ( UnV ) < c ( U ) +

+ c ( V ) = c ( V , UU2 ) + c ( U, ) .

- 1 4 -

On the other hand , i f we put U = U 2 we get c ( U2UV 1 , + c ( V2 ) � c ( UuV ) + c ( Unv ) < c ( U ) + c ( V ) = c ( U2 ) + + c ( V 1 uV2 ) ·

Addi ng the i nequal i t i e s g ives

wh ich proves the l emma for n = 2 .

Assume now that the f ormula has been proved f or n . We then prove i t for n + 1 . Put

U 1 = n u U. ,

j= 1 J

n u V. ,

j= 1 J

2 U = Un+ 1

The case n = 2 then g ives

c ( U 1 UU 2 ) + c ( V 1 ) + c ( V2 ) < c ( V 1 UV2 ) + c ( U 1 ) + c ( U2 )

and the i nduct ion assumption

Hence

n n n n c ( U U . ) + E c ( V. ) < c ( u V . ) + E c ( U . ) .

j = 1 J j= 1 J j= 1 J j = 1 J

n+ 1 n+ 1 n+ 1 n+ 1 c ( u U . ) - E c ( U . ) - c ( u V . ) + E c ( V . ) -

j= 1 J j= 1 J j = 1 J j= 1 J n n+ 1 n

- ( U V. ) - c ( V n+ 1 ) + c ( u V . ) + C U U. ) + c ( Un+ 1 ) -j = 1 J j= 1 J j= 1 J

n+ 1 n+ 1 n+ 1 E c ( U . ) - c ( u V . ) + E c ( V. )

j=1 J j=1 J j = 1 J

n n n + E c ( V. ) - c ( u V . ) - E c ( U . ) < 0

j= 1 J j= 1 J j=l J

n = c ( u U . ) +

j= 1 J

- 15 -

b y the i nd uc t i on ass umpt i on .

Proof o f the theorem . I t is e nough t o prove t hat i i ) h o l ds

true f o r c* . So ass ume that E """ E , s-++oo . G i ve n IS > 0 , s

fo r eve r y i there is a n open s et U. 1 conta i n i ng

that

so

c( U . ) - c*( E . ) < IS 1 1 - 2i

By Lemma 1 1 1:1 we have

n n n n c( u U . ) + 1: c*( E . ) < c*( u E. ) + 1: c( U . ) . 1 1 . 1 1

.

1 1 1'=1 1 1= 1= 1=

n n n

E . 1

c ( u U . ) - c* ( u E . ) < 1: c ( U . ) - c* ( E . ) < E: . i=1 1 i=1 1 i=1 1 1

H ence

00 n 0 < c( u U.) - c*( E ) < l i m ( c(u U . ) - c*( E ) ) = i=1 1 n-++oo i 1 n

n = lim ( c( u U.)

. 1 1 n-++oo 1 =

n c*( u E . ) ) < IS. . 1 1 1=

H ence c*( E ) < l im c*( E ) wh i ch pr oves the theorem . n n-++oo

s uc h

Let M b e a s et o f pos i t i ve measu r es o n S w i th mass

less or eq u a l to 1. W e f i n is h t h is s e ct i o n by prov i ng a

theorem that gi ves s u f f i c i en t cond i t i on o n M s o that the s e t

funct io n su p �( E ) is outer regu l a r on i ts z e r os ets . ]JEM

Def i n i ti on . L et N be the s et of pos i t i ve , l ower s em i cont i nu -

o us f u n c t i ons � on S w i th the property t h a t t o every IS > 0 there is a n open s et A w i th s up �(A) < E: a n d s uch that the

�EM

r es t r ic t i o n o f � to S \ A i s c on t i nuous .

Lemma 111 : 2 . Let Q be a downwa rd d i rec ted f am i l y o f pos i t i ve

func t i on s s u c h that

Then i n f s UE f�dU = � EQ u EM

M 3 � -+ f �du

s uE. i n f f �du . u EM � EQ

i s cont i nuou s f o r every

Proof . It i s c l e a r that s UE i n f f�dU sup i n f J �d]..l . G iv e n � E M t h e r e i s a � E Q w i th ]..l EM � EQ

f�d� < a s o A = {u EM; fCP d � < a}, � E Q i s then a n open �

� E Q .

coveri ng o f and s i n ce M is c hoos e T M c ompac t we c an ( �. ) . 1 T 1 1=

s o that u A :JM . But s i nc e Q i s downwar d d i r ec ted t h e r e i s i=1

� . 1

a n � E Q wh i c h i s dom i nated by a l l � . , 1 < 1 - i < T . T he r e f o r e

f�d� < a, \:f� E M s o i n f sUE f cp d]..l < a wh i c h p r ove s the l emma . �EQ ]..I E M

Theorem 111 : 2 . A s s ume that � i s bounded , po s i t i ve a nd l ower

sem ic on t i nuous . Then � E N i f and o n ly i f

M :3 u f+ f �d �

i s cont i nuou s .

Proof of Theorem 1 1 1 : 2 . � ) A s s ume that cp E N a n d that lJ s�]..I .

G i ven € > 0 c hoos e O€ a s i n the de f i n i t i on and extend �

to �€ ' cont i nuous o n U . Then

- 17 -

s o l im l f�dU s - f�du l = O . ( Note that s .... O

].)(0 ) < E: E: - by Lemma I: 1 ) .

�) A s s ume that M 3 u �f�dU i s c on t i nuou s and le t N l b e the

pos i t i ve c on t i nuous f u nc t i on s dom i nated b y � . By Le mma 111:2

we have o = s UE uEM

i n f f ( � -If' ) dU == If' E N l

We c a n

c hoose If', E � i E IN, N 1 ' s u p f� uEM

to be a n i ncr ea s i ng s eq uenc e of func-t i on s w i t h _ If',d]J <

1 ] 22j+l

, j = l ,2, . .. , and w i th l i m i t � .

co Then � == If'1+ L

j==l If', l-If', J+ ]

a nd i f we put

then ]J(OK)

Then c ( E )

wh ic h s hows

c ont i nuous

co .s. f2K(,L If',+,-If',)dlJ

J=K ] ] 2K

< 22K+2 -

< 1 a nd CE have m=T on we

2 co

that L If', 1 - If' ' c onverges j = l J+ ]

o n CE wh ic h c omple t e s the

co OK == { x: L

j=K

< 1 Let 2K

co L If' '+l-If', j==K ] 1

If'

 -' } 2K

co E OK· == u

K=m

K > K' m , 2

CE; � i s

Theorem 1 11 : 3 . Let M and N b e de f i ned a s above . A s s ume that

N c on t a i n s a c on vex c one R of f u nc t i on s w i th the f o l l ow i ng

prope rt ie s .

i )

i i) 1 E R .

co I f ( � , ) , 1 i s a u n i forml y b ounded a n d monotone s eq uence

] J== � n R , then � O ( l im �, ) * E R and l i m f�, d]J == f� O dU ' j .... +co ] j .... +co ] �u E M whe re � O i s the l a rg e s t l owe r s em ic ont i nuous

m i no r a n t of l i m � , . j .... +co ]

i i i ) I f � , If' E R then i n f ( � , If' ) E R .

iv) I f co

( A , ), 1 J J== i s a dec re a s i ng s eq ue nc e o f o pen se t s w i th

lim sup ]J ( A , ) == 0 then j .... +co ]JEM J

- 18 -

l im in f { s up fCPdW i cP E R , cP � 1 o n A.} = o . J j-++co lJEM

v } I f K is a compact s u b s et of S w ith s up lJ ( K } = 0 then lJEM

there i s a s equence co (A . ). 1 J J = o f open s e t s conta i n ing K

s uch that

l im sup lJ(A.) = O . j-++co lJEM J

Then G(E) = in f { sup fCPd lJi cP E R , cP > 1 o n E} is a n outer regulJEM

l a r capa c i ty .

Corol lary 1 11 : 1 . As s ume t hat M , N a nd R are a s in Theorem

111:3. Then , to every Bore l s e t E in S with s up lJ(E) = 0 lJEM

there is a dec rea s i ng s equence

E w ith

l im sup lJ ( A . ) = O. j-++oo lJEM J

co ( A ) . 1 J J = o f open s e t s conta in i ng

Proof . S i nce a l l f u nc t ions in N and hence in R are l ower

semicont inuous , the set f u nc t io n

G(E) = i n f { sup fCPdUi cP E R , cP > 1 on E} lJEM

i s "outer" in the s e n s e that

G(E) = inf{G(A)i ECA open } .

Th i s proves that G s a t i s f ie s ax i om i i i ) and a l so that the

corollary f o l l ows f rom the theorem .

co Let now (CPj)j=l be a decreas ing s equencee o f funct i o n s

1 n R , cp. < I J - a nd l et be the l a rge s t l owe r semi continuou s

- 1 9 -

mi norant o f l im � . . We cla im that to every £ > 0 there i s a j-++oo J

�£ E R such that �£ = on { l im �j > �O} and such that sup f� dw < £ . Let £ > 0 be g i ven . S i nce a l l the funct ions w E M E: 00 �O ' ( �j ) j= l belongs to N , there i s a decrea s i ng sequence o f

00 open sets ( A. ) . 1 with l i m sup w ( A . ) = 0 and such that al l J J = j-++oo wEM J

the funct i ons are conti nuous on CA . , j E m. By i v ) , there i s J a � ,.. E R such that sup f � dw < £/3 , � > 1 on A . for c. W EM £ £ - J £ some A . . Then { x E CA . ; � . > �O + l} = K� i s a decreas i ng J £ J £ J - v J

00 sequence o f compact sets and sup w( n K�) = 0 by i i ) . Hence

WEM j=l J by v ) and i v ) there i s a sequence ( �v ) oo of functions i n v= l R with �v > on

00

00 n K� and such that

j = 1 J

Then T = in f ( L �v , l ) E R by ( i i ) for i ncrea s i ng sequences , v=l

and T > on { ql 0 < l im ql. } nCA . . J J £ Furthermore sup f Tdw < -}

wEM

so � + T > 1 £ on proves the c la im .

00 Let now ( E . ) . 1 J J= 00 S with E = u E . .

j=l J

Choose ql� E R , ql� �

We be an want

and

i ncreas i ng

sup fql + T < £ whi ch wEM £

sequence o f subsets of to prove that l im G ( E . ) = G( E) . j-++oo J

on E. so that SUP fql�dW---"G(E.), K-++oo , J wEM J

�j E m where we can assume that j j �K+ l < qlK ' j , K E m . We denote by �� the largest l ower semi conti nuous mi norant o f j l im �K .

K-++oo

- 2 0 -

We can as sume that min { cpli I > j , I + m m - < j

j+1 CPo ' j E ill + K} ) . Let

( for we can replace CP� by € > 0 be g i ven and choose

cpj as above so that E where lim K-++oo

and so that

00 Then'!' . = Cp� + i n f ( � cp s , l ) E R , '1' . < '1" + 1 and'!' . >

J s= 1 € J - J J on E . . Hence G ( E ) � sup I l im 'I' .d� = sup lim I'!' . d� = J � EM j-++CXl J �EM j-++oo J

= sup �im I ( CP6 + i nf ( � cp� , 1 ) ) d � � l im sup ICP6d� + E . But �EM J-++oo s= 1 j-++oo �EM

s i nce a l l funct ions j 00 ( CP O ) j = 1 cont i nuous we have by i i )

and are lower sem i -

sup I CP�d� = sup i n f I CP�d� � i n f sup ICP�d� � G ( E . ) . � EM � EM K K � EM J

Hence G ( E ) < l im G ( E . ) + E wh i ch proves the theorem . - j-++oo J

Remark. If R and M satisfies i ) -v ) then M can be replaced, by its weak*-c losure .

Notes and re ferences

Exampl e 111 : 1 i s due to B . Fuglede, Capacity as a subl i near functional general i zing an i ntegral . Der Konge l i ge Danske Vi den-skabernes Selskab . Matematisk - fysiske Meddelel ser . 38 . 7 ( 1 97 1 ) .

The exi stence o f a Go - set A with propert ie s 1 ) and 2 ) i n Example 111 : 3 was proved by Roy O . Davi es, A non-Prokhorov space , Bul l . London Math . Soc . 3 ( 1 9 7 1 ) , 3 4 1 - 3 4 2 . The use o f A ln th i s context was observed by C . Del l acher i e , Ensembles

- 21 -

a na l yt iques , capa c i t es , me sures de Hausdor f f . Spr i nger LNM . 2 95 ,

1 9 7 2 . pg. 1 06 Ex. 4 .

Theorem 1 1 1 : 1 i s a var i a nt o f a theorem due to Choquet .

See the r e f e r e nc e s i n Sect i on I I .

Repre s e ntat i o n o f s t r o ng ly subadd i t i ve capac i t i e s by

mea s u r e s has been s t ud i ed by Bernd Anger , Repre s e nta t i on of

c a pa c i t i es . Math . A nn. 2 2 9 ( 19 7 7 ) , 2 45- 2 5 8 .

III b Outer Regularity (Co nt.)

In th i s section , we continue our study of outer regularity but in a more special s i tuat i on . Many problems in complex func-tion theory are related to outer regu lar capac i ties - in par-ticular outer regularity o f zero sets . We there fore proceed as fol lows .

Assumptions . Let i n what fo l lows F be a convex cone of posi-tive and lower semicont inuous funct ions ( l.s . c . ) defined on a compact and metric space U. h == in f { cp E F; g < cp} . g -

H == sup {0i 0 continuous , 0 < g} g where we assume that E F for every bounded and pos i t i ve function g and that LS continuous if g is .

Let 6 be a g iven probab i l i ty measure on U such that fhgd6 for all bounded positive funct ions gi we a l so

assume that fcpd6 < +00, �cp E F .

Furthermore , we assume that i f sequence of functions in F with lim ep . E F . . 1 1-*+00

{m.}� 1 '1"1 1== is an increasing lim Jep.d6 < +00 then . 1 1-*+00

Observe that Hh g = h for a l l l . s . c . g and that g

Hcp == hep == ep for al l ep E F . Note a l so that ep l , ep2 E F implies

- 2 3 -

For i f 9 i s l.s.c . , choose g,"'g , g . cont i nuous . J J

Then h = Hh < Hh < h g . g . J g . 9 J

Now h E F· , h � so l im h E F and s i nce lim h g . J

g . J g . J

we get that h = l im h E F and that h 9 g . 9 J

The "f i ne" problem i s now to dec ide i f

= Hh . 9

E f+h ( z ) XE

g . J

i s a capac i ty for every f ixed z E U ( X E i s the character i s t i c funct ion for E ) .

The "coarse" problem 1S to dec1de i f i s a capac i ty.

E f+ fh ( z ) d o ( z ) XE

>

Assumi ng a l l th i s about F , we de f i ne a class of pos i t ive measures M , M = { w � 0; f�dW � f�dO , �� E F } .

It i s c lear that M i s convex and s i nce every funct ion i n F i s l . s . c . , M i s compact by Lemma 1:1. We now def i ne c , c ( E ) = sup w ( E ) wh i ch i s a capac i ty by Theorem 1:1, and the

lJEM

connec t i on wi th outer regular i ty i s that c outer regular i f and only i f E f+ f h do i s a capac i ty ( c f . Propos i t i on 111:1 XE below ) .

We now turn to the study of the f ol lowi ng statements . 1 ) Every bounded funct i on i n F i s a member of N . 2 ) c i s outer regular . 3 ) c ( E ) = fh do for every Borel set E . XE 4 ) I f E i s a Borel set with c { E ) = 0 then c * { E ) = O .

9

5 ) c { { h > Hh } ) = 0 for every pos i t i ve and bounded funct i on g . 9 9

- 2 4 -

Lemma 111 : 3 . De f i ne for bounded funct ions g : c ( g ) = sup J9dIJ. and L ( g ) = J hgd o .

IJEM

Then 1 ) c ( g ) � L ( g ) . 2 ) Equal i ty holds i n 1 ) i f 9 is upper or lower semiconti nuous. 3 ) L { g ) = i n f { L ( � ) ; 9 < � E l . s . c . } = i n f { L ( � ) ; 9 < � E F } .

Proof . 1 ) Assume 9 � o . S i nce J hgdO = J Hh d o , there i s , by 9

Choquet 's lemma , � . > h , � . E F, i E ill, a decreasing sequence 1 - 9 1

of funct i ons such that f � id o � J hgdO , i�oo .

c ( g ) Thus , i f IJ E M; J 9dlJ � J hgdIJ � J � i dIJ � J � i dO = Sup J 9dIJ � J h do = L ( g ) .

IJEM 9

wh i ch g i ves

2 ) It is clear that the funct i onal L has the fol lowing propertiel i ) L ( ag ) = a L ( 9 ) , a > O.

i i ) L ( gl+g2 ) i L { g 1 ) + L ( g2 ) · i i i ) If 0 � g 1 � g 2 then L { g 1 ) < L ( g 2 ) '

From i ) , i i ) and the Hahn-Banach theorem it follows that to every cont i nuous funct i on g there is a measure s such that

J gds = L ( g )

J �ds < L ( � ) , 'tj cont i nuous � .

Thus i f + s = s -s is the decomposit ion o f s i n positive and negative parts , i t follows f rom i i i ) that f �ds+ � L ( � ) , 'tj cont i nuous � .

- 2 5 -

Assume that � E F : choose { �i }7=1 and i ncrea s i ng

sequencp o f cont i nuous f unction s w i th l im i t == � . Then f �ds+ =

l im f �i ds+ � l im L ( �i } < L ( � ) we have proved that s+ E M

so c ( g ) = L ( g } , for all con t i nuous g . I f 9 i s upper

sem i cont i n uous , choose {g. } � 1 1 l= to be a decreas i ng sequence

of cont i nuous f unct i on w i t h limi t = g . Then

1 im c ( 9 . ) < 1 im )..l. ( g . ) < 1 im )..l l' ( gk ) < )J ( gk ) . 1 - . 1 1 1 1

L ( g } � l im L ( g . } == . 1 1 where we can assume

that )..li �)..l . Th i s g i ve s L ( g ) � )..l ( g } � c ( g } . I f 9 i s l ower

semi cont i nuou s

L ( g ) = f hgd O =

then gi--"g i

l im f h do . g . 1 1

Hh = g . 1

= L ( g i )

h ..A h = Hh so g . 9 1 9

= c ( gi } < c ( g ) , i -++00. -

F i na l l y , i n order to pr ove 3 ) use Choquet's l emma and

choos e �. E F , i E m, � . � g so that i f 8 i s l . s . c . w i th 1 1

8 < i n f �. then 8 < h . I n part i cular Hh = H . f so i EN 1 9 lD �.

g i EN 1

s i nce Hh < h < h . f = in f �. we get 9 lD � . 1 9 i EN

f Hi n f do = f i n f � . do � i EN 1 i EN

I n other words : f hgd O

proof .

1 i EN

so

l im j-++oo

h = i n f �. 9 i EN 1

f i n f � . do l < i < j 1

Proposi t ioD III: 1. 2 ) = 3).

a . e . ( do ) .

wh i ch completes the

Proof . 2 ) � 3 } ) By Lemma 111 : 3 , c ( E ) = L ( XE } i f E i s open

or c l osed . So i f c i s outer regular then c ( E ) � L ( X E } � i n f L ( X O } = ECO o open

i n f c ( O ) == c ( E) . ECO

- 26 -

On the other hand , 3) together with Lemma 111: 3 3) proves

that 2 ) holds .

Corol lary 111 : 1 . 3) � �).

Lemma 111: 4 . We have that c*( E ) = O � there is a � E F w i th

f cpdo<+oo but cp I E= +Cfl

Proof . ¢:) Put 0 ={ z EU ; E:cp ( z » l } , E:> O . E: Then 0E: i s open , conta i n s E and c(0E:)�E: f cpdo -+ O , E: -+O .

�) Choose for every s EThl , O s open wi th C ( O ) < 1 • Then s 2 s hO =Hh ' wh ich we denote by f s ' i s i n F .

s ° s 00

Then c ( O ) s so cp= L: f s= 1 s i s the requ i red function .

Proposition 111 : 2 . 4 ) + 5 ) � 3).

It i s enough to prove that is a capacity s i nce

C(K)= f h do for a l l compact set s K by Lemma 111: 3. Let XK

E /'IE be a g i ven i ncrea s i ng sequence o f sets and (> 0 . By s Lemma 111: 4 there i s a Ii' E F s such that

where we can a s sume that f Ii' do <. .�. Put s 2 s

{ h > H } c {1i' =+=} XE XE

s s s 00

F= l: Ii' • s' s= l we then

have + EF s o � Hh � 1 im Hh XE s-++oo XE s

+ E: F .

Hence lim f Hh d6� f Hh d6.s.l im f Hh d6 + E: f FdO . Thus XE XE

s XE s s

l i m f h d6 = f h do whi ch means that s-+oo XE XE s

and the proposit ion fol l ows .

is a capac i ty

- 2 7 -

Propo s i t i on 1 1 1 : 3 . 1 ) + 5 ) � 3 ) .

Proof . Fol l ows f r om Theorem 1 1 1 : 3 . I n t h i s s i tuati on , there i s

a s i mpler proo f . I t f o l l ows f rom the proof o f Propos i t i on 1 1 1 : 2

that i f c * ( { h > Hh } ) = 0 f o r a l l pos i t i ve a nd bounded 9 - 9 9

then 3 ) holds true . Choose � . E F to be a decrea s i ng sequence 1

wi th I i m � . = Hh a . e . ( d o ) . G i ven s > 0 , choose 0 c- open i E N 1 9 "-

wi th c ( O s ) < s s uch that CP i ' i EJN and Hh

i s con t i nuous on 9

U \ O . Then s

co { h > Hh } c O U U N

t: where 9 g t: s = ' s

Nt: = { x � 0 c- : h ( x ) > Hh ( x ) + l } . s "- 9 - s 9

Each is c l osed so C * ( NS ) = O s wh i ch means t hat

{h > Hh } c a n be cove red by open sets of arbi trary sma l l 9 9 capa c i ty wh i ch proves the p r opo s i t i o n .

Theorem 1 1 1 : 4 . A s s ume that cP i s a bounded f unct i on i n F .

Then cp E N i f a nd o n l y i f there a r e two sequences o f con-

t i nuous f u nc t i on s i n F , and s uch that

00 i i ) F 3 ( uP + L Ut ) � cp , p�+oo out s i d e a set E w i t h c * ( E ) = O .

t=p

Proof . A s s ume f i r s t that 00

cp E Fn N n L . A s i n the proof o f

Theorem 1 1 1 : 2 w e can choo s e an i ncreas i ng s equence

con t i nuous funct i o n s i n F w i th

sup f ( cp - If' . ) < -' -. •

\lEM J 2 J Lemma 1 1 1 : 3 2 )

l im If' . = cp and j�+oo J

g i ve s

00 ( If' . ) . 1 J J = o f

D ef i ne uP = 'I' an d

u p

p

= H h

( 'I' '1' ) p+ 1 - p

- 2 8 -

Then al l the fu nctions ar e c ontinu ou s and in F.

Sinc e '1' . 1 - 'I' . < Hh J + J - '1' . - 'I'.

J + 1 J

we ge t

00 = uP+ 1 - uP - u = 'I' - 'I' - H < 0 s o t h a t

P p+ 1 P h'l' - 'I' p + l p

00 uP + E u

t is a decre as i ng s equence . Furthermore t=p

00 uP < Cjl < uP + E ut s o

t=p

00 00 { l im uP + E u > Cjl } c { E u = +oo } wh i ch comp l et es the proo f i n

t=p t

t= l p

00 th i s di re c ti on s i nce f E U . do < +00.

j = 1 J

On th e oth er hand , i f and h ave t h e

p r ope r t i es above , w e wish t o prove that Cjl E N . G i ven E > O ,

choose 0 E wi th c ( O ) < E E an d such t hat Cjl 00

l i m (uP + I ut

) p-++oo t=p

t h e n f Cjld ]J � l im f Cjld]Js

< S-4-+OO U

outs ide I f

< l im S-4-+OO

f Cjl d ]Js < E sup Cjl ( x ) + l im x E U S-4- +OO f Cjld ]J S

c a E U

< E su p Cjl { x ) + x E U

< E sup Cjl ( x ) + x E u

- 2 9 -

+ l i m s ___ +oo

00 00 S i nce L: ut E F , L: f utd 8 < +00 so l et t i ng p tend to +00

t=p t=p U

the r ight hand s i de tends t o € sup � ( x ) + f �dw wh i ch prove s x E U U

the theor em , us i ng Theorem 1 1 1 : 2 .

Note s and re feren c e s

A proo f o f " Choquet ' s l emma " can b e found i n Doob , J . L . ,

Clas s i cal poten t i al theory and i t s probab i l i st i c counterpar t ,

Spr i nger -Ver l ag ( 1 9 8 4 ) .

IV Subharmonic Functions in IR"

Le t B be the u n i t ba l l i n Fn , l et F be the r e s t r i c-

t i on t o B o f a l l pos i t i ve s uperharmon i c f unct i o n s o n RB ,

whe r e R i s a f i xed number >1 . I f we take 0 t o be t h e

Lebesgue mea s u r e on B , i t i s we l l known that U = B and 0 s at i s f i e s a l l the a s sumpt i o n s made i n Sect i o n I II ; the " coar s e "

probl em has a pos i t i ve s o l ut i on . But much more c a n b e sa i d : t he

f i ne probl em has a pos i t i ve s o l ut i on .

F i x x E U a nd de f i ne d ( K ) = i n f { cp ( x ) E F ; cp � 1 on K } f or x

K compact . Con s i de r the c l a s s M o f pos i t i ve mea s u r e s o n x

B , mx= { )J � O ; S cpd p � cp ( x ) , Iicp E F } . The n , by Lemma 1: 1 and

Theorem 1 : 1 ,

c ( E ) x

M g i ve s r i se to a capac i ty : x

= sup )J ( E ) \.lEM x

a nd by a proo f , s im i l a r to that o f Lemma 1 1 1 : 3 we have that

c ( E ) < d ( E ) w i th equa l i ty i f E i s compact or ope n . x - x

Furthermore , i t i s a con sequence o f the max i mum pr i nc i p l e

+ dx ( K 2 ) s o b y Theorem I I I : 1 , capa c i ty and there fore c =d x x

d extends to a n oute r r eg u l a r x

Moreover , the fol l ow i ng s t r onger ver s i on o f Theorem 1 11 : 4

hol d s true i n th i s ca s e . I f f i s a bounded and pos i t i ve

superharmon i c f unct i on on R B then

00 f E f .

j = 1 J

- 3 1 -

whe r e 00

( f . ) . 1 J J = i s a s eque n c e o f pos i t i ve a nd cont i nuous s uper-

harmo n i c f u nc t i o n s on B .

Not e s a n d r e ferences

Brelot , M . , E l ement s de l a t heor i e c l a s s i que du poten t i e l .

C e n t r e documentat i o n U n ive r s i ta i r I e Sorbonne , 1 9 6 5 .

Choquet , G . , Theory o f capac i t i e s . Ann . I ns t . Four i e r 5

( 1 9 5 3 - 5 4 ) .

Choquet , G . , Lectures on a n a l y s i s . W. A. Ben j am i n . New York

a nd Amsterdam 1 9 6 9 .

Landkof , N . S . , Foundat i on o f modern potent i a l theory , Spr i nger-

Ve r l ag , 1 9 7 2 .

V Plurisubharmonic Functions in en -The Monge-Ampere Capacity

Let B be the u n i t bal l i n [n and l et F be the

rest r i ct i o n to B of a l l pos i t i ve p l ur i s uperharmon i c f u n c t i o n s

o n RB , where R i s a f i xed numb e r > 1 . We take 0 to be

the Lebesgue mea sure on B a nd form c as i n Sect i on I I I : b .

I t i s then true that F and 0 meet a l l the requ i r emen t s i n

Sect i o n I I I : b and we a r e go i n g t o see that F h a s prope r t y 1 )

and c has property 5 ) ; thus 2 ) and 3 ) h o l d true by Propo-

s i t i on s 1 1 1 : 3 and 1 1 1 : 1 .

The d i f f er en t i a l operato r s a and a are d e f i ned by

n a a = , E � d ZJ'

J = 1 J and

n E a

j = l ai , J

d o , J

so that

a . De f i n i t i on o f the Mon g e - Ampe r e operator

d= a + a

Let U be a n open and bounded s u b s et o f [n o I f

a nd

1 n E c 2 , ) V , • • • , v \ U , we de f i ne 1 n MA ( v , . . . , v ) to be the symme t r i c

and n - l i near operator

1 n c 1 c n MA ( v , . . . , v ) = dd v A . . . A dd v .

I f moreover 1 n v , . . . , v E PSH ( U )

pos i t i ve mea sure .

c 1 , c n then dd v fl • • • dd v i s a

- 3 3 -

Theorem V : 1 . I f 2 i i 00 P S H n C ( U ) 3v . � v E L ( U ) , j-++oo , l < i < n then J

1 n MA ( v . , . . . , v . ) i s a weak l y convergent sequence o f pos i t i ve J J mea s u re s . Moreove r , the l i m i t i s i ndependent o f the part i cu l a r

cho i ce o f t h e decrea s i ng sequences l v . , 1 < i < n ; j E N . J

Proo f . The s tatemen t i s pure l y l o c a l , so we can a s s ume that

1 i V 1 �V j ' 1 �i�n , j aN out s i de a f i x compact subset K o f U�B ;

the u n i t ba l l i n �n .

G i ve n we wan t to prove that

l i m 8dd v . /1. • • • /l dd v . f c 1 A C n

j-++oo J J ex i s t . Take

and choose ll � l near the support o f 8 , u n i on K .

Then by Stokes formu l a ,

f C 1 A " c n 8dd v . 1\ • • • 1 \ dd v . J J

f 1 c A c 2 A c n We con s i de r llv . dd 8 1 1\ dd v . . . . dd v . J J J

a nd observe that

f 1 A c A c n f 1 c A c n ll v j + 1 dd 8 1 /\ " . dd v j + 1 < n V j dd 8 1 . . . dd v j + 1 <

f 2 c " c j A 1\ c n < v j + 1 dd 8 1 dd ( n v 1 ) /I • • • dd V j + l

- 3 4 -

c n dd v j + 1 ' s i nce n = l n e a r

we g e t f 2 2 c A C j A A c n ( v j + 1 -V j ) dd 8 1 /1 dd v 1 II • • • /1 dd v j + 1 +

f j A c /\ c 2 A C 3 A A c n < nV 1 dd 8 1 dd V j /I dd V j+ 1 /\ • • • dd v j + 1 •

Repeat i ng t h i s , we get that

is dec rea s i ng in j wh i ch proves tha t t he l i m i t ex i s t .

We now prove that the l i m i t does not depend on the pa r t i cu-

l a r cho i c e o f the approx imat i ng sequence s . We f i r s t note that

i v . , J p

i t fol l ows f rom the proo f above that i f

subsequences o f i v . , J

to the s ame l imi t a s

i t l S e nough to show

t he n c 1 A C n dd v . II • • • dd v . J p J p

c 1 /\ 1\ c n dd v . . . . dd v . . S i nce

that

J J

ddc v ; A • • • A ddcv� J

l < i < n , j E m a r e - - p

tends wea k l y

MA i s n - l i n e a r ,

tends to t he s ame

l im i t a s c l " A c n dd v . . . . II dd v . J J

i f �n v . J h a s the s ame propert i e s a s

n v . • But i f we have that J

- 3 5 -

and s i nce n �n vk -vk -+ 0, k-++oo t he r i ght hand s i de tends to z ero

when k-++oo for every f i xed j wh i ch prove s the theorem .

Def i n it ion . I f v1 E PS H n Loo ( U ) , l < i < n we de f i ne

1 n c 1 r, A c n MA ( v , . . . , v ) = dd V 1\ • • • 1\ dd v to be the weak 1 i m i t o f 2 i i P S H n C ( U ) 3v . � v , j-++oo , l < i < n . J

Def in i t i on . I f E i s a B o r e l subset o f U , we put

d ( E ) =sup {fMA ( U , . : . , u ) : u E PSH ( U ) , O <u < l } . It i s easy to see

E n t 1.mes

that the f o l l ow i ng propos i t i o n hol d s .

Proposi tion V : l .

00 00 i i ) d ( u E . ) < E d ( E . )

i = l 1. - i= l 1.

i i i ) I f E . ./I E : j-++oo then J l i m d ( E . ) = d ( E ) . J

( We wi l l see l ater that d i s a capa c i ty i n Choquet ' s s e n se ) .

Theorem V : 2 . I f 1. 1. 00 PSH3v . ,;v E PS H n Ll ( U ) , j-++oo , O < i < n then

J oc

o c 1 A C n 0 c 1 . c n v . dd v · A . . . I\ dd v . z.v dd V fl • • • dd v . J J J

Proof . The s t ateme n t i s purely l oc a l , s o we c a n a s sume that o i v 1 =v j ' O�i�n ; j Elli out s ide a f i xed compact set K o f U=B .

As sume f i r s t that i 2 v . E PS H n C ( B ) . J

I f 00

O,S,n E C O ( B ) , n= l near

we f i nd as in the proo f of Theorem IV : 1 , that

f 0 c 1 nv . dd v . J O J 1

K ,

- 3 6 -

i s decrea s i ng i n ( j o ' · . . , j n ) . Ther e f o r e

O n t h e other hand , s i nce o

n v . J

1 S d e c r ea s i ng , we g e t

-1 . f °dd

c 1 A A ddC n -1 ' f °

ddc 1 1\ 1\ dd

cv

n < 1 m nv . v . /\ . • . /\ V . � 1m nVK v . • . j-++oo J J J k-++co

< nv dd v 1 \ • • • 1\ dd V • f 0 C 1 A A c n

The r e f o r e l i m nv . dd v . . . . dd v . = nv dd v 1\ • • • dd v f ° c 1 1\ 1\ c n f ° c 1 A 1\ c n

j-++oo J J J

so i f � i s any ac cumu l a t i on po i nt f or

�

o c 1 c n v . dd v . /I. • • • dd v . , j ElN J J J

o C 1 1. c n has the s ame mas s a s v dd v /\ . . . dd v

upper s em i cont i nuous i f 00 o � G E C O ( 8 )

a nd s i n c e o

Gu

Thus o c 1 A C n Z < u dd V 1 \ • • • dd v and they have the s ame ma s s s o

they have to b e equa l .

00

1 S

Now , i f i

v . J

are i n P S H n Ll ( 8 ) oc

o n l y , we con s i de r the

regul a r i z ed func t i on s

Then , i f 00

( e ) . 1 J J =

i 00 v . E PS H n C J , E: i n a sma l l er ba l l .

1 S any s equence decrea s i ng to z er o we

- 3 7 -

get f rom above that

o c l c n 0 c l c n v . dd v . 1\ • • • dd v . Z, v dd v 1\ . • • dd v . 1 , ( . 1 , ( . 1 , ( . 1 1 1

On the other hand , g i ven

( so that P j

we can to every

f 0 c 1 ... A C n l 1 - 8v . dd v . 1\ • • • / \ dd v . < J.

J J J

j f i nd

because the theorem i s a l ready proved for C2- funct i ons .

There fore o c l A A c n 0 c n \ A c n v . dd v . (\ . . . I\ dd v . �v dd v I • • • / \ dd v wh i ch J J J proves the theorem .

b . Quas i cont i nu i ty with re spect to d .

Theorem V : 3 . To every £ > 0 and every vE PSH ( U ) there i s an

open set o ( such that d ( O ) < £ ( and v l u- o ( i s cont i nuou s .

Proof . Assume f i rst that O <v< l and that K i s a g iven

compact subset o f

We c l a i m that

1 irn j-++oo

U . Choose 2 v . E PSHnC J

Assumi ng t h i s for a moment , c hoose k . J

near K, V j "'lI. V 1 , j-++oo .

to be a f undamental

sequence of compact subsets of U and l et £ > 0 be g i ven .

Choose for every

sup uEPSH ( U ) O <u < l

j , v ' J

- 3 8 -

such that

f ( v j -v ) ( dd 1 u ) n k� J

and v ' 1 <v , J + - J on k " j Em and put J 1 0 o , = { v , -v).-} nk " J J J J Gk= u 0 , .

j�k J co .L Then Gk i s open , d ( Gk }� E I 2 J' j =k loca l ly un i forml y , j�+co .

and on U-Gk , v j� v

I t rema i n s to prove the c la im . Th i s i s purely loca l so we can assume that U==RB where R) l , K=B , v=v , J Take � 2 uE PSHnC x { RB ) , u=const . near B and Then , for k�j , Stokes � formula g ives

f c � c n- l == - d { v j -vk ) A d ( u-u ) /\ ( dd u )

f c� A c n - l s i nce d { v j -vk ) /\ d u ( dd u ) =0 .

outs ide B · ,

s ince { ddcu ) n- l i s a pos i t ive ( n- l , n- l ) - f orm we get

I fd ( V j -Vk ) /\ dC ( u-u } /\ { ddcu } n- l I � 1 f A C A C n- l 2

< [ d ( v j -v k ) 1\ d ( v j -v k ) f\ ( dd u ) ] •

1 . [ fd( U-� } /\ dC ( u-� } /\ ( ddcu ) n- l ] 2

1 f C C n- l '2 = [ ( v j -vk ) dd ( v j -vK ) ( dd u ) ] .

1 • f ( U-� ) ddc ( u-� ) /\ ( ddcU ) n- l ) 2 <

1 1 < ( f ( v j -vk ) ddC ( v j +Vk ) A ( dd cu ) n- l ) '2 . ( 2d ( L ) ) 2"

- 3 9 -

Cont i nu i ng i n th i s manner , we get an est i mate o f the f orm

where c i s a constant and where a > O . Let k�+oo , then by

J c n J c n Theorem V : 2 ( v j -Vk ' dd ( v j +vk ' � ( V j -V l dd ( V j +V ) .

j�+oo . Then aga i n by Theorem V : 2 ,

( v . -v ) dd ( v . +V ) � 0 J c n J J

which proves the c l a i m .

c . D i r i ch l et problem

Let now

Theorem V : 4 . Let U be a strictly ps eudoconvex and bounded

doma i n i n �n . I f h EC ( a U l and f E C ( U l , then there ex i st s a un ique p l ur i s ubharmo n i c f unct i on uEC ( U ) such that

a u

o n U

d . Cont i nu i ty o n i ncrea s i ng sequences

Theorem V : 5 . Assume that where 00 ( u . ) . 1 J J =

i s a non-decrea s i ng sequence such that l im u j=uO ( a . e . ) when

j�+oo .

Then MA ( u j , . . . , u j l � MA ( u 0 ' . . . , u 0 ) . n t imes

- 4 0 -

Proo f . Let 8 E C� ( U ) be g i ve n . We have to prove that

Take

1 im J 8ddcu j A • • • !\ ddcu j = J 8ddcu O !\ . . . " ddcuO · j-++oo

w i th 1l= 1 on supp 8 and

I f we knew that

C A A C A C C A A C A C * dd u j 1\ • • • 1 \ dd uj 1\ dd 8 1 4dd U o /\ . . . / \ dd U o / \ dd 8 1 ( )

we would have for k�j ,

by monoton i c i ty , so s i nce uk are quas i cont i nuous :

Thus J ll ukddcu o /\ . . . A ddcu O A ddc8 1 �

J C A " C A C -'- J C " c " C llu . dd U . /\ • • • I \dd u . dd 8 1 � 1 1m llu . dd u . A . . . / ldd u ' l \dd 8 1 < ] ] ] j-+ +00 ] ] J -

J TlU OddCu jA . . . A ddCu jl\ddc 8 1 = J TluOddcu o

A . . . A ddcu O /\ ddc 8 1

where the l a s t i nequa l i ty fol l ows f r om the fact that U o i s

qua s i cont i nuous .

- 4 1 -

From Lemma V : l be low we get that

l im f Tjukddcu o !\ . . . " ddCuO!\ ddce 1 = J Tjuoddcuo /I. • • • !\ ddcuo /\ ddc e 1 k-t-+oo

and fu . ddcu . 1\ . • • /\ ddcu . 1\ ddce2 can be handled i n exactly the

J J J same way prov ided we know that

ddCU j 1\ • • • A ddCU j 1\ ddc ,¥ � ddcu O /I. • • • /\ ddcu o 1\ ddc,¥ for eve ry 00 '¥EC ( U ) .

Thi s we could do a s above i f we knew that

ddcu . A • . • A ddcu . /I. ddc ,¥ 1\ ddCy � ddcu . 1\ • • . 1\ ddcu . II ddc,¥ /I. ddCy , J J J J

00 \i'¥ , yEC ( U ) .

I f we repeat th i s n- l t imes we get a s tatement wh ich i s

true . Thus ( * ) and Theorem V : 5 i s proved by the follow i ng

Lemma .

Lemma V : l . I f J u . J < cons t . u . E P S H ( U ) \:i j ElN , and i f a . e . on U J - J one has u j .?l u O E P SH ( U ) , then

c l A c n c l " u j dd v /I. • • . 1\ dd v � u Odd v 1\

vi

E P S H ( U ) n Loo

, l u . J But

uO > l i m u . w i th equal i ty a lmo st everywhere with respect to the - 1

Lebe sque measure so i t i s enough to prove that u and c C A A C n uOdd v 1\ • • • /l dd v have ( l ocal l y ) the same mas s . We can assume

. . 2 that v J =v J E C E out s i de some f i x compact subset

v� '::l. v j , E-t- O on U .

K o f U where

- 4 2 -

CX) f c 1 c n n I f n E C o ( U ) , n= l near K then nU j dd v " . . . /\ dd ( v -vE: ) =

J c 1 A c n n f n n c 1 ,. A C n - 1 A A c u . dd v A • • • dd ( v - v ) = ( v -v ) dd V 1\ • • • I \dd v 1\ • • • l\dd u . � o , J E: E: J uni formly i n j s i nce n v i s quas i cont i nuous . Thus

such that nu . dd v . . . 1\ dd v > f c 1 1'1 A C n J E: 1 -- f c l fI c n f c 1 A c 2 A " c n - E: + n u

J' dd v " . . . 1\ dd v E: > - nE: + nu . dd v dd v 1\ • • • 1\ dd v - J E n

E n - 1 E: l

where . f c 1 A c n 1 1m n u . dd v . . . /\ dd v � j�+CX) J

J c 1 1\ c n nUOdd v A . . • dd v - E: n . E: n E 1

I f we now let E:� O and use t he

quas i cont i nu i ty we get the des i red concl us i on .

e . Compa r i son theorems

Let U be and open and bounded subset of �n .

Lemma V : 2 . I f CX) u , vEPSHnL ( U ) and i f u=v near au then

Proof .

K , u=v

fMA ( U , . . . , U ) = U

f MA ( v , • • • , v ) U

G i ven K , compact i n U , choose

on u \ K . Then J x MA ( u , . . . , u ) U

CX) X E C O ( U ) , X == 1 near

= J Xddcu 1\ A ddeu

= f uddcuX " ddcu A . . . A ddcu by Stokes formula . But u=v on

supp ddC X so the r i ght hand s ide equal f vddc X 1\ ddcv A • . • ddcv = = fX ( ddcV ) n = f xMA ( v , . . . , v ) .

CX) Lemma V : 3 . If u , vEPSHnL ( U ) , u<v on U and i f l i m u ( z ) -v ( z ) =O

\ixE a U then

f MA ( v , . . . , v ) < f MA ( u , . . . , u ) . U U

z�x z EU

Proo f . G i ven £ > 0 , put

so by Lemma V : 2

S i nce

f MA ( v £ ' • • • , V £ ) = U

a s £ ":11 0 ,

- 4 3 -

v =sup ( v- £ , u ) . £

f MA ( u , . . . , u ) . U

Then v =u £

we have by Theorem V : 5

MA ( v £ ' . . . , V £ ) 4 MA ( v , . . . , v )

wh i ch proves the c l a i m .

near

Theorem V : 6 . 00 I f u , vEPSHnL ( U ) and l im ( u ( z ) -v ( z » O then

Z4 a U z E U

f MA ( v , . . . , v ) < f MA ( u , . . . , u ) . { u< v } { u< v }

Furthermore , i f l im u ( z ) -v ( z » o f o r s ome 0 > 0 then Z4 d U z E U

f MA ( v , . . . , v ) < { u< v }

f MA ( u , . . . , u ) . { u<v }

au

Proof . We f i rst note that the s econd statement fol lows f rom the

f i rst by c on s i der i ng u and v+ £ and l et t i ng £ decrease to

zero . We a l so note that i f u a nd v are also cont i nuous , the

f i rst s tatement fol lows f r om Lemma V : 3 . To prove the theorem ,

it i s no loss o f general i ty t o a ssume that {u<v}cU ( cons ider

otherw i s e u+£ i nstead of u ) .

00 We f i rs t c l a i m that i f ( w . ) . 1 J J =

of plur i subharmon i c funct i on s on U i s a decreas i ng sequence

w i th 00 l im w . =wE PSHnL ( U ) J

then

f ( ddcw ) n < l im

j -++oo { u < v } and

f ( ddcw ) n l im > { u <v }

j-++oo

- 4 4 -

f ( ddcw . ) n J

{ u < v }

f { u < v }

c n ( dd w j ) .

For let 0 > 0 be g i ven . S i nce u and v are qua s i cont i nuous

there i s an open set 0 0 w i th sup f( ddcw . ) n < o . J J 0 o

and the r e a r e

two conti nuous func t i ons u a n d v such that { u�u } u { v�v } c O o .

There fore

f ( ddcw ) n < l im { u<v }

wh i ch g i ves the f i r st part o f the c la i m . Mor eover

� im f ( ddcw . ) n < � im f ( ddcw . ) n + 8 � f ( ddcw ) n + 2 0 wh i ch

J-+ +oo J J-++oo _ _ J { u< v } { u < v } { u<v }

g i ve s the second par t o f the c l a i m .

We can now f i n i sh the proof o f the theorem . Choose U o

and v to the decrea s i ng s equences o f cont i nuous plu r i subE: harmon i c func t i ons de f i ned i n a n e i ghborhood o f {u<v 1 } and

such that {u < v 1 }cu .

- 4 5 -

Then f ( ddcv ) n < (the ) < l im f ( ddcv ) n < c l a im (-+ 0 ( { u <v } { u <v }

< l im l im f ( ddcv ) n < l im l im f ( ddcv ) n < ( Lemma V : 3 ) < (-+ 0 0-+ 0

( (-+ 0 0-+ 0 ( { u o <u } { u o <v( }

< l im l im f ( ddcvo ) n < l im l im f ( ddcu o ) n < ( the cla im ) < (-+ 0 0-+ 0 { u o <v( } (-+ 0 0-+ 0 { u<v } - (

< l im f ( ddcu ) n = f ( ddcu ) n . E:-+ 0 { u<v } { u< v } - E:

Hence f ( ddcv ) n � { u+n < v }

for a l l sma l l but pos i t i ve n .

Let now n decrease to z ero a nd we have the desi red i nequa l i ty .

Corollary V : 1 . I f <Xl u , vEPSHnL ( U ) w ith l im u ( z ) -v ( z » O z-+ (l U z E U

i f MA ( u , . . . , u ) �MA ( v , . . . , v ) then u > v o n U .

and

Proof . 00 Let O > p E PSHnL ( U ) such that f ( ddcp ) n> O for every

V

Borel set V i n U with pos i t i ve Lebesgue measure . I f there i s a z O EU w ith u ( z O ) <v ( z O ) take n > O s o sma l l that u ( z O ) <v ( z O ) +n p ( z O ) . Then the Lebesgue measure of T= { z EU ; u <v+np } i s str i ctly pos i t ive and so is f ( ddcp ) n . By

Theorem V : 6 we have that f ( ddC ( v+n p ) ) n � T

right hand s ide i s assumed to be smal ler than

Hence f ( ddcV ) n+nn f ( ddcp ) n < f ( ddcV ) n so T T T

wh ich i s a contrad i ct ion .

but the

f ( ddcV ) n . T f ( ddcp ) n = 0

T

- 4 6 -

Corol lary V: 2 . I f 00 u , vEPSHnL ( U ) , l im u ( z ) -v ( z » O and i f z-+ a u z E U

o then u > v o n U .

Proof . I f { u< v } � 0 choose P as i n the proof o f Corol lary V : l V : l and n > O so that

{ u <v+nP } � 0 .

Then by Theorem V : 6

f ( ddcv+n p ) n < { u<v+n P }

f ( ddcu ) n = O . { u<v }

Agai n f ( ddcp ) n=O whi ch i s a contrad i ct ion . { u<v+nP }

Theorem V : 7 . I f K i s compact i n B and i f uK=sup { �EPSH ( RB ) ; - l��O ; � I K=- l } then fMA ( UK , · . · , UK ) =O .

CK

Proof . I f i s cont i nuous , i t fol lows f rom Theorem V : 4 and Corol lary V : l that SuppMA ( UK , . . . , uK ) CK .

I f K i s a g iven compact set , we can choose a decreas i ng sequence { Ks } ;= l o f compact sets decreas i ng to K where each

UK s i s cont i nuous ( cover K w ith f i n i tely many bal l s ) . Then

l im uK =uK s-++oo s the proo f .

a . e . and an appl i cat i on o f Theorem V : 5 completes

Remark . I f K . '-1 K , J

subsets o f B then

- 47 -

j Em i s a decreas i ng sequence of compact u * /f u* so K . K

J

by Theorem V : 6 and Theorem V : 7 .

Theorem V : 5 now g ives that

Thi s together w ith Propo s i t i on V : l proves that d i s a capac ity .

f . Quas i cont i nu i ty 1 n the system F , c

Propos i t ion V : 2 . I f K i s compact i n B then d ( K ) = f MA ( uK , · · . , uK ) where uK=sup { uE PSH ( RB ) ; - l �u�O ; u I K= - l } .

RB

Proof . G i ven K compact i n B we know from Theorem V : 7 that

f MA ( uK ' . . . , uK ) = f MA ( uK ' . . . , uK ) . K RB

Let �EPSH ( RB ) ; - l <� < O be g i ven . S i nce we want to est i mate f MA ( � , . . . , � ) , it i s no restr i ct ion to as sume that K l im � ( z ) =O , �yE a RB . z-+y

- 4 8 -

Assume f i r st that i s cont i nuou s . By Theorem V : 6 we

have f MA ( <p , . . . , <p ) < K

f MA ( uK , · . . , uK ) = { uK< <p }

f MA ( uK ' . . . , uK ) = K

f MA ( uK ' . . . , uK ) . B

I f i s not cont i nuous , choose to be a decrea s i ng

sequence o f compact sets w i th i nter sect i on equal to K . Then

a . e . s 0 f MA ( <p , . . . , <p ) = 1 i m f MA ( <p , . . . , <p ) < s-++oo

< l im s-++oo

K K s f MA ( uK , . . . , uK ) = f MA ( uK ' . . . , uK ) where the l a s t B s s RB

equal i ty comes from Theorem V : 5 .

Corol lary V : 3 . I f 0 I S an open subset o f B then

d ( 0 ) = f MA ( uO ' • • • , uO ) •

RB

Proof . Choose an i ncrea s i ng sequence o f compact sets { k } oo s s= l

00 w i th u k . =O .

j= 1 J

Then d ( k s ) = f MA ( Uk , . . . , uk ) by Propos i t i on V : 2 . Thus RB s s

d ( O ) =l im d ( ks ) =1 im f MA ( uk , . . . , uk ) ::=; f MA ( uO ' . . . , uO ) by s-++oo RB s s RB s-++oo

Propos i t i on V : 1 and Theorem V : 5 .

We are now i n pos i t ion to prove the results s tated i n the

i ntroduct i on to th i s sect i on . We f i rst prove that 1 ) holds .

Reca l l that F denotes the p l u r i superharmo n i c f unct ions on

RB , R> l .

- 4 9 -

G i ven u E F . By Theorem V : 3 there i s a decreas i ng sequence

of open sets { o }'X> s s= l such that l im d ( O ) =0 and s uch that s�+oo s

u l co s i s cont i nuou s . By Cor o l lary V : 3 , MA ( UO , . . . , uO ) � o ,

s�+oo .

I t fol lows f rom Corol l a r y V : l that l im

s s

Uo ( z ) = O s

00

a . e . so ,

i f nece s sary , by pas s i ng to a subsequence �= E Uo E P SH ( RB ) . s= l s

The restr i ct i on o f u t o KT= { ��-T } i s always con t i nuous

and C ( CKT ) i * f �dO� O , T�+oo . Th i s proves that every u E F 1 S B

qua s i cont i nuous with respect to c .

g . Cond i t i on 5 ) i s ful f i l L ed

Let 0ig be a bounded f unct i on . We w i sh to prove that

c ( { h > Hh } ) = O . 9 9 ( 8 ) , i�+oo and

Choose h . E F : h . >g so 1 1-

put A = n { h . > r . > Hh } r . . 1 J J 1 9

that h i � Hh a . e . 9

wher e { r . } � 1 a re the

J J = 00

rat i on a l number s i n ( 0 , 1 ) . Then { h > Hh } c u A so i t i s 9 9 j = 1 r j enough to prove

a g i ven compact

c ( k ) =O . S i nce

that Hh < 1 on 9

i s compact and

that

subset

h . �> 1 r .-J

k so

c ( A } =O , tf j ElN. F i x j r . J

o f A ; i t i s enough r . J Hh

on k but --2< 1 on r . J

00 kc 1 u { H < 1 - - } . h - s Now

s= 1 9

A

1 k Assume that Hh < 1 -- on k s s

s

and let k be

to

r . J

prove that

i t 1 S c l ear

rpE F ; l im rp ( z ) =O , z� �

'tf [, E aRB and that

- 50 -

Then 1 Hh 2. ( 1 - s ) cp k s on k . s Further-

more , MA ( Hh , . . . , Hh k k ) =0 on RB \k by Theorem V : 7 . Hence , s

s s 1 by Coro l l ary V : 2 Hh � ( 1 - s ) cp k on RB s o we have that

Hh k s

h .

1 « l -- ) h - s k s

I t fol l ows

and so

that

s Hh k s

hk =0 s

Some consequences

1 ( d o ) . « l - - ) h < h =H a . e . - s k - k s hk s s

a . e . ( d o ) wh i ch proves the c l a i m .

We have now seen that i n the plur i superharmon i c case ,

F and c sat i s f i es cond i t i o n s 1 ) and 5 ) respect ively ;

here we l i st some consequences o f th i s .

Theorem V : 8 . Let { u j } j E I be a fami l y o f plur i subharmon i c

funct ions , loca l l y bounded above . Then there i s a plur i subhar-

mon i c f unct i on � such that

{ sup u . « sup u . ) * } c { � = -oo } . j E I J j E I J

Proof . Propos i t ion 1 1 1 : 3 g ives 3 ) s o the statement fol l ows

from Corol lary 1 1 1 : 1 and Lemma 1 1 1 : 4 .

Theorem V : 9 . The capac i t i es c and d are outer r egular and

have the same z ero sets . Furthermore

for every Bor e l set E .

- 5 1 -

Proo f . That c i s outer regu lar follows f rom Propos i t i on 1 1 1 : 3 and Propo s i t ion 111 : 1 . By Lemma 1 1 1 : 4 c ( E ) =0 � 3uEPSH such that u=-oo on E . There fore c ( E ) =O � d ( E ) =O . Assume for a moment that we have proved that

d ( E } = f ( ddc-h ) n , for al l Borel sets E . XE

I f d ( E ) =O then by Corol l ary V : I -h =0 a . e . so c ( E ) =O . X E

I t rema i ns to prove t hat d ( E ) = f ( ddc-h ) n for a l l Borel XE sets E and we know th i s for E compact or open by Propo s i t ion V : 2 a nd Corol lary V : 3 . I f E i s a g i ven Borel set then , s i nce

00 i s outer regular , we can f i nd a decreas i ng sequence ( 0 . ). 1 J J =

at open sets conta i n i ng E such that ( -h } *= ( i n f h ) * . XE j X O . ]

Hence d ( E } < l im d ( O . ) =l im S ( ddc-h ) n= s ( ddc-h ) n by - J X XE O . J Theorem V : 5 . On the other hand , s i nce

c ( E ) = Sh d o XE

i s a capac i ty , we can f i nd an i ncrea s i ng sequence compact sets conta i ned i n E w i th

( -h ) * -� ( -h ) * . XK . XE ]

Then

Theorem V : 2 .

l im c ( K . ) =c ( E ) ]

00 ( K .)

. 1 J J= so

by

o f

c

- 5 2 -

Thus d ( E ) = f ( ddC ( -h ) * ) n wh i ch completes the proof o f X E

Theorem V : 9 .

Propos i tion V : 3 . The set f u nct ion

G ( E ) = i n f - l <u< O uE PSH ( RB ) u=- l on E

[ s up - l < v < O v E PS H ( RB )

i s an outer regular capac i ty .

f c n - u ( dd v ) ]

Proof . The results i n th i s sect i on show that Theorem 1 1 1 : 3

app l i e s .

00 Theorem V : l 0 . Assume that O >uE PSHnL ( RB ) when R> l . Then

there are two sequences and of negat i ve ,

conti nuous and plur i subharmo n i c f unct ions on B such that

i ) on B

00 i i ) uP + L Ut /'f u

t=p out s i de a set o f c-capac i ty z ero .

Proo f . Theorem 1 1 1 : 4 .

I n th i s s ec t i on , we have seen the solut i on o f the " coar s e "

probl em ( c f . Sect ion I I I ) .

The poi ntw i s e ( " f i n e " ) behav i our o f E f+ H ( z ) , X E

z f i xed in B i s not c l ear ; we do not know i f t h i s i s a capac i ty i n

Choquet ' s sense . The m i s s i ng part i s proper ty i i ) : I f E ?l' E , s s�+oo we woul d l i ke to know i f h ( z )� h ( z ) , s�+oo . What XE XE s we have i s the fol l ow i ng theorem .

- 53 -

Theorem V : l l . Assume that K /f K , s-++oo are compact sets of s the un it bal l . Then for every f ixed z E B , l im h ( z ) =h ( z ) .

s-++oo XK XK s

Proof . Fol l ows f rom Lemma 1 1 1 : 1 , s i nce E � c { XE ) i s a capac i ty i n Choquet ' s sense .

Notes and references

General references : Hormander , L . , An i ntroduct i on to complex analys i s I n several var i ables . Van Nostrand , 1 9 6 6 .

Krantz , S . G . , Function theory o f several complex var iables . Wi l ey- Intersc i ence ser i es , 1 9 8 2 .

Lelong , P . , Plur i subharmoni c funct i ons and pos i t ive d i f ferent ia l f o rms . Gordon and Breach , 1 9 6 9 .

Theorem V : 8 as wel l as many other results i n th i s sect ion was f i r st proved by E . Bed ford and B . A . Taylor i n : A new capac i ty for plur i subharmon i c funct i ons . Acta Math . Vol . 1 4 9

( 1 9 8 2 ) .

Theorem V : 4 i s proved by the same authors i n : The D i r i ch let problem for the complex Monge-Ampere operator . I nvent . Math . 3 7 ( 1 9 7 6 ) .

See a l so L . Caf farel l i , J . J . Kohn , L . Ni renberg and

J . Spruck . , The D i r i chlet prob l em for non-l i near second-order

- 5 4 -

e l l i pt i c equat i on s I I . Comp l ex Monge-Amper e , and u n i formly

e l l i pt i c equat i on s . Commu n i c at i on s o n Pure and Appl i ed

Mathema t i c s ( 1 9 8 5 ) .

S . -Y . Cheng and S . T . Yau , On the e x i stence o f a comp l ete

Kahl er met r i c on non-compact comp l ex man i fo l d s and the regu-

l a r i ty o f F e f ferman ' s equat i on . Commun i cat i on s on Pure a nd

App l i ed Mathema t i c s 3 3 ( 1 9 8 0 ) .

U . Cegrell , O n the D i r i c h l et prob l em for the Monge-Ampere

operator . Math . Z . 1 8 5 ( 1 9 8 4 ) .

The qua s i cont i nu i ty h a s a l so been proved by A . Sadul laev ,

Rat i onal approx i mat ion and p l u r i p o l a r s e t s . Ma th . USSR Sbor n i k .

Vo l . 4 7 ( 1 9 8 4 ) . No . 1 .

A s t ronger ve r s i on o f Theorem V : l 0 i s proved i n : U . Cegrel l ,

S ums o f cont i nuous p l ur i s ubharmon i c funct i o n s and the comp l ex

Monge-Ampere operator . Mat h . Z . 1 9 3 ( 1 9 8 6 ) .

In contrad i st i nc t i on t o the s ubharmon i c c a s e , E � h ( x ) i s X E

not s t rong l y subadd i t i ve . Th i s i s s hown by Johan Thorbi6rnson

i n : A counterexampl e t o the s trong subadd i t i v i ty of extrema l

plur i subha rmon i c funct i on s . To appear i n Mh . Math .

Theorem V : I I was f i r s t proved i n : U . Cegre l l , Capac i t i e s

and extrema l p l ur i subharmon i c f un c t i o n s o n subsets o f [ n o

Ark . Mat . 1 8 ( 1 9 8 0 ) .

- 5 5 -

A s e t E i s ca l l ed � n -pol a r i f to every z O E E there i s

a ba l l B ( z O , r ) a nd � E P SH ( B ( z O , r ) ) such that � j -oo and

EnB ( z O , r ) c { �=-oo } . It wa s shown by B. Josefson : On the

equ i va l ence between l oc a l l y p o l a r and g l oba l ly pola r set s for

plur i s u bharmo n i c funct i ons in [n o Ark . Mat . 1 6 ( 1 9 7 8 ) that

one a lway s can t ake n �EPSH ( � ) .

For a s tocha s t i c po i nt o f v i ew s e e M. Fukushima and M . Okada ,

On D i r i ch l e t forms for p l u r i s u bharmon i c funct i on s . Acta Math .

1 5 9 : 3 - 4 , 1 9 8 7 .

VI Further Properties of the Monge-Ampere Operator

Let B be the un i t bal l i n [n and let P be the re-str i ct ion to B of all negat ive plur i subharmon i c funct ions on RB where R> l . We saw in Sect i on V that F=-P and 0 , the Lebesgue measure on B , g i ve r i se to two natural capac i t i es d ( E ) =Sup{f ( ddcU ) n , - l <U< O , UE P } and

E C ( E ) =-fu do=sup � ( E ) E lJEM

where uE ( Z ) =SUP { �EP ; : � I E�- l } ( =-hXE) and where M i s de f i ned

to be the weak * -closed convex set of pos i t i ve measures � on B such that

We can of course equal ly wel l cons ider the f unct i onals

and

c ( f ) =sup f l f l d � lJ E M

where f is any bounded funct ion on B .

We do not know i f { ( ddcu ) n ; - l �u�O , u E P } i s compact or convex but i f we de f ine N to be the pos i t i ve measures on B that are domi nated by d ( i . e . O <vEN � V ( K ) �d ( XK ) � compacts K i n B ) then obv iously N i s convex . But N i s a l so compact f O i f � . EN , � . :A � then , g i ven ( > 0 and K compact , choose J J an open subset A conta i n i ng K such that d ( A ) <d ( K ) + E . Then � ( K ) < � ( A ) < l im � . ( A ) <d ( A ) <d ( K ) +E by Lemma 1 : 1 , so � E N . - --. - J - -J-++oo

- 5 7 -

Propo s i ti on VI : l . a ) Every measure i n M i s absolutely con-t i nuous w i th respect to a measure i n N . b ) There i s a pos i t ive constant K such that KNCM .

Proo f . a ) S i nce N i s weak * -compact and convex every Borel measure � has a un ique decompo s i t ion U = � l + �

L. ' where iJ 1 i s absolutely conti nuous with r espect to a measure i n N and iJ 2

i s car r i ed by an Fa-set E such that sup { V ( E ) , vEN } =O . I f u E M i t fol lows from Theorem V : 9 that � ( E ) =O so � 2 = O wh ich proves the cla im . b ) Fol l ows from Propos i t ion V I : 2 below .

Propo s i tion VI : 2 . There i s a constant c such that

Proof .

f - CP ( ddcU ) n � C f - cpd o , 'dcpE P ; 'duE P , - l u on RB\ rB and v <u on B . I f we put n- r-

1 - cp ( ddcu ) n � f - cp ( ddcU 1 )n =

B rB

= f - cpddCVr II ( ddCu 1 )n- l + f - cpddC ( u 1 -vr ) /I ( ddCu 1 )

n- 1 :=;:

rB

f C C n- 1 f c C n- 1 = - cpdd v r 1\ ( dd u 1 ) - ( u 1 -v r ) dd cp ,\ ( dd u 1 ) < rB

� f - cpddcvr /\ ( ddcU 1 )n- 1 � ( repeat n- l t imes ) <

rB � f - cp ( ddCvr ) n = cr J - cp ( w ) da ( w )

rB I w l =r

- 5 8 -

where a i s the s u r face mea su r e on the boundary o f r B i n

[n and where c i s a con s tant , o n l y depend i ng on r and n . r

Now , s i nce � i s subharmon i c we know

f �d o < 2n f �d o < 2n f �d a . B r r B r

Ther e fore

c f- �d o < r r 2 n f - cpd o r - r

B

wh i ch proves Propos i t i on V I : 2 .

Note that i f K i s a c ompact s u b s e t o f B then o f cou r s e

d ( XK ) �d ( UK ) a n d we can have st r i ct i nequa l i ty .

and

Then

and

For l et

2 3 L = { z E a: ; I z I � "4 } .

uK =

uL =

{ log l z l max l og 4 '

1 02 1 z 1 max { l og 4/ 3 '

- 1 } ,

- 1 }

- 5 9 -

c dd u ::::; K

= log 4 i og 4/3 J u L dd c [ max ( l og I z I , - l og 4 ) ] dd c [ max ( log I z I , -log 4 / 3 ) } =

::::; 1 0g4 i094/3 J uL ddc [ max ( log I z I ' - 10g4/ 3 ] 2 =

= 10g4 1�94/3 Jddc [ maX { log l z l , - 1 094 ) ] 2 >

> 1 2 Jddc [ maX { log l z l , - 1 09 4 ) ] 2 = J { ddCUK ) 2 = d ( UK ) .

( 10g 4 ) Note that i n v i ew o f the results i n Sect ion V ,

and

are outer regular capac i t i es .

We now turn to the prob lem o f est imat i ng the Monge-Ampere mas s of plur i subharmon i c funct ions de f i ned outs ide a compact set .

Let � be an open and bounded pseudoconvex set i n [n , n> 2 and K a compact subset o f n so that n\K i s connected . By Hartogs exten s i on theorem , every analyt i c funct ion on � \ K extends to � . When i t comes to plur i subharmonic funct ions the s i tuat i on i s d i f ferent .

Propos i t i on VI : 3 . As sume that � E P S H n Lool ( n\ K ) oc where K is a removable s i ngular i ty set for the plur i subharmon i c funct ions . Then J ( ddc� ) n <+oo when Kccn ' cc� .

r.l ' \K

- 6 0 -

Proof . From Section V we know that ( ddc� ) n i s a wel l -de f i ned pos i t ive measure on �\ K . S i nce � extends to a plur i subhar-mon i c funct i on on � , we can choose a sequence near TI' such that

� . E PSHnC ]

00

Le t W= { z E n l i d ( z , K ) > �d ( a n I , K ) } . Then 8 = i n f � ( z » -oo z EW

and i f we put ¢ . =sup ( � . , 8 ) ] ]

then ¢ . =� .

] ] on W .

S o f ( ddc¢ j )n= f ( ddc� j )

n by Stokes I theorem . n l n l

f c n = ( dd sup ( � , c ) ) <+00 s i nce 00 sup ( � , 8 ) E PSHnLl ( � ) . oc n l

Theorem VI : l . Let ¢EPSHnC 2 ( n ) and assume that n ' = { ¢< l } and that K= { ¢�s } for an s , O < s < l . I f �EPSHnLool ( �\K ) then oc

and

f I � I ( ddc� ) n- 1 ,\ ddc¢ < + 00

{ s < ¢ < l }

J c n ( ¢- s ) ( dd � ) < + 00 .

{ s < ¢ < l }

Proof . S i nce � i s pseudoconvex and ¢ plur i subharmon i c , there i s to every z O E { s < ¢ < l } an ana l yt i c funct ions such that f ( z } f f ( z O ) ' � z E K . I f we restr i ct � to { f ( z ) = f ( z O ) } and apply the max imum pr i n c i pl e , we conclude that � i s uni formly bounded above on TI'\K . I t i s there fore no restr i c t i on to

- 6 1 -

assume that �iO .

Let s < r < l . Then we have by Stoke s ' theorem

f - � ( ddc� ) A ( ddc� ) n- 1 + f ( �_r ) ( ddc� ) n < { r <�< l } { r < � < l }

� l im f - �ddcmax ( � , t ) " ( ddc� ) n- l + f ( �-r ) ( ddc� ) n = t r { r <�< l } { r < � < l }

+ f d max ( � , t ) 1\ dC � /I ( ddc� ) n- l + f ( IV-r ) ( ddc� ) nJ = { r < �< l } { r < IV < l }

+ f ( max ( IV , t ) - r ) dc� " ( ddc� ) n- 1 -{ �= 1 }

f C C n- l ( max ( 1jJ , t ) -r ) d � 1\ ( dd � ) + { IV=r }

+ f ( r-max ( � , t ) ) ( ddc� ) nJ � f - �dC� 1\ ( ddc� ) n- l + { r < � < l } { �= 1 }

+ f ( �-r ) dc� A ( ddc� ) n- l . { �= 1 }

S i nce the r i ght hand s i de i s uni formly bounded i n r , s < r < l and s ince �iO we get

- 6 2 -

f I qJ l ddc1jJ J\ ( ddcqJ ) n- 1 + f ( 1jJ- s ) ( ddcqJ ) n < +oo { s < 1jJ < l } { s < 1jJ < l }

where each number i s non-negat ive .

An example

The fol l ow i ng example s hows that the convergence factor 1jJ-s real l y is needed in the theorem . There are funct i on s qJ and 1jJ such that

f c n ( dd qJ ) = +00

{ s < 1jJ < l }

where qJ can be taken to be plur i subharmon i c and bounded .

De f i ne u ( z ) =- li z I 2 - � then u i s subharmon i c on 1 2 1 { zE a: j 2 < l z l < 1 } , - - <u< O

12 and tlu = 1 - 1 z 1 2

( 1 z 1 2 _ � ) 3/2 .

2 2 8 1 De f i ne 1jJ ( z , w ) = l z l + } I w l and let O < n <20 and put

qJ n ( Z , w ) =max [ - � I( 1 +n ) 2 1 Z 1 2 - � + 1 w i 2 , 1 w 1 2 + ( 1 +n ) 1 w 1 4 -I � ( n + i 2 ) ] . Then qJ n i s plur i subharmon i c on {.l < 1jJ < (� ) 2 } for i f 2 2 1 ( l ' :- n ) 2 1 z 1 2 .s. 1 we have , when 1jJ ( Z , w ) > .;.

2 2 1 8 1 1 2 1 - 2 1 2 n+n = } w i > 2 - 1 z > 2 ( 1 - ( 1 +n ) ) = 2 ( 1 + n ) 2

2 n n + "2' ( 1 + n ) 2

so

- 6 3 -

Assume now that � < l z I 2 + 1 I w I 8 « �� ) 2 and that

} I n + �2

4 2 3 9 In + i2

15 1 + n < I w I < I } 40 1 + n .

which i s a wel l de f i ned doma i n i f

2 2 8 2 2 /3 2 4 2 8 n + r- - } I w l ( l + n ) < 6 ( n + r) -8 2" ( n + r ) ( l +n ) I w l + 4 ( l +n ) I w l

o < 5 ( n + i2 ) + � 4 ( 1 +n ) 2 1 w 1 8 - a ll in + i

2 ( 1 +Tl ) I w 1 4

whi ch holds wi th I w l 4 i n the i nteral above - the r i ght hand s ide i s then str i ctly larger than

2 n 2 n + 2 3 9 2 5 ( n + !l- ) + J....! ( 1 +n ) 2 1 - - a • - ( n + !l- ) = O . 2 3 5 ( 1 1 n ) 2 4 0 2

There fore f ( ddcCP n ) 2 � c f 2

dz �w _ l ) 3/2 1 2 2 8 2 0 2 A « 1 +n ) I z I 2 2< l z l +} I w l « 21 ) n

- 6 4 -

where

1 2 8 2 2 2 1 (/3 2 4) 2 An = ;2 r-n -� ;'---n

�2 1 (- - 3 1 w i ) ( 1 + n ) < ( 1 + n ) 1 z 1 < 2+ 4 l 2 ( n + t-) - ( 1 + n ) 1 w 1

+ � n + 2 4 2 1 o. < l w l < B + n + n

and where we have chosen 13 3 9 0. Y 5 < o. < 8 < 4O" Y 1 and where c i s a str i ctly pos i t ive constant . The r i ght hand s ide i s not sma l l e r than

2 3 3 9 12. c { ( a - - ) + - Y � - 8 } 5 4 0 3 2

( n + .!L)dw 2 2 4 3 + t-) - ( 1 + n ) I w i )

>

d ---=2---- �+oo , n� O ( d pos i t i ve constant ) ( n + .!L) 1 / 4 2

where n = /� : f } . 1 Furthermore , - 2 < <P n � 2 so i f we put

<P = 00 ,, 1 4 l. 2 <P l /]' j = 2 0 j

and f ( ddc<p ) 2 = +00 .

�< I z I 2 + i I w 1 8 < ( � � ) 2

- 6 5 -

Remark . I t i s pos s i ble to mod i fy the cp : S n

to get the funct ion cp o f class can mod i fy to be C

Notes and references

00 C on 00 on

We do not know i f one but st i l l have f ( ddccp ) 2 = +00 .

Q\K

For the decompos i t ion o f measures used i n the proof of Propo s i t ion VI : 1 a ) see Sect i on XI .

Propo s i t i on V : 2 was f i r s t proved by Demai l ly , J . -P . ,

Mesures de Monge-Ampere et caracter i z at ion geometr ique des var i etes algebr iques a f f i nes . Bul let i n de l a Soc iete Mathematiques de France , Memoi re 1 9 , 1 1 3 ( 1 9 8 5 ) , 1 - 1 2 5 .

Propos i t ion VI : 3 i n the case K equal to one po i nt was proved by Gri f f ith , P . , Two theorems on extens ion o f holomorph ic mapp i ng s . I nv . Math . 1 4 ( 1 9 7 1 ) , 2 7 - 6 2 .

Theorem VI : 1 i n the plur i subharmon ic and c2 case was proved by Fornaes s , J . E . and S i bony , N . , Pluri subharmon i c funct ions o n r i ng doma i n s . Krantz , S . G . , Ed . , Complex ana lys i s . Semi nar Un i vers i ty Park PA , 1 9 8 6 . Spr i nger Lecture Notes i n Mathemat i c s . Vol . 1 2 6 8 .

Propo s i t i on VI : 1 and Theorem VI : 1 wi l l appear i n Cegrell , U . , Plur i subharmon i c funct ions outs i de compact sets . Proc . AMS .

The above example shows that there are funct ions that can be subextended and st i l l have unbounded Monge-Ampere mas s near a compact set .

I n smooth doma i ns , there are plur i subharmon ic funct ions that cannot be subextended : Bedford , E . and Taylor , B . A. ,

Smooth plur i subharmon ic funct i ons with no subextens ions . Manuscr ipt 1 9 8 6 .

VII Green's Function

An open subset D o f [n i s cal led pseudoconvex i f -log d ( z , CD ) i s plur i subharmon i c . ( d ( z , CD ) = i n f j z -w l ) .

wECD

I f there i s a conti nuous and plur i subharmon i c funct i on � < O on D so that

{ z E D ; � ( z ) <a } ccD , �a< O

then D i s sa id to be hyperconvex .

I f there i s a C 2 - funct i on p : [n�ffi, grad p � O on D ,

D= { p < O }

n 1:

i , j == l a 2 p 2 ( P ) w .� > c l w l ' � P E a D , �wE[n

a Z . dZ . 1 J 1 J

for some c > O then D i s s a i d to be str i ct ly pseudoconvex .

Def i nition . Let D be an open subset o f n [ , n> l and assume that �ESH ( DxD ) . Then � i s cal led 2 -plur i subharmon i c ( 2 -PSH ) on DxD if

�Bz � � ( z , w ) E PSH ( D ) , 'dwED

n 3w � � ( z , w ) E P S H ( D ) , 'd z E D .

Def in i t i on . Green ' s funct ion relat ively t o D i s

W ( z , w ) =sup { u ( z , w ) E 2 PSH ( DxD ) ; u�O

u ( z , w )�log l z -w l - log max [ d ( z , CD ) , d ( w , CD ) ] } .

- 6 7 -

Lemma VI I : l l . I f uEPSH ( D ) , u < O , wO E D and i f u ( z ) -log ! z -wo l

i s bounded above near t hen

u ( z ) �l og l z -wo l - l og d ( z , CD ) , � z E D .

Proof . Con s i der for T E [ a nd z E D

Then l EV and s i nce z

l S subharmon i c on V we get z

g ( l ) �sup g ( T ) �SUp - l og I T ( Z-wo ) 1 = T EV T E av z z

There fore u ( z ) �log ! z -wo l - log d ( wO , CD ) , z E D .

Corollary VI I : l . I f u�O , u ( z , w ) - l og l z -w l loca l l y bounded

near the d i agonal �cDxD and i f uE 2 - P SH ( DxD ) then

u ( z , w ) �l og l z-w l - l og max [ d ( z , CD ) , d ( w , CD ) ] .

Corollary VI I : 2 . I f f EHoo ( D ) , I f l � l , ftcon s t . then

I f ( Z ) - f ( W ) I v f ( z , w ) =l og _

� W ( z , w ) . l - f ( z ) £ ( w )

- 6 8 -

Proof . I t i s c l ear that v f E 2 -PSH ( DXD ) ( but v f i s not PSH ( DxD ) ! ) , vf�O . Furthermore , v f ( z , w ) - l og l z -w l i s l ocal ly bounded above on 6 so by Corol lary VI I : 1 , v f�W '

Propos i t i on VI I : l . 1 ) W ( z , w ) =W ( w , z ) 2 ) WE 2 -PSH 3 ) I f furthermore 0 i s str i ct l y pseudoconvex then

l im W ( z , w ) =O , �wE D . z-+ a D

Proof . 1 ) S i nce l og l z -w l - log max [ d ( z , CD ) , d ( w , CD ) ] i s symmetr i c s o i s w . 2 ) Denote by W* the sma l l e s t upper semi cont i nuous ma jorant of W. Then W i s subharmon i c on DxD and we c la im that W* I S 2 -PSH ( DxD ) . I f we take � to be a funct i on only depend i ng on I I f z - � w-n z and con s i der WE , c ( z , w ) = W ( � , n ) � (--E- ) � (--C- ) then W 1: � w* when E , 0 ':. 0 and W 1: i s 2 - PSH . Furthermore E , v ( , v

W* ( z , w ) < log l z -w l - Iog max [ d ( z , CD ) , d ( w , CD ) ] s i nce the r ight hand s i de is upper semi cont i nuous . There fore W = W* wh i ch proves 2 ) . 3 ) I f 0 i s str i ctly pseudoconvex i t i s wel l known that to

there is a f unct ion f . P

ana l yt i c on every pE a D cont i nuous on D such that f ( p ) = l and I f 1 < 1 on

By Corol lary VII : 2 I f ( Z ) - f ( W ) I v f ( z , w ) = 1 og

_ .s. W ( Z , w ) so therefore p l - f ( z ) f ( w )

D and D .

o > l im W ( z , w ) � l im W ( z , p ) � l im vf ( z , w ) =O wh i ch completes z-+p z-+p z�p p

the proo f o f Propos i t ion VI I : l .

- 6 9 -

Theorem VI I : l . Let f : D�D ' be a hol omorph i c map between two

open sets Dc�n , D ' c�n . Then

WD ' ( f ( z ) , f ( w ) ) �WD ( z , w ) .

Proof . I t i s c lear that WD ' ( f ( z ) , f ( w » 15 negat i ve and

2-PSH ( DxD ) .

Furthermore WD , ( f ( z ) , f ( w ) ) �l og l f ( z ) - f ( w ) 1 -

-log max [ d ( f ( z ) , C D ' ) , d ( f ( w ) , CD ' ) ] =log I z -w 1 + l og 1 f ( z ) - f ( w ) I -I z -w l

- log max [ d ( f ( z ) , CD ' ) , d ( f ( w ) , CD ' ) ] s o Wp , ( f { z ) , f ( w ) ) - l og { z -w )

i s l oca l ly bounded above near 6 . There fore WD , ( f ( z ) , f ( w ) �

� WD { z , w l .

K l i mek and Dema i l l y have stud i ed the followi ng funct i on .

u ( z , w ) =uD ( z , w l =sup { u ( z l E PSH ( D ) ; u�O , u ( z ) - l og l z -w l bounded

above near w } .

I t i s c l ear that W<u .

Propos i t i on VII : 2 . As sume that D i s str i ctly pseudoconvex .

The fol l ow i ng cond i t i ons are equ i valent .

i ) l i m f ( dd� max ( W ( z , � ) , t » n= ( 2 TI ) n , � z E D t�-oo

i i ) W=u D

i i i ) u ( z , w ) =u ( w , z ) , � ( z , w ) E DxD

i v l D3w � u ( z , w ) E PSH ( D l , � z E D .

- 7 0 -

Proof . i i ) � i i i ) fol l ows f rom Propos i t i on V I I : l .

i i i ) � i v ) s i nce z .... u ( z , w ) E P SH ( D ) , tlwE D . i v ) � i i ) . I t fol lows f rom Corol lary VI I : l that u ( z , w ) < log ! z -w ! - log max [ d ( z , CD ) , d ( CD ) ] so u <w . On the other hand , it i s always true that u>W . i i ) � i ) ( ddcu ( z , w ) ) n�O outs ide w . There fore

l im f ( ddc ( maX u ( z , w ) , t ) ) n�l im f ( ddcmaX ( 1 09 ! Z -W ! , t ) ) n� ( 2 1T ) n . t .... -oo D t .... -oo D i ) � i i ) . F i x wED . S i nce max ( W ( z , w ) , t )imax ( u ( z , w ) , t ) and s i nce both funct i ons have boundary va lues z ero , it fol l ows from Lemma V : 3 that

f c n f c n n ( dd max ( W ( z , w ) , t ) ) � ( dd max ( u ( z , w ) , t ) ) � ( 2 1T ) •

o 0 Therefore i ) means that C n ( dd W ( Z , W ) ) =0 for z�w because , s i nce W ( z , w ) - log ! z -w ! i s bounded near z =w , l im f ( ddcmax ( W ( z , W ) , t ) ) n= ( 2 1T ) n for every neighborhood E o f t .... -oo E w . Then ( l - E ) W ( Z , W ) I S equal to z ero at a D , l arger than u ( z , w ) near w . There fore , Corol lary V : l g i ves that ( l - E ) W ( Z , W ) �U ( Z , W ) for every E > O wh i ch completes the proof o f Propos i t ion VI I : 2 .

Def i n i t i on . Let D be a doma i n i n �n . I f ( z , w ) E DxD we def i ne the Caratheodory di stance

CD ( Z , W ) =Sup { P ( F ( z ) , F ( W ) ) ; F : D .... U , F hol omorph i c } F

- 7 1 -

and the Kobayashi d i stance

m KD ( z , w ) = i n f { L 0 D ( 2 , , 2 ' - 1 ) ; z = z z =w } j = l J J 0 ' m

where 0 D ( z , w ) = i n f { p ( � , n ) ; 3 f : U4 D w i t h f ( � ) = z , f ( n ) =w ,

f holomorph i c } where U i s the u n i t d i s c i n � and where p

i s the hyperbo l i c d i s ta nce on U .

Propo s i t i on VI I : 3 . l og tanh C ( 2 , w ) �W ( z , w ) �u ( 2 , w ) �1 0g tanh o ( z , w ) .

I f D i s con vex , then equa l i ty holds .

Proo f . We have that

l og tanh C ( Z ' W ) =sUP { l Og l f ( Z ) - f�W ) I ; f : D4 U , f holomorph i c } 1 - f ( z ) f ( w )

so by Cor o l l a r y V I I : 2 t he f i r s t i nequa l i ty fol l ows . The s econd

i s c l ea r a nd the t h i rd fol l ows f r om Theorem VI I : 1 s i nce

uU= l og tanh ( p ) . I f D i s convex , i t i s a r e s u l t o f Lempert

that C=o .

When n= 1 , we have f o r every compa ct set KCU , the un i t

d i sc

where u i s a u n i quely determ i ned pos i t i ve mea s ure on K .

When n ) 1 , thi s i s no l onger true s i nce there are compact

plur i po l a r s e t s s uppor t i ng pos i t i ve mea s u r e s such that

fW ( z , � ) d U ( � ) i s bounded on D .

- 7 2 -

Note s and r e f e rences

The funct i on u i s i nt r oduced by M . Kl i mek i n the a r t i c l e

Ext r ema l p l u r i s ubharmon i c funct i on s a nd p s e udod i s t ances . Bu l l .

Soc . Math . France 1 1 3 ( 1 9 8 5 ) .

I n Me s u r e s de Mong e-Amp e r e et me s ur e s p l u r i harmon i que by

J . -P . Dema i l l y , Math . Z . 1 9 4 ( 1 9 8 7 ) , 5 1 9 - 5 6 4 , th i s s t udy i s

cont i nued , i nc lud i ng a representat i on formul a and expl i c i t

examp l e s .

That CO=KO for convex s e t s 0 wa s proved by L . Lempert

in Analy s i s Mathemat i ca 8 ( 1 9 8 2 ) .

VIII The Global Extremal Function

o n

Denote by L t h e c l a s s o f p l u r i subharmon i c funct i on s f

ern s uc h that

where i s a con stant ( depend i ng on f ) . I f Ecern we de f i n e

and V* E

VE ( z ) = s up { f ( z ) ; f E L ; f�O on E }

to be the sma l l e s t upper s em i con t i nuous ma j orant o f

Propos i t i on VI I I : l . VE E L � E i s not plur i pol a r .

I f V * E L E then the funct i on i s c a l l ed the g l oba l extremal

p l u r i subharmo n i c f u nc t i on r e l a t i ve l y to E .

Theorem VI I I : l . A s s ume that E i s a r e l a t i ve l y compact subset

o f the u n i t bal l . Then ( ddc h * ) n t 1 1 E are mu u a y

a b s o l u t e l y con t i nuou s . ( See V I for the de f i n i t i on of hE ' )

Let uEL and de f i ne

Y r ( u ) = f u ( rw ) da ( w ) - l og r

I w l = l

where a i s the norma l i z ed Lebesgue mea sure on the un i t sphere .

The n i s convex and bounded a t +00 so Y ( u ) r i s de -

crea s i ng a n d we denote by y ( u ) the l i m i t l i m Yr ( u ) . I f r-++oo

- 7 4 -

furthermore u�log+ l z l then y ( u ) �O .

Lemma VI I I : l . There i s a constant n C = ( 2 1T ) n so that

Propos i t i on VI I I : 2 . I f 8 i s a d - c l osed ( n - l , n- l i - f orm t hen

I uddClog + I z I II 8 = S log + I z I ddc u ,\ 8 + I z I <R I z I <R

+ I udClog + I z 1 ,\ 8 - l og R I dCu II 8 .

I z l =R I z l =R

Proo f . Stoke s ' formul a .

Theorem VI I I : 2 . I f 1 0g+ l z l �uEL then

I 1 I + I I C C + I I n - 1 y ( u ) = u ( w ) da ( w ) -C n l og z dd u j\ ( dd log z ) .

l u l = l [ n

Proof . Take 8 to be ( ddC log+ l z l ) n- l i n Propos i t i on V I I I : 2 .

Then

C n I u ( Rw ) da ( w ) - l ogR I ddCu A ( ddClog+ l z l ) n- l = I w l = l I z l �R

= C n I u ( w ) da ( w ) - I 1 0g+ l z l ddcu A ( ddClog+ l z l ) n - 1

I w l = l I z l <R

and t h e l e ft h a n d s i de i s , by Lemma V I I I : l n o t l e s s than z e r o .

Hence ,

So

and

s i nce

- 7 5 -

o < f 1 0g+ l z l ddcu /\ ( ddCl 09+ l z l ) n - l ..s. Cn I u ( w ) do ( w ) . �n l u l = l

Cn I u ( zw ) do ( w ) - l ogR f ddcu A ( ddCl og+ l z l ) n - l I w l � l I z l <R

= cn f U ( RW ) dO ( W ) - 1 09R ) +1 09R [ f ddcu A ( ddC log+ l z l ) n - l _

I w l = l [n

S ddcu A ( ddC l og + I z I ) n- l ] ==

I z I ..s.R

Cn [ S u ( Rw ) do ( w ) - l ogR ] + l ogR f ddcu A ( ddC log+ l z l ) n- l I w l = l I z l �R

S C 1\ C + I I n- l 1 0gR dd u ( dd log z ) -+ 0 ; R -+ +00

I z I �R

f log + I z I A ddcu A ( ddc log + I z I ) n- l < +00 �n

wh i ch proves the theorem .

Corol lary VI I I : l . I f 1 0g+ l z l ..s.uEL then

f + I I C C + I I n- l log z /\ dd U /\ ( dd l og z ) < +00 .

- 7 6 -

Lemma VI I I : 2 . I f u . 1 i s a decrea s i ng s equence o f funct ions i n

L such that l im u . =uo > l og+ l z l 1 - then

Proo f . It i s c l ear f r om the de f i n i t i on that l im y ( u , » y ( uO ) . J -

On the other hand , we have f rom Theorem VI I I : 2 that

l im S j-+ +oo <rn

+ 1 1 c c + 1 1 n l og z dd u j A ( dd log z ) �

S S + 1 1 c c + 1 1 n - l � uOda ( w ) - c l og z dd U o A ( dd log z )

I w l = l n I z l <R

by Theorem V : 2 .

I f we let R-++oo and app l y Theorem VI I I : 2 aga i n we get the

des i red conclu s i on .

Def i nit ion . Let uEL , we then de f i ne f ( u ) = sup u ( z ) - log r r 1 z I =r and f ( u ) =l im f r ( u ) ( s i nce f er ( u ) I S convex and bounded at

r-++oo +00 , fr ( u ) IS decreas i ng i n r ) .

I t i s cl ear that y ( u ) � f ( u ) . I f ECB , B the un i t bal l

I n <r , then y ( VE ) = f ( VE ) . The fol l ow i ng example shows that

i f n=2 , we can have str i ct i nequal i ty .

- 7 7 -

Example 1 . 2 F i x a> l and take K= { ( z l , z 2 ) E [ ; 2 2 1 z 1 I +a I z 2 I � 1 } •

Then VK ( z ) 1 + I 2 I 2 = 2" l og ( z 1 I +a z 2 1 ) so

y ( VK ) = � f log ( I w I 2+a l w2 1 2 ) dO ( w ) < � l og a = f ( VK ) · I w l = 1

Propos i tion VI I I : 3 .

log+ l z l <uEL .

Proo f . I f we f i rst take j = l , Propo s i t i on VI I I : 2 g ives

+ f udCl og+ l z l f\ ( ddcu ) n- l _ log R f dCu /\ ( ddcu ) n- l � Cn rR ( u ) + I z I =R I z 1 =R

+ log R f ( ddcu ) n -+ Cn f ( u ) , R-++oo s i nce I z I �R

10g R f ( ddcU ) n-+ O , R-++oo .

j + 1 .

I z l > R

Assum i ng the propos i t ion for j , we want to prove i t for

f f C n- j - l C + 1 I j + l u8 i + 1 == u ( dd u ) f\ ( dd l og z ) =

- 7 8 -

f C n- j C + 1 I j . 2. u ( dd u ) A ( dd log z ) + Cn f ( u ) 2. ( assumpt I on )

f C + 1 I n . f C + 1 I n 2. u ( dd log z ) + Cn J f ( u ) + Cn f ( u ) ::: u ( dd log z ) +

Corol lary VII I : l . I f 10g+ l z l EuEL then

f u ( w ) da < �n f 10g+ l z l ( ddcu ) n + y ( u ) + ( n- l ) f ( u ) . I w l ::: l

In part i cular i f ECB , the un i t ba l l , then

f VE ( w ) da ( w ) 2. y ( VE ) + ( n- l ) f ( VE ) · I w l ::: l

Proo f . Take j =n- l i n Propos i t i on VI I I : 3 . Then , by Theorem VI I I : 2 , we get

fU ( W ) da ( W ) - y ( u ) ::: � J 10g+ l z l ddCu ( ddClog+ l z l ) n- l <

I w i ::: 1 n C n

which proves the f i rst statement . The second statement fol l ows from the fact that i f E c B then C n supp ( dd VE ' C B .

- 7 9 -

Remarks and re ferences

A proof o f Propos i t i on V I I I : 1 is i n J. S i ci ak , Extrema l plur i subharmon i c funct ions i n �n , Proceed i ngs o f the f i rst F inn i sh-Po l i sh Summerschool in Complex Analys i s in Podles i ce , 1 9 7 7 , pg . 1 2 3 - 1 2 4 .

Theorem VI I I : 1 i s due to N . Levenberg , Monge-Ampere Measures Assoc i ated to Extremal P l ur i subharmon i c Funct ions i n �n . Trans . Am . Math . Soc . 2 8 9 ( 1 9 8 5 ) , 3 3 3- 3 4 3 .

Lemma VI I I : 1 i s proved by B . A . Taylor , An est imate for an extremal plur i subharmon i c funct ion . Semi nare d ' Analyse P . Le long , Dolbeault-H . Skoda , 1 9 8 1 / 1 9 8 3 . Spr i nger Lecture Notes i n Mathemat ics 1 0 2 8 . Th i s paper a l so contai n s a somewhat weaker vers ion o f Corol lary VI I I : 1 .

I n S . Kolodz i e j , The l ogar i thmi c capac i ty i n �n ( To appear - f ( VE )

i n Ann . Pol . Math . ) , i t was proved that C ( E ) =e i s a -y ( VE )

capac i ty i n Choquets sense . That e i s a capaci ty was proved by the same author i n : Capac i t i es as soc i ated to the S i c i ak extremal funct ion , Manusc r ipt . Cracow . 1 9 8 6 .

The relat ionshi p between y and f has also been stud i ed by J . S i c i ak , On logar i thmi c capac i t i es and plur ipolar sets i n �n . Manuscr ipt , October 1 9 8 6 .

U s i ng Corol lary 6 . 7 i n E . Bedford and B . A . Taylor , Plur i subharmon i c funct ions with l ogar i thmi c s i ngular i t i e s ,

- f ( VE ) Manuscr ipt 1 9 8 7 , one can prove that e i s an outer regular capac i ty .

v . P . Zaharjuta has studied capac i t i es and extrema l plur i subharmon i c funct i ons i n connect ion w i th trans f i ni te diameter

- 8 0 -

and the Bernste in-Wal sh theorem : Trans f i n i te d iameter Cebychev constants and capac ity for compact i n �n . Math . USSR Sborn ik , Vol . 2 5 ( 1 9 7 5 ) , No . 3 .

, Extremal plur i subharmon i c funct i ons , orthogonal pol ynomia l s and the Bernste i n-Wal sh theorem for analyt i c funct ion s o f several complex var iables . Ann . Polon . Math . 3 3 ( 1 9 7 6 ) .

For result s and more re ferences , see : Nguyen Thanh Van and Ahmed Z er i ahi , Fami l Ies de polynomes presque partout bornees . Bul l . Sc . Math . 2 c Ser i e 1 0 7 ( 1 9 8 3 ) .

w . Plesniak and W . PawXucki , Markov ' s i nequal i ty and COO funct ions on sets with polynomial cusps . Math . Ann . 2 7 5 ( 1 9 8 6 ) , 4 6 7 - 4 8 0 .

A . Sadullaev , P lur i subharmon i c measures and capac i t i es on complex man i folds . Russ i an Math . Surveys 3 6 ( 1 9 8 1 ) .

IX Gamma Capacity

Def i n i t ion ( the Choquet I ntegral ) . Assume that f 1 S a nonnegat ive funct ion and c a capac i ty . Then ffdc i s de f i ned by

co

ffdc = fC ( { x ; f ( x » s } ) ds . o

Def i n i t ion . Let p be a precapac i ty . A non-negat ive funct ion f is sa id to be p-capac itable i f

f fdp = sup f gdp ; g� f , g upper semi -cont i nuous .

Lemma IX : l . Assume that E v ' vEN i s an i ncrea s i ng sequence of p-capac i table sets . Then

00

ex> u E v v= l

i s p-capac i table .

Proof . We have p ( U E v ) = s u p p ( Ev ) ' Let £ > 0 be g iven . V= 1 v E N

Choose v so that

p ( U E ) < p ( E ) + E/2 v= 1 v v

and a compact subset k o f E v such that

co Then k i s compact i n U E v and

v= l

00

co p ( U Ev ) < p ( k ) + £

v= l

so U Ev 1 S p-capac i table . v= l

- 8 2 -

Theorem IX : l . I f f i s p-c apac i ta b l e then { X i f ( x » s } i s

p-capa c i table . .

Proof . A s s ume t hat f i s p -capac i ta bl e . Then there i s a n i n

crea s i ng s equence { fn } �= l o f upper sem i -cont i nuous funct i on s

wh i ch a r e sma l l e r o r equa l t o f w i t h

l im n-++oo

If d = Ifd . n p p

I t i s no rest r i ct i on t o a s sume that every f h a s compact n

support .

I t i s easy to see that

p ( { X i f ( x » s } ) = p ( { X i l i m f ( x » s } ) \:1 s > O . n n-++oo

Pu t E = { x ; f ( x ) > s + 1 } . m , n n - m Every E m i n i s a compact s u b s e t

o f { X i f ( x » s } and

00 00 { X i l im f ( x » s } = u u E n m i n n-++oo m= l n= l

so i t f o l l ows f rom Lemma I X : 1 that { X i l im f ( x » s } i s p-capa-n n-++oo c i table a nd { X i f ( x » s } has to be p-capac i ta b l e .

Theorem IX : 2 . A s s ume that c i s a s t ro ng l y subadd i t i ve capa-

c i ty . Denote by t he character i s t i c f u n c t i on of

n n B 1 ( f ) = i n f ( l: a . c ( A · ) i

1. =l: l a i XA1.� f ) .

i = l 1 1

A . P u t

- 8 3 -

Then

Corol lary IX : l . Assume that c is a capac i ty as in Theorem I X : 2 . Then the Choquet i ntegral i s subadd i t ive ( and therefore a semi norm on the non-negati ve funct ions ) .

Corollary I X : 2 . Assume that c i s a capac ity as i n Theorem IX : 2 . Then

f fd = i n f fgd ; f < g , g i s lower semi -cont i nuous . c c -

Swarms . Product capac i t i es .

Def i n it i on . A c lass o f func t i on s ( LF ' EcV

i s cal led a swarm i f a )

s ) for every f i xed xEU , E�LE ( x ) for every compact subset K of

i s a capac i ty on v , v , LK ( X ) i s a bounded ,

upper semi -cont i nuous funct ion wi th compact support .

Example . Choose U=v=� n and denote by X E the character i s t i c funct ion of E . Then ( X E ) Ec �n i s a swarm .

Theorem IX : 3 . Assume that c i s a capac i ty on U and ( LE ) ECV a swarm . Then C ( E ) =fLEdC i s a capac i ty on V . Furthermore , i f LE ( x ) i s subaddi t i ve for a l l x l n U and i f c i s strongly subadd i t i ve then C i s subadd i t ive .

- 8 4 -

Proof . i ) c l ear .

i i ) Let Ev ' vEJN f be a non-d e c r e a s i ng s equence o f s u b s e t s o f V .

Then

s up C ( E v ) = l i m fL E dc = l i m fC ( { XEU ; LE ( x » s } ) ds = v Em v-+ +00 V v-+ +00 0 V

00 00

= f l im c ( { xE U ; LE > s } ) ds = fC ( { xE U ; L 00 ( x » s } ) ds = O v-++oo V 0 U E v v= l

00 = C ( U E) .

v= l

i i i ) Let K v ' vElli, be a decrea s i ng s equence o f compact s u b s et s

o f V . Then

00

i n f C ( Kv ) = l i m fC ( { XEU v E IN v-+ +00 0

LK ( x ) > s } ) ds = v

00 00

= l i m fC ( { xEU ; LK ( x ) ::.s } ) d s =f l i m c ( { xE U ; LK ( x ) ::.s } ) d s = v-++oo 0 v 0 v-++oo V

00

= fC ( { XE U ; L 00 o n K v v= l

00 ( x » s } ) d s = C ( n Kv ) .

v= l

The l a s t s tatement i n the theorem f o l l ows f rom Coro l l ary I X : 1 .

Theorem IX : 4 . Let ( LE ) ECV be a swarm . Then LE i s a u n i ve r

s a l l y capac i ta b l e func t i on f o r every u n i ve r sa l l y capa c i t a b l e

s e t EE P ( V ) .

- 8 5 -

Proof . S i nce , by Theorem I X : 3 , C ( E ) = f LEdc i s a capa c i ty for

eve ry capa c i ty c , we h ave for any u n i ve r s a l l y capa c i table

set E

fL d = C ( E ) = E c s up KCE

K compa c t

C ( K ) = s up KCE

K compact

S i nce LK i s uppe r s em i -con t i nuous and l e s s or equa l to LE ,

LE i s c -capac i tab l e . But c wa s an a r b i t r ar i ly chosen capa

c i ty so i t f o l l ows that LE i s u n i ve r sa l l y capac i ta b l e .

Theorem IX : 5 . A s s ume that c i s a c apa c i ty on V . Then

LE { x ) =c ( { yE V ; ( x , y ) E E } ) , ECU x V i s a swarm . Furthermore , i f c

i s s ubadd i t i ve then LE ( X ) i s subadd i t i ve for every f i xed x

i n U .

Proo f . a ) c l ea r .

8 ) Let a c ompact subset K o f Ux v be g i ve n . I t i s c lear

that LK has compact s uppo r t so it rema i ns to prove that LK

1 S upper s emi con t i nuou s . G i ve n a > O and x O E {x E U ; LK ( x ) �a} .

We have to prove that LK ( xO ) �a . Choose x n w i t h LK ( xn ) �a

such that xn�x O ' n�+oo . Put

and

o = { yE V ; ( x , y ) E K } . n n

I t i s eas i ly ver i f i ed that

co co D O :> n ( u D . ) .

i = , j= i J

- 8 6 -

S i nce c i s a capa c i ty we h ave

co

co co

= l i m c ( u D . ) > l im c ( D . ) i�+co j = i J j�+co J

= l im LK ( X j ) > u . j�+co

The l a s t s tatemen t i n the t heorem i s obv i ou s and the proof i s

comp l ete .

Def inition ( Product Capa c i ty ) . Let c and d be capa c i t i e s

on U a nd V respect i ve l y . The product capa c i ty c x d on

UxV i s de f i ned by

where

LE ( X ) =d ( { yEV : ( x , y ) E E } ) .

By Theorems IX : 5 and IX : 3 , c xd i s a capac i ty . Furthermor e ,

i f c i s strong l y s ubadd i t i ve and d i s s ubadd i t i ve , i t

f o l l ows f rom Theorem IX : 2 that c x d i s s ubadd i t i ve .

Example I . Let U=V=� and denote by c the Newton i an c apa

c i ty . Con s ider in �2 the set E= { ( z " z 2 ) ; I z , I + l z 2 1 = ' ,

Im z , =O } . I t i s eas i l y seen that c x c ( E » O . But i f we i n te r -

change t h e va r i a b l e s a nd i . e . con s i der the set

E ' = { ( Z l , z 2 ) ; I z , I + l z 2 1 = 1 : Im z 2= O } i t i s c l ear that

LE , ( z 1 } = O , \:fz l E� , s o c x c ( E ' } = O .

on

- 8 7 -

Let now U be a n open s u b s et o f � and c a capa c i ty

U . We c o n s truct

c =c 1

c =c x c 1 n n -

on U

o n by i nduct i on

I t i s c l ea r that i s a c apac i t y on Un , i f c i s strongly

subadd i t i ve , then c n i s s ubadd i t i ve . Put for EcUn , x E Un-p

By Theorem I X : 5 , ( Ln-p ) i s a swarm . E Ec Un

Remark . I t f o l l ows f rom Theorem I X : 4 and Theorem I X : l that

{ xE Un -p ; L�-P ( x » s } i s u n i ve r s a l ly capa c i ta b l e for every s > O

and every u n i ve r s a l l y capa c i ta b l e set E .

Let c be a capa c i ty o n a n open s u b s e t U o f � . We

de f i n e a precapa c i ty o n by i nduct i on

=c on

. n - l P n ( E ) =c ( { xE U ; Pn _ 1 ( { yEU ; ( x , y ) E E } » O } ) , ECUn .

Theorem IX : 6 . 1 ) P ( E ) =O � n c ( E ) = O , n

2 ) 3 )

4 )

1 LE ( X » O } ) ,

P i s a precapa c i ty on Un , n

every u n i ve r s a l l y capa c i t a b l e i s P -capac i ta b l e , n

5 ) i f c i s subadd i t i ve , t h e Pn i s s ubadd i t i ve .

- 8 8 -

Proof . 1 ) , 2 ) i nduct i o n . n = l c l ea r . A s s ume that 1 ) and 2 )

hol d for n - 1 . Prove 1 ) and 2 ) for n .

Thus

n- l P n ( E ) =C ( { xE U ; Pn_ l ( { yEU ; ( x , y ) E E } » O } =

n - l = c ( { xE U ; cn _ 1 ( { y E U ; ( x , y ) E E } » O } ) =

= c ( { xE U

and i t i s c lear that 1

c { xE U ; LE ( x » O } = O i f and o n l y i f

3 ) i ) - i i i ) a r e c l ear s i nc e

whe r e c i s a c apac i ty a nd i s a swarm .

4 ) A s s ume that E i s un i v e r s a l l y capac i ta b l e . By Theorem IX : 4 , L� i s u n i ve r s a l l y capac i t a b l e and

wher e Kv ' vErn, i s a n i nc r e a s i ng sequence o f compact s u b s e t s

o f E . Hence

c ( { xE U ; L� ( X » S } ) = c ( { xE U ; l i m L� ( x » s } ) = v-++oo v

= l im c { x E U ; LK ( x » s } ) v-++ro V

- 8 9 -

for a l l s�O , s o P ( E ) =l im P ( K ) = sup { P ( K ) i K compact , Ke E } n n \I v� +oo wh i ch mean s that E i s P n- c apac i tabl e .

5 ) I nduc t i on . n= l c lear . A s s ume that Pn - 1 i s s ubadd i t i ve .

n - l Then Pn ( E l u E2 ) =c ( { xE U i Pn _ 1 ( { y E U ; ( x , y ) E E 1 U E2 } » 0 ) =

n- l n - 1 = C ( { XE U i Pn _ 1 ( { y E U i ( X , y ) E E 1 } U { yE U i ( X , y ) E E 2 } > 0 ) } <

n - l < C ( { XE U i P n _ 1 ( { y E U ; ( x , y ) E E 1 } » 0 } U

n - 1 U { xE U ; P n_ 1 ( { y E U i ( x , y ) E E 2 } » 0 } ) �P n ( E 1 ) +P n ( E 2 ) and i t

f o l l ows that Pn i s s ubadd i t i ve .

Corollary IX : 3 . A s s ume that 2

c i s a capac i ty on U . I f

C ( E \I ) =O , \1= 1 , 2

property .

� c ( u E ) = 0 , \I \1= 1

then cn and Pn has the same

Proof . A s the proof o f Theorem I X : 6 , 2 .

Theorem IX : 7 . Let c be a ( pr e ) c apac i ty o n V and ( a i ) i E I

a comp l et e norma l f am i l y o f c o n t i nuous f u n c t i on s , a . : U-+V . 1 Then

C ( E ) = SUp c ( a . ( E ) ) i E I 1

i s a ( pr e ) capac i ty on U . I f c 1 S s ubadd i t i ve , then C i s

subadd i t i ve .

Proof . i ) , i i ) c l ear .

i i i ) Let E\I

' \lEill , be a n i nc r e a s i ng sequence o f subsets o f U .

G i ve n £ > 0 . Choose a . 1 £ such that C ( E ) < c ( a i ( E ) ) + £ / 2 and

(

then such that c ( a . ( E ) ) < c ( a . ( E ) ) + ( / 2 . 1 ( 1 ( \I Then

- 9 0 -

C ( E ) < c ( a . ( E ) ) + E < C ( E , , ) + E 1 V v E

and i i i ) i s proved .

S i nce a i ( E 1 u E2 ) =a i ( E 1 ) u a i ( E 2 ) ; i E I , we have , for c

subadd i t i ve

wh i ch proves that C i s s ubadd i t i ve .

A s s ume now that c i s a capac i t y . Let Kv ' v Eill, be a

decrea s i ng sequence o f compact subset s o f U . We have to prove

that

Choose

00 i nf C ( K ) = C ( n K ) . v E N v v= 1 v

so that

C ( K ) < c ( a ( K ) ) + 1 n n n n

S i nc e ( a i ) i E I i s norma l , a nd compl ete , we c a n a s s ume that

an� a O ' u n i forml y on compact subset s of U . We c l a im that

G i ve n

00 00 00 a o ( n Kn ) :::> n ( u a ( K ) ) .

1 . 1 . n n n= 1 = n= 1

00 00 z E n u a ( K ) ) . . 1 . n n 1 = n = 1

Then

00 z E U 0. ( K )

. n n n = l

- 9 1 -

\:l i E N .

Choose z � E a.n ( � ) ( Kn ( � ) ) whe r e n ( � » � such that z �� z , �� +oo .

Then z l=a.n ( l ) ( x � } where x l E Kn ( � } and we can a s s ume that

00 xQ,�x E n K v ' ��+oo . Now

v= 1

so z =o. O ( x ) and s i nce z was a n a r b i t r a ry e l ement i n

00 00 n u a. ( K } ) we have proved that . 1 . n n 1= n=1

00 00 00 o.o ( n K } :::;) n ( u 0. ( K ) ) .

n= 1 n i = 1 n= i n n

To f i n i s h the proof , we can now a rgue a s i n the end o f the

proo f o f Theorem I X : 5 .

Ronk i n ' s g amma -capa c i ty and Favorov ' s capa c i ty

Denote by c aP 2 and the i nter i or and exter i or

loga r i t hm i c c apac i ty o n � , r e spect i ve l y . i s then de f i ned

as f o l l ows

- 9 2 -

Ronk i n ' s gamma-capa c i t y i s then by de f i n i t i on

r ( E ) =sup { y ( a ( E ) ) ; a complex u n i tary t r a n s f ormat i on } . n n

Propos it ion IX : l . y ( E ) =P ( E ) n n f o r a l l u n i ve r s a l l y capa c i ta b l e

sets E , where Pn i s formed w i th r e spect t o caP 2 .

Proof . I nduct i on . C l ea r l y Pr opo s i t i on I X : 1 i s true for n= l .

A s s ume i t � s true for n - 1 . Prove i t f or n .

A s s ume that E i s u n i ve r s a l l y capa c i t a bl e .

I t i s c l e a r that n - 1 { z E cr ; { z l , z ) E E } � s u n i ve r sa l l y capa c i t a b l e .

Hence

for a l l z l E cr . Then

n- 1 = ( Theorem I X : 6 , 1 ) ) = caP2 { { z l E cr ; cn _ 1 ( { z E cr ; ( z l , z ) E E } » O } ) =

( by d e f i n i t i on )

= ( Remark p . 6 6 )

- 9 3 -

and the propos i t i on i s prove d .

Corollary IX : 3 . I f E i s u n i ve r s a l l y capac i ta b l e then

Corollary IX : 4 . By Corol l ar y I X : 3 , rn

{ E1

) = rn

{ E2

) = 0 i mpl i es

that

Proposi t ion IX : 2 . Eve ry u n i ve r s a l l y capac i ta b l e set i s

r - c apa c i ta bl e . n

Proo f . As s ume that E i s u n i ve r s a l l y capac i table and l et ( > 0

be g i ven . Choos e a comp l ex u n i tary t r an s format i on a such that

r ( E ) < y ( a { E ) ) + ( / 2 . n n

a { E ) i s u n i ve r s a l l y capa c i ta b l e s o by Theorem I X : 6 there i s a

compact s u b s e t K o f a { E ) such that

Thus r { E ) < r ( a- 1

( K » + E and s i nc e a- 1

( K ) is a compact sub-n n

set o f E , the proo f i s comp l e t e .

Favorov ' s capac i ty i s b y d e f i n i t i on

F r

n{ E ) = sup { c

n{ a { E » i a comp l ex u n i tary tran format i on } ,

wher e c i s the l oga r i thm i c capa c i ty on � .

- 9 4 -

Corollary IX : 4 . The a n a l ogy o f Corol l a r y I X : 5 holds t ru e f o r

rF n '

Proof , Fo l lows f rom Theorem I X : 6 and Corol l ary I X : 4 .

Propos i ti on I X : 2 . I f E i s �n -pol a r then r� ( E ) =O .

Proof . The property be i ng �n -polar i s i nvar i ant under comp l ex

u n i tary t r an s f ormat i on s . I t i s t he r e f ore enough to prove that

yn ( E ) = O , A s s ume the propos i t i on to be true for n - 1 . Then by

Theorem I X : 6 we have

where � t - oo i s a plur i subha rmo n i c f u n c t i on s uch that

EC { �=-oo } . But a subharmon i c funct i o n ( t-oo ) on � i s > -00 out s i de a s et o f caP 2= O wh i ch prove s the propo s i t i o n .

n The fol l owi ng examp l e s hows that there are non-� -po l a r

s e t s o f z e r o gamma -cap ac i ty .

Example 2 . Put It i s c le a r that

r n ( H » O . Denote by g the b i holomorph i c map

Then

- 9 5 -

Any complex l i ne cuts g ( H ) i n at most four po i nts so

g Now , by Propos i t i on IX : 2 , H i s not �2 -polar and s i nce

i s b i holomorph i c , g ( H ) cannot be � 2 -polar . Hence , there are non-� 2 -polar subsets of �2 w i th van i sh i ng r 2 -capac ity .

To get a capac i ty wi th null sets i nvar iant under holo-morphi c mappi ngs we proceed as fol l ows .

Let B denote an open and bounded subset of [ . Denote by An the holomorphi c mapp i ngs f , f : Bn�Bn .

Def i ni tion . h ( E ) =sup r� ( f ( E » , n fEA n Theorem IX : 8 .

i ) ECBn ==> h ( E » h ( f ( E » , n - n f EA , n van i shes on n � -polar subsets o f

Proof . i ) follows from the de f i n i t i on of h . n Observe that thi s means that h i s i nvar i ant under biholomorph i c mapp i ngs n of Bn onto i tsel f . i i } As sume that N i s a n � -polar subset o f It fol lows from Propo s i t i on IX : 2 that Let now f EAn be g iven . We have to prove that

r� ( f ( N } ) = O .

Denote by T ( f ) the Jacob ian o f f . I t i s clear. that

- 9 6 -

so by Coro l l a r y I X : 5 i t rema i n s to prove that

rn ( f ( N n { l ( f ) = O } ) ) = O . Th i s fol l ows f rom Coro l l ary IX : 6 be l ow .

Def i n it ion . A s ubset E o f [n i s c a l l ed a ( proper ) l oc a l l y

ana l yt i c s e t i f f o r every w i n E there i s a n e i ghborhood

° of w such that Eno i s a ( proper ) analyt i c s et i n ow . w w

Theorem IX : 9 . Let U be an open subset o f [n and

F= ( f 1 ' • • • , f n ) a holomorph i c map n F : U-+[ . Then F ( { t ( F ) =0 } ) i s conta i n ed i n a denumerable u n i on o f proper loca l ly a n a l yt i c

sets .

Proof . Put J= { Z E U i l ( F ) = O } . We c l a i m that

P a n a l yt i c o f d im<m ==) F ( pnJ )

i s conta i n ed i n a denumerable u n i o n o f proper l oca l l y ana l yt i c

sets .

Th i s i s c le a r l y t rue for m=O , f or an a n a l yt i c set o f

zero d imen s i on i s denumerab l e . Now , i f the s t atement i s t r ue

for m- l we have to prove i t for P wi th dim P=m . We c a n

a s s ume that P n J i s connected .

Choo s e ( a f i P ) a z .

1 lq p= , . . . , s q= 1 , • • • , s

w i th

( a f . J G=de t a z �p t o l q

where s=max rank (::� ) . J P n J . . 1 1 , J = , . . • , n

- 9 7 -

Put Q= ( J n P ) n { G�O } . reg By the r emark i n Remmer t ( see the

note s b e l ow ) , F ( Q ) i s conta i ned i n a denumerable u n i on o f

pr oper l oc a l ly a n a l yt i c s e t s . Q 1 = ( Jn p ) n { G= O } i s o f reg

d i me n s i on <m- 1 so , by a s s umpt i o n , F ( Q ' ) i s conta i ned i n a

denumerab l e u n i o n o f p r ope r l oc a l l y a n a l yt i c s et s .

Fur thermore , s i nc e

for F ( Jn P ) . ) . S i nc e s I ng

d im ( J n P ) . < m- 1 , s l ng-

F ( Jn P ) CF « J n P ) . ) u F ( Q ) u F ( Q ' ) s I ng

the s t a teme n t i s proved .

the same i s true

To p rove the theorem , it i s e nough to choo s e P=U .

Corol lary IX : 6 . I f N i s a: 2 -po l a r i n U , where U i s open

i n a: n and i f F : U-+ a:n i s a n a n a l yt i c map , then F ( N ) i s

n a: - po l a r i n a:n.

Proo f . I t i s c l ea r t ha t F ( Nn h ( F ) �O } ) i s n a: -polar . From

Theorem I X : 9 i t f o l l ows that F ( Nn h ( F ) =O } ) i s n a: - polar ,

s i nc e proper l oc a l ly a n a l yt i c s e t s are n a: -pol a r .

Not e s and r e f er e n c e s

Theorem I X : 2 i s due to F . Tops¢e , On con s t ruct i on o f

mea s ur e s . K¢be nhavn s Un i ve r s i te t , Mat . I ns t . Prepr i nt s er i e s

1 9 7 4 : 2 7 .

Corol l a r y IX : 1 i s due t o G . Choquet , Theory o f capac i t i e s .

Ann . I n s t . Fou r i e r 5 ( 1 9 5 3 - 5 4 ) .

- 9 8 -

The notat i on o f swarm i s c l os e l y r e l ated to that o f � noyau

capac i t a i re r egul i e r � as de f i ned i n C . Dellacher i e , Ensemb l e s

a n a l y t i que s . Capac i te s . Mes u r e s d e Hausdor f f . S pr i nger LNM . 2 9 5

( 1 9 7 2 ) .

v

Examp l e 1 i s due to V . Sei now ( s ee Ronk i n s book b e l ow ) .

Example 2 i s due to C . O . Ki selman . Man u s c r i pt . Upp s a l a 1 9 7 3 .

The gamma capa c i ty was i nt r oduced i n L . I . Rank i n , I n t ro

duct i on to the theory of ent i r e f u nct i o n s of s eve r a l var i a b l e s .

Amer . Math . Soc . Prov i dence . R . I . 1 9 7 4 , a nd the mod i f i ed g amma

capac i ty i s i n S . Ju . Favorov , On capa c i ty character i za t i o n s o f

sets i n �n . Charkov 1 9 7 4 ( Ru s s i an ) . The r ema rk by Remme r t ,

u sed i n the proof o f Theorem I X : 9 i s i n R . Remmert , Hol omorphe

und mer omorphe Abb i l dungen komp l ex e r Raume . Math . Ann . 1 3 3

( 1 9 5 7 ) .

The set funct i on s Y n a nd r n h a s been used i n connec

t i on w i th removable s i ngul a r i ty s et s ; c f . U . Cegrell , Removab l e

s i ng u l a r i ty s e t s f o r a n a l yt i c f un c t i o n s hav i ng modu l u s w i t h

bounded Lapl ace mas s . P roc . Ame r . Mat h . Soc . Vo l . 8 8 ( 1 9 8 3 ) .

P . Jarvi , Remov a b l e s i ng ul a r i t i e s for HP- funct i on s .

P roc . Ame r . Math . Soc . Vol . 8 6 ( 1 9 8 2 ) .

J . Ri i hentaus , An exten s i on theorem for meromorph i c func

t i ons o f seve r a l va r i ab l e s . Ann . Acad . Sc . Fenn . S e r e AI .

Vo l . 4 ( 1 9 7 8 / 7 9 ) .

Some o f the mater i a l o f th i s sect i o n has been publ i shed

in s emi na i r e P i er r e Le l ong Hen r i Skoda ( An a l ys e ) 1 9 7 8 / 7 9 .

Spr i nger LNM 8 2 2 . 1 9 8 0 .

X Capacities on the Boundary

Let U be an ope n , bounded and connected s ubset of � n

cont a i n i ng z e ro . The n H ( U ) ( Hoo

( U ) ) i s the c l a s s o f ( bounded )

a n a l yt i c f u n c t ions on U a n d A ( U ) con s i s t s o f the funct i on s

i n H ( U ) that extends cont i nuou s l y t o U . I f w i s a pos i

t i ve mea sure o n a u we de f i ne HP ( W , a U ) ( 1 �p< +00 ) t o be the

c l o s u r e of A ( U ) i n LP ( w , a U ) .

and

For z EU we d e f i ne the c l a s s e s o f probab i l i ty measures

M = { w > O ; f { z ) = f fdW , � f E A ( U ) , s upp w c a u } z -

( PS H and C are t he p l u r i su bharmon i c f u nc t i on s a nd the

con t i nuous funct i o n s respec t i ve l y . )

We wr i te B for the u n i t ba l l i n �n and a i s the

norma l i z ed Lebesgue me asure on a B o Then f E HP ( a , a B ) i f and

on l y i f

whe r e

s u p f l c [ r z , � ] f ( � ) da ( � ) I Pda ( z ) < +00 O < r < 1

I S t h e Cauchy kerne l . Hence , each

f E H1

( a , a B ) extends to H ( B ) .

The space LH 1 cons i s t s o f the fun c t i on s f i n H ( U )

- 1 0 0 -

such that I f I has a plur i s u bharmon i c ma j orant . A n orm on

LH 1

( U ) i s

I l f l l 1 = i n f { h ( 0 ) ; hE PH ( U ) , I f I �h } LH

whe r e PH denotes the p l u r i s ubha rmon i c f u n ct i on s . Th i s norm i s

c a l l ed the Lume r norm .

For each po i nt z EU we d e f i n e the capa c i t i e s

Q ( E ) =sup u t E ) z u EMz

R ( E ) = s up u t E ) . z E N \J z

Observe that R < Q a nd that R c a n van i sh at s e t s o f pos i t i ve

Q-capac i ty . See examp l e be l ow . ( We wr i t e R=R o and Q=Qo ' )

We now wi sh to study Q and there fore extend i t to a funct i onal . Put P=Re A ( U ) a nd l e t P be the c l osure o f P

i n 1 Ll ( U ) . oc

Let � be a r e a l -va l ued f u nc t i on on a u and put for z EU

� ( z ) = s up f*�dU .

u EM z

Theorem X : l . I f � i s upper s emi - cont i n uous on a u , then

� ( z ) = i n f { h ( z ) ; hE P , h�qJ on a u } , z E U .

Proof . The proof o f Lemma 1 1 1 : 3 appl i e s .

- 1 0 1 -

Lemma X : 1 . I f O�cp i s upp e r s em i -con t i nuous , then -cp i s

con t i nuous a n d p l u r i s ubharmon i c on u .

Proo f . By Theorem X : 1

- cp ( z ) = s up { -h ( z ) ; h E P , h�cp } =s up { h ( z ) ; hE P , h�- cp } ,

so i t f o l l ows ( - cp ) * ( z ) = l i m - cp ( z ' ) i s p l u r i s ubharmon i c ; i t A z ' � z

r ema i n s t o prove that -cp i s cont i nuou s , and the next propos i -

t i on shows t h i s .

Propos i tion X : l . Let ( h i ) i E I be a f am i l y of ha rmon i c

funct i on s o n U ,

cont i n uous .

u n i formly bounded a bove . Then s up h . i E I 1

i s

Proof . Wi thout l o s s o f g e n e r a l i ty , we c a n a s sume that a l l

h . , i E I , a r e negat i ve . 1

Put H = s u p h . . i E I 1 S i nc e each h . 1 i s con t i nuous i t i s c l ear

that H i s l owe r s emi -con t i nuous so we have to prove that H

a l so i s upper s em i -con t i nuous . Let Z o be g i ve n . I f H i s not

upper s em i -co n t i nuous at Z o there i s a sequence

w i th l i m i t

c a n choose h v w i th

Take r O > O s o that B ( z O , r O )

put

for some E > O . We

i s rel a t i ve l y compact i n U a nd

- 1 0 2 -

We then have l im r\) = r O and B ( z v , r\) ) , vETIiJ, are relat i ve l y v-++oo

compact i n U for v l arge enough . S i nce

we have

l im h ) z O ) = l im m ( B ( Z� , r o ) ) J h ) z ) dz = v-++oo

= 1 i m v-++oo

> l im v-++oo

= l im v-++oo

v-++oo

m ( B ( z , r ) ) v v m ( B ( z O , r O ) )

m ( B ( z , r ) ) v v m ( b ( z O , r O ) )

m ( B ( z v , rv ) ) m ( B ( z O , r O ) )

B ( z O , r O )

m ( B ( z1

, r ) ) J v v B ( z O , r O )

m ( B ( z v , r v ) ) J B ( z , r ) v v

h ( z » H ( Z O ) + E: . \) v -

Hence , there i s a v s o that

hv ( z ) dz

hv ( z ) d z

wh i ch i s a contrad i ct i on and the theorem i s proved .

>

=

Example . Let E= { ( z , 0 ) Ecr:2 ; I z 1 = 1 } C S= Cl B where B i s the u n i t

bal l lO n � 2 . Then R ( E ) 0 '" ( ) = , Z , w � Otw ( s i nce E i s pol a r ) .

On the other hand , by Theor em X : l and Lemma X : l , Q ( ) ( E » O , Z , w � ( z , w ) EB .

Lemma X : 2 . Let ( In ) be a sequence o f negat i ve subharmon i c 'f' i i E I f unct i ons o n U . I f there i s a po i nt z O E U such that

i n f -00 i E I l

- 1 0 3 -

t h e n t h e r e i s a s u bh a rmon i c f un c t i on

00

a n d a

s u b s equ e n c e ( (Il . ) . 1 't' l . J = s o t h a t F u r t he rmor e , i f the J

f u n c t i on s <P i ' i E I , a r e h a r mon i c t h e n h i s ha rmon i c and t h e

c o n v e rg e n c e i s u n i f orm .

Proof . S i n c e U i s a s s umed to be c o n n e c t e d , the c o nd i t i on

i n f h . ( z 0 ) > -00 i E I 1

i n f f i E I K

i mp l i e s t h a t

h . ( z ) d z > - 00 l

f o r e v e r y compact s ub s e t K o f U . H e n c e ( h i)

i E I c o n t a i n s a

wea k l y c o n ve r ge n t s ub s eque n c e 00

( h . )

. l ' 1 . J = J

I f t h e n

l i m j-+ +oo

s o the weak l i m i t i s a s ubha rmon i c f u n ct i on .

Th i s c omp l e t e s t h e proo f , b e c a u s e i t i s a we l l - known f a c t that

a weak l y c o n v e r g e n t sequence of h a rmo n i c f u n c t i on s c o n v e r g e s

u n i f o r m l y o n compac t s .

Lemma X : 3 . I f <p= s up <P whe r e v E JN v

s em i co n t i n u o u s on a u a n d i f

a n d p l u r i s ubha r mo n i c o n U .

Proo f . By Lemma X : 1 ,

t i n uo u s f o r e v e r y m . Moreover

<P v are n o n - n eg a t i ve a nd upper

<p ( 0 ) < +00 then - <p i 5 con t i nuous

i s p l u r i s u bharmon i c and con-

- � ( z ) = - 5 up f 5 up <p d w � - <p , m-+ +OO l < v <m v W E Mz

l < v <rn v

5 0 s i nc e - <p ( 0 » -00 , -<p i s p l u r i s ubha rmon i c on U .

- 1 0 4 -

I t rema i n s to prove con t i nu i t y . We put A= { hEP ; h�-� }

and c l a im that A I S non -empty a nd

s up h = -� hEA

wh i ch w i l l prove the cont i nu i ty , by Propo s i t i on X : l .

By de f i n i t i on , s up h < - � a nd Lemma X : 1 prov i de s h E A

----- �

u s w i t h

1 a sequence h E P so t hat hm < - s up �v and - - -� ( o ) < hm ( 0 ) . m l < v <m m

Lemma X : 2 shows that we can s e l ec t a subsequence 00

( h ) . l ' m . J = J

converg i ng un i f ormly on compact s u b s e t s . I t i s c lear that t h i s

l im i t h O be longs t o A a n d that h O ( O ) = -� ( O ) . Thus we have

shown that A i s non -empty and that - � > -oo everywhere . So we

can repeat the a bove argument for any po i nt i n U .

Theorem X : 2 . Let � be c non-negat i ve a nd u n i ve r s a l l y mea s u r -

able f unct ion o n a u . I f � ( O ) < +00 then there i s an i nc r ea s i ng

sequence o f upper sem i -cont i nuous funct i ons �v ' v E lli , such that

� < � v - and l im v-+ +00

and con t i nuous on U .

Fu r the rmore , - � I S p l u r i s u bharmon i c

Proof . S i nce � i s u n i ve r s a l l y mea s u r a b l e we have

cp ( z ) =sup \.l E M z

fCP ( � ) d\.l ( E� ) = s up sup g�� \.l E M z

where g va r i es a�ong the upper semi -cont i nuous funct i on s .

Hence ,

-cp ( z ) =- sup g ( z ) = i n f - g ( z ) g�cp g��

- 1 0 5 -

so -� i s upper semi -con t i nuous by Lemma X : l .

By Choquets Lemma there i s a denumerable subset 00 ( g . ) . 1 J J =

o f { g�� } so that i f 1 i s lower semi -conti nuous and i f

l < i n f -g . then 1 _< -� . Now - . J J

-sup g . ( z ) = - sup f sup g . d 11 � - sup sup fg . d 11 < j J I1EM j J I1EM j J

z z

< i n f - sup f g j dl1 j I1 EM z

= i n f-g . .

j J

By Lemma X : 3 the l e ft-hand s ide i s a conti nuous funct i on so i t

fol l ows that

--------sup g . ( z ) < -� ( z ) .

J -j

But on the other hand ,

�. - � = - s up g . . J

� sup g . < � so s up g . < � and there fore J - J -j

I f we take �m= sup g . we get a sequence of funct i ons with l < v<m J

the r equ i red propert ies .

Corol l ary X : l . I f � j " a non-nega t i ve and un iversally mea sur-

able funct i on on au then � ( z ) = i n f { h ( z ) i hEl' , � � h } .

Proo f . Use Theorem X : 2 to f i nd an i ncreas i ng sequence

of upper sem i - cont i nuous funct ions such that � < � and v -l im � v = � . Appl y now the argument i n the end o f the proof o f v-++oo Lemma X : 3 .

- 1 0 6 -

Theorem X : 3 . Let E be a F -set i n a a u o f van i sh i ng Q-capa-

c i ty . Then there . i s a sequence ( f ) oo of funct i ons in A ( U ) v v= l w i th I f v l � 2 s uch that

1 )

2 )

3 )

l im

l im v-++oo

l im V-+ +OO

f ) z ) = O ,

f ( � ) = 1 v

z E U ,

out s i de a set o f van i sh i ng

on E .

Q-capa c i ty ,

Proof . We know that E= u K v v= l where is an i nc r ea s i ng

un i on o f compac t set s . By Lemma X : 1 , we can choose g E A ( U ) , v 1m g ( 0 ) =0 , v h =Re g v v such that h > X K ; v - h ( 0 ) < 1 I v 3 . Put v

- vg P =e v

v Then I P 1 < 1 v -

l im P v ( z ) = l im v-++oo

v s i nce h > 0 and i f z E E we get v -

- v h ( z ) e v = O .

Furthermore ,

- vh ( 0 ) O < j ( l -Re P ) d].l = l -Re P ( O ) = l -e v < 1 / v 2

- v v

for every ].l EMO and every vE N . I t fol l ows that

l im Re P ( z ) = l a . e . v ( ].l ) s o l im Pv ( z ) = l a . e . v-++oo

( ].l ) s i nce V-++OO

l im V-++OO

S i nce ].l was any measure i n MO i t fol l ows that

P = 1 v out s i de a set o f Q-capa c i ty z ero . I f we put f = l -P v v we get a s equence o f funct i on s w i th property 2 ) and 3 ) and s i nce

l -Re P > 0 ; l - l im Re P ( 0 ) =0 , Lemma X : 2 prove s that v - v v-++oo

- 1 0 7 -

l - l im Re P = 0 on U . Aga i n , s i nce I p 1 < 1 , v v -l im f ( z ) = l im 1 - P ( z ) = 0 v V on U , wh i ch completes the proof v-++co v-++ao o f Theor em X : 3 .

Theorem X : 4 . Let � be a non-neg at i ve subharmo n i c funct i on

on the un i t bal l . Then � has a p l ur i harmonic ma j orant i f and

o n l y i f

s up sup f � ( r � ) d � ( � ) < +co . O < r < l \.l EMO

Proof . I f � � hE PH then � ( r U � h ( r U so

s up � ( rg ) ( 0 ) � h ( a ) •

O < r < l

On the other hand , i f � i s subharmon i c then � ( r � ) l a s i s

upper semi con t i nuous . Hence , by Lemma X : 1 ,

so choose

� � ( r � ) ( z ) = i n f { h ( z ) i h ERe A ( B ) , � ( r U � h ( U }

w i th sup O < r < l

h ( 0 ) < +co . r

co Appl y Lemma X : 2 a nd choose a subsequence ( hr . ) j = l con-J

verg i ng un i formly on compact subsets o f B to h . S i nce � i s s ubharmon i c ,

on B s o

� ( r . z ) - h ( z ) < O J r . -

J

r Z o Z o ] � ( z O ) - h ( z O ) = � im I � ( r . - ) -h ( - ) < 0 , 'tj z E B . J-++co - J r j r j r j - 0

- 1 0 8 -

For the rest o f th i s sect i on we rest r i ct ourselves to the un i t

bal l al though some o f the r e s u l t s are true on mor e general

doma i n s .

Reca l l that LH 1 ( B ) i s the Banach space o f ana l yt i c

funct i ons with modul u s hav i ng a p l ur i subharmon i c ma j orant .

The norm on LH 1 i s

I l f l l 1 :;;; i n f { h ( 0 ) ; h E PH ( B ) ; I f l.s. h } . LH

Observe that Theorem X : 4 shows that f E H ( B ) i s i n LH 1 i f and

on ly i f s up I f ( r U I ( 0 ) < +00 . O < r < 1

Theorem X : 5 . I f f E LH 1 then

� I l f l l 1 :;;; l im I f ( r U I ( O ) .

LH r-+ 1

Proo f . I f h E PH with h � l f l then h ( r � ) � l f ( r � ) 1 so .----.

h ( O ) � l f ( r � ) I ( O ) . Hence

wh i c h g i ves that

----------I l f l l 1 � l i� I f ( r U I ( 0 ) .

LH r-+ 1

On the other hand ,

-----------l im I f ( r f,; ) I ( z ) > fQ ( z , U 1 i m I f ( r U I d lJ ( U > I f ( z ) I r-+ 1 r-+ 1

- 1 0 9 -

� so Coro l lary X : 1 proves that 1 i m 1 f ( r t;; ) 1 ( 0 ) � I I f II 1 wh i ch

r� l LH completes the proo f .

I n one var i abl e , the func t i on s i n A ( B ) are den s e I n H l

( take d i l atat i on s ) .

Example . We are goi ng to construct a bounded analyt i c funct i on

f on Bc� 2 such that

1 ) l im f ( r t;; ) ex i sts �t;; E S .

2 ) f ( r t;; ) do not converge to f i n LH 1 .

Let t;;K = (� , /1 - �2 ) , KEJN. Then i G { e t;; K ' GEm} are closed

and d i s j o i nt sets so we can choose open d i s jo i nt sets VK i n

B where Vk conta i ns i G { e �K ' GElR} .

For each K choose nK so that

co nK and put f ( z ) = L < z , t;;K > K= l

The n I f ( z ) 1 � 1 + € s i nce z EVK for

at most one K .

t h e n Let 0 < r 1 < 1

0 < r 2 < 1 w i t h be g ive n , choose K

n K 1 r 2 > 2" s uc h t h a t

For wE � ; I w l = l we now have

so that nK r 1 < 1 /4 and

i f 1 € = 16

- 1 1 0 -

Therefore sup f l f ( r 1 � ) - f ( r 2 � ) I d� � � s i nce we can take � � EMO

to be the norma l i zed Lebesgue measure on i 8 { e �K ' 8t= :ffi } . Th i s

prove s that f has property 2 ) .

To prove 1 } take � E S f i x , then

with un i form convergence , s i nce � E VK for at most one K .

However , we have the f o l l ow i ng character i z at i on o f ALH 1 ,

the funct i ons i n the c l osure o f A ( B ) with respect to the

Lumer norm .

Theorem X : 6 . Assume that f E H ( B } . Then f E ALH 1 ( B ) i f a nd

only i f

and

where

Ref \) \) = l: <P 1 - L <P 2 \)

1m f = L \jJV - L l./J \) 1 2 v

v \) cp� , \jJ� , v E JN , � = 1 , 2 ,

are non-po s i t i ve and p l u r i harmon i c on B , conti nuous up to the

boundary .

Proof . � ) Choose

choose t . so that J

f E A ( B ) n

I I f - f i l l < 1 / j 2 , m n LH

so that

m , n) t . , - J

I l f - f l l 1 -+ 0 , n LH n-++oo , and

and put

Then

- 1 1 1 -

I I f t - f t II 1 < 1 2 s s- l LH ( s - l )

so we can choose plur i harmon i c ma j orants 1 h s ( O ) < - 2 · ( s - l )

and

I f we de f i ne

v v <il l = Re ( ft - f t ) -h " , <il = -h , v v - ' v 2 v

h with s

we get f unct i on s w i th the requ i red propert i e s .

G i ve n

c ) As s ume now t h a t f E H ( B ) ha s the r epresentat i on

f ( z ) = l: <il� - l: <il� + i ( l: 1\!� - l: 1\!� ) .

( > 0

- (

v v v v

choos e N s o t hat

00 l:

v=N (

(

v v v v <il , + <il2 + 1\! , + 1\! 2 ) ( 0 ) < ( •

Then

- 1 1 2 -

N N I f ( r � ) - f ( s � ) I V I v v v v - 1jJ2 ( s � ) ) -

V�N( � l ( rt'; ) +� l ( s t,; ) +� 2 ( r � ) +� 1 ( s U +

E

The f i r st term at the r i ght-hand s i de converges u n i formly to

zero when r , s � l . The second sum is a p l u r i harmon i c funct i on

whi ch i s l e s s than 2 E at z e r o . Hence ( f ( r � ) ) a < r < l i s a

Cauchy s equence i n LH 1 and we have proved that f ( r � ) tends

to f in LH 1 and the proo f i s comp l et e .

Remark . By Theorem X : 6 we can to every f E ALH ( B ) de f i ne i t s 00

boundar y va l ues : The term i n the ser i e s v E � 1 are non-po s i -v= l

t i ve and cont i nuous up to the boundary . I f ).l EMa then

v so E � 1 ( � ) > -00 out s i de a set ( on a B ) o f van i sh i ng Q-capa c i ty .

So we put

f * ( z } t: E a B ,

- 1 1 3 -

we get a funct i on def i ned out s i de a set o f van i sh i ng Q-capac i ty . Furthermore , l im sup f l f * ( � ) - f ( r � ) I dw=O .

r-+ 1 \.lEMO We now use th i s last property to prove a result rel ated to

i nner funct ions . Observe that we do not assume I f I to be bounded .

Theorem X : 7 . ( n > 2 ) Assume that f EALH 1 ( B ) and that f I f * I dw= 1 , tI \.l EMO · Then f ::: constant .

Proof . I f \.l = a we get that

fQ ( Z ' � ) I f * ( S l l da ( � ) 1 0 = 1

where Q i s the clas s i ca l Po i s son kerne l . To prove that f ::: const . i t i s enough to prove that I f ( 0 ) I = 1 because t h a t

forces I f I to be harmon i c a nd therefore constant .

Choose for O < r < l a measure \.l rEMO supported on

We have

{ � E a B ; f ( r � ) = f ( O ) } .

I f ( o ) 1 "'" l im f l f ( r � ) l d JJ r ( S l > l im f l f * l dWr -r-+ 1 r-+ 1

l im f l f * ( S l - f ( rU l d\.l r ( � ) = r-+ 1

s i nce the last term van i shes by remark fol lowing Theorem X : 6 .

- 1 1 4 -

We now return to MO ; we have seen that to every funct i on fEALH 1 there i s a " boundary va lue funct i on " f * so that 1 ) f ( O ) = Jf *d\.l , 'd \.l EMO

2 ) l im sup J l f * ( � ) - f ( rU l d\.l=o . r/1 1 I.l EMO

The above example shows that 2 ) need not hold i f ALH l i s <Xl repl aced by H ( B ) . We do not know i f 1 ) i s true for <Xl H ( B ) .

But f * i s wel l de f i ned for every f i xed \.lEMa !

A complex measure on S i s cal led an A-measure i f for every sequence , l im f . ( z ) =O tl z E B ]

L EA ( B ) , I f . ( z ) I <M , 'd j EJN , 'd z E B ] J -

i t fol lows that l im f f . d\.l=O . . J

with

'rheorem X : 8 . I f \.l i s an A-measure and i f f . EA ( B ) , ] I f J. ( z ) I �M , 'd z EB , j EJN such that l im f . ( z ) 3 t1 z E B then f . d \.l i s

j-++<Xl J J weakly convergent ( i . e . l imJ� f jd\.l 3 t1�ECO ( S ) ) .

In par t i cular , i t fol l ows from Theorem X : 8 that i f \.lEMO ' <Xl f E H ( B ) then f ( r� ) d\.l ( � ) I S weakly convergent so there I S a

funct i on ( determi ned <Xl I.l-a . e . ) f * E L ( S ) such that

Theorem X : 9 . Let I.l be a probabi l i ty-measure on S such that there i s a constant c w i th

sup f l f ( rU I 2d\.l ( � ) � cJ l f ( t;; ) 1 2d\.l ( t;; ) , tl fEA ( B ) . O < r < l

I f <Xl

f E H ( B ) then ( f ( rU ) O < r < l converges i n 2 L ( I.l , S ) , r """ 1 .

- 1 1 5 -


Theorem X : 3 i s due t o F . Forel l i , Analyt i c measures . Pac i f i c J . Math . 1 3 ( 1 6 3 ) .

Theorem X : 4 i s a genera l i z a t i on o f a theorem o f G . Lumer ,

Espaces de Hardy en plus i eurs var i ables complexes . C . R . A . S . Pa r i s 2 7 3 ( 1 9 7 1 ) .

The example i s a s i mpl i f i ca t i on o f an example o f W . Rudin ,

Func t i on theory i n the un it ba l l o f �n . Spr i nger Verlag 1 9 8 0 , pg . 1 5 0 .

I t has been proved by W . Rudi n i n " New construct i ons o f funct i on s hol omorph i c i n the u n i t bal l o f �n " , AMS reg i onal con ference ser i e s in mathema t i c s No . 63 on page 64 that ALH 1 ( B ) conta i ns no i nner funct i ons .

Theorem X : 8 i s due to G . M . Henk i n , Banach spaces o f anal yt i c f unct i ons on the ba l l and on the b icyl i nder are not i somorph i c . Funct i onal Ana l . Appl . 2 ( 1 9 6 8 ) .

Theorem X : 9 i s proved by U . Cegr e l l i n On the strong convergence of d i l a ta t i ons of bounded anal yt i c f unct i on s i n the un i t ba l l of �n . Manusc r ipt . Toul ouse . 1 9 8 6 .

For more references , see U . Cegrel l , Sma l l sets i n �n , Ann . Pol . Math . ( 1 9 8 5 ) .

XI Szego Kernels

We keep the notat i on from Sect i on x .

1 . Approx imat ion o f the ident i ty

Lemma XI : l . Let U be a pos i t i ve measure on a � and l et 00 ( P i ( z ' � ) ) i = l ' ( z , U E a � x a � be a fami l y o f funct i ons i n

L 1 ( U®U , a � x a � ) such that

2 ) P . > O , 1- i E JN

b ) sup fP i ( Z , � ) dU ( Z ) =M<+oo i , �

Then fP i ( Z , � ) � ( � ) d U ( � ) tends to � i n LP ( U , a � ) , i�+oo for every � E LP ( U , a n ) .

Proof . Assume f i rst that � i s cont i nuous . Then

and s i nce

l im i-++oo

SUI? l fp i ( z , � ) � ( U d u ( f; ) I � sup l � ( � ) I Z , l �

by 2 ) the lemma fol lows by domi nated convergence . I f �E LP and i f ( > 0 i s g iven , choose � E E C ( a � ) with f l �-�E I PdU « / ( M+ 1 ) . Then

- 1 1 7 -

+ ( f I fp . ( z , U ( cp ( � ) - cp ( z » d � ( � ) I p d � ( z ) ) . l £ £

Fub i n i ' s theorem and property 1 ) show that the f i rst two i ntegra ls are sma l l er than £ and the last i ntegral tends to zero when i�+oo by t he f i r st part of the proo f . S i nce £ > 0

was arbi trary , the lemma fol lows .

Lemma XI : 2 . 00 Let ( P i ( z , � ) ) i = 1

be a fami ly of funct i ons as i n Lemma X I : 1 . Assume that 00

( ,I , . ) . 1 't' l l == l S a sequence o f funct i ons such that l\J . dw � I\Jd� where l

then I\J . l tends to I\J i n 1 L ( w , a � ) .

1 I\JEL ( � , a � ) .

1 L ( � , a � ) -

I f

Proo f . f l l\J ( z ) -l\J i ( z ) I d � ( z ) 2.. f l l\J ( z ) -fP i ( z , U I\J ( U d� ( U I d� ( z ) + + f ( f p i ( z , � ) I\J ( � ) d� ( � ) -l\J i ( Z » d� ( Z ) .

By Lemma X I : l , the f i r s t i ntegral tends to zero when i�+oo . By weak convergence , we have the same conclus ion for the second i ntegra l .

- 1 1 8 -

2 . s z ego and Poi sson kernels

Let � be a pos i t i ve measure on a n such that to every

compact subset K of n there i s a constant CK such that

( * ) s up z E K

I f ( z ) I < CK ( f I f I 2

d � ) 1 / 2 ,

a n f EA ( n ) .

Choose an ex> ON-system ( e v ) V= l ' e E A ( n )

I n A ( n )

ON-ba s i s

v ( and i n

2 Comp l ete so H ( � i a n » .

ex> ex> ( e V) v= l u ( dv ) v= l i n 2 L ( � , a n ) .

then has a unique representa t i on

For mE 1N ,

s o b y ( * )

g ( z ) = � fg ( � ) e ( � ) d � ( � ) e ( z ) + v=O v v

CD + 1: fg ( � ) d ( � ) d � ( � ) d ( z ) , z E a n .

v=o v v

sup z E K

o f

the

Every

funct i on s dense

system to an 2 gE L ( � , a n )

m 1 , m 2 -+ +CD •

CD Thus 1: fge e converges u n i form l y on every compact subset

v=O v v

o f n and there fore represents an analyt i c funct i on there .

I f f EA ( n ) , we can app l y the s ame a rgument on

- 1 1 9 -

m f - L ffe due to conclude that

v=O v v

f ( z ) CX)

= L ffe du e ( z ) , 1 v v z E Q ,

wi th u n i f orm convergence on c ompact subsets o f Q .

n nd

I f we con s i der the mapp i ng s

2 p CX) L ( u ) 3g 1-+ l:

V==O

z E Q ,

i t i s c l ea r that they are cont i nuous so the i r compo s i t i on

T ==V oP z z 1 S g i ven by a n e l ement i n

2 g E L ( u ) .

2 L ( u ) :

I t 1 S easy to see that

T ( � ) z

CX) L ev ( z ) ev ( � ) ' � E a Q

v= l

and we have seen that T extends to an analyt i c funct i on on Q , z

T ( w ) = z

CX) L e ( z ) e ( w ) ,

v= l v v

Now , a c r uc i al property o f

wEQ .

on i s : . Are the

l i near l y i ndependent as analyt i c funct i ons on Q?

e . S v ·

I n other words : I f

� must a = 0 , 't1 v f JN ? v .

- 1 20 -

Theorem XI : l . Let � be a pos i t i ve measure sat i s fy i ng proper-

ty ( * ) . The fol low i ng statements are equi valent .

1 )

2 ) I f f v EA ( � ) , f l f) 2d� � 1 , v E JN ,

then f vd � z,. O .

There ex i sts an ON-bas i s

and i f l im f v ( z ) = O , 't1 v E � , v-++oo

f llnct i on s i n A { � ) , l i nearly i ndependent on � .

i Proof . 1 ) => 2 ) . I f 2 ) i s f a l s e , put f . ( z ) = E a e ( z ) .

1 v = O v v

2 ) => 1 ) . Assume that

l im f v { z ) = O , z E � . V-++OO

f . EA ( � ) , I l f . 1 2d� < 1 1 J - and that

Select a weakly convergent subsequence ( wh i ch we aga i n

denote by ( f . ) wi th l im i t f E H 2 { � , d � ) . Now 1

f . { Z ) = f f . ( U 1 ( t;; ) d� ( t;; ) s o l I Z

By assumpt i on , ffevd � = O , 't1 z E JN , wh i ch shows that the weak l i m i t 00

o f ( E . ) . 1 i s zero and the theorem i s proved . 1 1 =

Corollary XI : l . Every pos i t i ve measure on d B wh i ch sat i s f i e s

( * ) and possesses a bas i s for H2 s o that

- 1 2 1 -

has the propert i es i n the t heorem .

Def i n it ion . The Szeg6 ker nel ( re l at i ve l y w and n ) i s

00 S ( z , � ) = L:

v=O e ( z ) e ( � ) , z E n , s E a n , v v

a nd the Poi s son kernel i s

P ( z , � ) = I s ( z , � ) 1 2 S ( z , z ) z E n , s E a n .

I t i s c l ear that P ( z , s ) � O . I f f EA ( n ) , then

so

f ( z ) = f S ( z , s ) f ( s ) S ( z , � ) dw ( � ) 8 ( z , z )

Note that s i nce P i s real ,

Re f ( z ) = fp ( Z , � ) Re f ( � ) d J..t{ s ) , fE A ( Q ) .

Theorem XI : 2 . As sume that the Choquet bo�ndary o f n rela-

t i ve l y A ( n ) equal s a n a nd let w be a pos i t i ve mea sure on

an hav i ng property ( * ) . Furthermore , a s s ume that there i s a

f am i ly 00

( F . ) . 1 � � = o f ana l yt i c mapp i ng s i nto n ,

de f i ned near TI such that l im F . ( z ) = z , � z ETI . . �

I f

s up f i EJN llE a Q

� .... +oo

2 I S ( F . ( z ) , l1 ) 1 � d w ( z ) < +00

each o f them

- 1 2 2 -

where S i s the S z eg6 ker nel relat i ve l y a � then � has the

fol l owi ng property . For eve ry s equence f EA ( � ) , I f I < 1 s s -l i m f s ( z } = O , � z E � i t fol l ows that f d� z:,. 0 , s-++oo . S s-++oo

wi th

Moreover , i f 00 f E H then 00 ( f ( F i ( � ) ) ) i = 1 converges i n 1 L ( w , a � ) .

Proof . Put P . ( z , � ) =P ( F . ( z ) , � ) where P i s the Po i s son 1 1

kernel relat i ve a � . We f i r st prove that 00 ( P . ) . 1 1 1 = i s an

approx i mat i on o f the i dent i ty in the sense o f Lemma 1 .

F i rst , each P . 1 i s i ntegrable on S i nce

f ( F . ( Z ) ) = fp . ( Z , U f ( U d � ( U , � f EA ( � ) , � z ETI , 1 1

i t i s cl ear that P . ( z , � ) d w ( � ) � 8 , i-++oo for every z E a � . 1 z

( Th i s i s so because every po i nt i n a � i s i n the Choquet

boundary . ) Thus 3 ) of Lemma X I : 1 i s va l i d . It is t r i v i a l that

2 ) and the f i rst part o f 1 ) hold . We a l so have

Hence

I S ( F . ( z ) , � ) 1 2 1 = fp l' ( Z , U d � ( � ) = f 1 d w ( � ) • S ( F . ( z ) , F . ( z ) ) 1 1

S ( F . ( z ) , F . ( z ) ) = f I S ( F . ( Z ) , n I 2 d � ( n ) 1 1 1

s o

- 1 2 3 -

sup fp · ( z , � ) d W ( Z ) =SUP f i ElN

1 i ElN

� E a � � E a �

= SUp f i E lN � E a �

2 I S ( F . ( z ) , � ) 1 1

2 I S ( F . ( z ) , � ) 1 dW ( z ) 1 S ( F . ( z ) , F . ( z ) ) 1 1

d W ( z ) < + 00

by a s sumpt ion s o the l a s t part o f 1 ) holds true .

=

Now , l et fS

E A ( � ) , I fS I � 1 be a g iven sequence w i th

l im f s ( z ) =O , � z E � . I f s-++oo

f dw does not converge weakl y to z ero , s

then we c a n s e l ect a subsequence ( wh i ch we aga i n denote by

f d IJ ) s such that f sd W � fdlJ where

We now w i sh to prove that

00 o t f E L .

l im fP i ( Z , U f s ( U d W ( � ) = fP i ( z , � ) f ( � ) dW ( � ) ' � z E � , s-++oo

because then

fP i ( z , � ) f ( � ) d IJ ( � ) = l im f s ( F i ( z ) ) = 0 , � z E � , s-++oo

by a s s umpt i on . On the other hand , we have shown that Lemma X I : 1 app l i es s o f = O a . e . ( IJ ) wh i ch i s a contrad i ct i on and the

f i r s t part of �he theorem wou l d be proved .

So f i x i Em a nd z E � and con s i de r

j S . ( F . ( z ) , � ) = L e ( F . ( z ) ) e ( U . J 1 v=O V 1 V

- 1 2 4 -

G iven ( > 0 choose j so that

I I s . ( F . ( z ) , � ) -S ( F . ( z ) , � ) 1 2d\.J( U < ( S ( F . ( z ) , F . ( z ) ) J 1 1 1 1

and then s so that

Then

I I l s . ( F . ( Z ) , � ) 1 2 ( f ( U - f ( � ) ) d \l ( U I < E: S ( F . ( z ) , F . ( z ) ) . J 1 S 1 1

( I S ( F . ( z ) , � ) 1 2

< I I S ( F . �Z ) , F . ( Z ) ) 1 1

2 I S . ( F . ( z ) , � ) 1 ) J 1 S ( F . ( z ) , F . ( z ) ) ( f ( � )

1 1

2 I l s . ( F . ( Z ) , U I

- f s ( U ) d\l ( � ) I + I J 1 ( f P'; ) - f s ( � ) ) d\l ( � ) I < S ( F . ( z ) , F . ( z ) ) 1 1

2 I I S . ( F . ( z ) , U - S ( F . ( z ) , � ) I < 2 J l S ( F . ( Z ) , F . � Z ) ) d \l ( � ) + € < 3 E: 1 1

and the proof of the f i rst part of the theorem i s complete .

It rema i ns to prove the l a st statement . I f f EHm then f . =f ( F . ( z ) ) i s a uni formly bounded sequence i n A ( Q ) . We can 1 1 f i nd a funct i on m

f E L ( \l , a Q ) and a sequence f . d\l � gd\l and by the proo f above 1 . J

m ( f . ) . 1 1 . J = ]

l im f p . ( z , � ) f ( F . ( � ) ) d \l ( U = I p . ( Z , � ) 9 ( � ) d \l ( U . i-++m J 1 J

so that

- 1 2 5 -

Hence f ( F . ( Z ) ) = fp . ( z , � ) g ( � ) dU ( � ) s o a nother appl icat i on o f J J Lemma XI : 1 completes the proo f o f the theorem .

3 .

We now return to the un i t bal l i n B and a , the norma l -

i zed Lebesgue measure on d B .

I t i s c l ear that a s at i s f i e s property ( * ) . The set a

(�) i s a n ON-bas i s f or H 2 ( a , d B ) where ca aE:lt'P

f I a l 2 ( n- 1 ) ! a ! c a = � da ( � ) = ( n- 1 + l a l ) ! d B

The Cauchy kernel i s then

a -a S ( z , U = C [ z , E;: ] = E z E;:

c =

n ' z E B , � E d B , a a ( 1 - < z , � »

and the corre spond i ng Poi s son YGrnel

Observe that

= I S ( z , U 1 2 = s ( z , z )

sup fp ( r z , � ) da ( Z ) = E;: E d B

O < r < 1

( 1 - l z I 2 ) n

1 1 _ < z , � > 1 2 n .

f ( 1 _r 2 ) n sup � E d B 1 1 - < z , rE;: > 1 2 n

O < r < 1

da ( z ) = 1

so P z ( z , E;: ) =P ( r z , � ) sat i s f i e s the cond i t i on s i n Lemma XI : 1 r

- 1 2 6 -

and the Theorem XI : 2 . Corol lary X I : l shows that a has the

equ i valent propert i e s i n Theorem X I : 1 . Th i s proves the f i r st

part in the fol l ow i ng propos i t i on .

Propos i tion XI : l . a ) I f f s E A ( B ) , f l f s l 2da � 1 , and i f

l i m f ( z ) = O , \:1 z E B then f da 4 0 , s�+oo . s s s�+oo

b ) Let v be the Lebesgue mea sure on B . The restr i c t i on map

L 1 ( " ) �A ( B ) � f r f l E H 1 ( � B ) v "J --+ a B 0 , 0

has a closed exten s i on r with

and

Dom r = { f E H ( B ) ; sup f l f ( r U l da ( U < +oo } O < r < l

Range - 1 r = H ( a , a B ) .

Proof . We f i rst reove ( the wel l -known fact ) that i f f E H ( B )

and sup f I f ( r � ) I do < +00 the n ( f ( d � ) ) 0 < r < 1 con v erg e s i n O < r < l

1 L ( a , a B ) .

Choose B E ] O , l [ and con s i der the u n i formly i ntegrabl e

f am i l y

1 T E L ( y ) ( I f ( r U I

B ) o < r < l ' � E d B . Choose

so that I f { r v � ) I B da � Tda .

r A 1 , v�+oo and v Let Q be the c l a s s i ca l

Poi s son kernel f o r the uni t bal l i n �n . Then

I f ( z ) IB

� fQ { z ' � ) T ( t; ) da ( u , z E B ,

- 1 2 7 -

wher e SQTdO i s the sma l l e s t harmon i c ma j orant o f I f 1 6 .

CXl There fore T i s i ndependent o f ( r v ) v= l and Lemma XI : 2 proves

that

By the R i e s z - F i scher theorem , we can sel ect a sequence

s A l , V-++CXl s o that v I i m I f ( s v � ) I 6 = T ( � ) V-++CXl

Fatou ' s l emma g ives

so and

a . e . ( 0 ) •

I f ( z ) I = ( I f ( z ) I 6 ) 1 / 8 < S Q ( z , � ) T ( � ) 1 / 6 do ( U

by Jensen ' s i nequa l i ty . Another app l i cat i on o f Lemma XI : 2 g i ves

the de s i red conclus i on .

Now , i f f E H ( B ) with s up S I f ( r � ) I do ( � ) < +CXl O < r < l

r t f ) as the L 1 ( O , a B ) - l im i t o f f ( r � ) . I f

and i f f -+ g i n H 1 ( o , a B ) then i t i s c lear s

sup S l f ( r � ) I do ( � ) < +CXl O < r < l

so r t f ) i s we l l de f i ned . By Lemma XI : 1 ,

f -+ f s that

we de f i ne

i n L 1 ( v )

Sp ( r Z , U g ( U dO ( U = l i m Sp ( r Z , U f s ( 9 ) dO ( � ) = f ( r U S-++CXl

- ' 2 8 -

so � ( f ) =l im f ( r � ) =g ( � ) i n L ' ( da ) by Lemma XI : ' . Thus r

i s a c losed operator a nd the proof i s compl ete .

4 . A-measures

In th i s sect i on , we study extens i on prob l ems related to

Theorem XI : 2 . We a l so study t he i r connect i on with the behav i our

of the S z eg6 kernel relat i ve l y a certa i n measure .

Con s i de r the restr i ct i on operator r ,

r Loo ( dw ) :;)Hoo ( n ) :;)A ( n ) �A ( n ) l a n = A ( a n )

and l et u be a g i ven measure on a n . ( Here dw i s the

Lebesgue mea sure on n . ) We thi nk of A ( n ) and A ( a n ) a s sub

spaces o f ( L ' ( dw ) ) I and ( L ' ( U , a n ) ) I r e spect i vely and make

the fol l ow i ng de f i n i t i on s .

Def i n i tion . The measure U l S s a i d to be c l osed i f r has a

c l osed extens i on r I n the weak * -topol ogy .

Def i nition . We def i ne 00 H ( U , a n ) to be the weak * - cl osure o f

A ( a n ) .

Def i n i tion . We say that n has property ( * * ) i f every

f E Hoo ( n ) i s the weak * - l i m i t ( re l at i ve l y dw ) o f a s equence o f

funct i on s i n A ( n ) .

Lemma XI : 3 . Con s i der the f o l l ow i ng cond i t i o n s .

- 1 2 9 -

i ) � i s a c losed measure ;

i i ) I f f EA ( &1 ) , s s E lN , converges

( re l at i ve l y dw ) to f , then

i n the weak* -topol ogy on &1

( f s ) := 1 1 S weak* -

i n 1 and there i s u n i f ormly bounded convergent L ( � , a &1 ) a 00

i n A ( &1 ) such l im 9 ( z ) ex i st s equence ( gs ) s= 1 that s s-++oo ever ywhere on &1 and equal to f and l im 9 ( z ) ex i st s s-++oo a . e . ( � ) on a &1 .

( i i i ) For every sequence f EA ( &1 ) , s ElN , sup i f s ( z ) i .s. 1 w i th s s ElN l im f s ( z ) =O , 'tt z E &1 , have z E&1 we s-++oo

l im IqJ ( Z ) f s ( Z ) d� ( Z ) = 0 , 'tt qJ EC ( a &1 ) ; s-++oo

Then i ) � i i ) � i i i ) and i f &1 has property ( * * ) then

i i i ) � i ) .

Proof . i ) � i i ) . As sume that i s c losed and that

s EJN , 1 S a weak* -convergent s equence i n 00

L ( dw , &1 ) . 1 L ( dw , &1 ) i s complete , ( f S ) := 1 i s u n i formly bounded on TI .

But s i nce 1 L ( � , a &1 ) i s s eparable and comp l ete , con-

ta i ns a subsequence that converges to a funct i on f in the

weak * -t opolog y i n 00 L ( � , a &1 ) . But the a s sumpt i on that i s

c l os ed now g ives that the sequence ( f ) oo i tsel f converges s s= 1 to f i n the weak* -topol ogy . By the Banach- Saks theorem , we

can s el ect a subsequence s o that 1 m 9 = - r f m m j = l S j

tends to f i n 2 L ( � , a &1 ) and by the R i e s z - F i scher theorem , we

- 1 3 0 -

can select another subsequence 00 ( gm . ) j == 1

J so that

a . e . ( � ) . To complete the proo f we observe that

a u n i formly bounded sequence s i nce ( f s ) := 1 i s .

That i i ) � i i i ) i s c lear .

l im g = f . m . J-++oo J 00 ( gm . ) j = 1 J

i s

i i i ) � i ) . Assume that � has property ( * * ) . Thus , i f

f E Hoo ( Q ) there i s a sequence f E A ( Q ) converg i ng to f i n the s weak* -topology . S i nce L 1 ( dw , Q ) i s complete ,

s up s E1N z E Q

so , by i i ) ,

l im s-++oo

I f ( z ) I < +00 s

ex i sts for every � E L 1 ( � , a Q ) and the l im i t i s i ndependent o f

the par t i cular cho i ce o f the s equence 00 ( [ s ) s= l - We can now

de f i ne r t f ) a s th i s l im i t a nd i t i s c l ear that r so def i ned

i s a c l os ed operator .

Def i nit ion ( c f . Sect i on X ) . Let v be a complex measure on

a Q . Then v i s cal led an A-mea sure i f for every uni formly

bounded sequence

l im f s ( z ) =O , � z E Q , s-++oo

( f ) oo of funct i ons i n A ( Q ) w i th s 8= 1 i t fol l ows that l im ff sdV=O .

s-++oo

Examples of A-measures are mea sures i n MO and c l osed

measures .

Theorem XI : 3 . Let � be a regular ( compl ex ) Borel measure and

M a weak* -compact and convex set of r egul a r Borel probab i l i ty

- 1 3 1 -

measures on a compact Hausdor f f spaces . Then U has a uni que

decompos i t i on

where u 1 i n M and

i s absol utely continuous w i th r espect to a measure

i s car r i ed by an F -set E such that a

sup { U ( E ) i u E M } = O .

Theorem XI : 4 . Let U be a n A-measure . Then

Jl = gdv+n

where 1 v E M

O' g E L ( v ) and n i s carr i ed by an Fa- set o f

van i sh i ng Q-capac i ty . Furthermore , n .l A ( � ) ( i . e . Ifdn:;; o , 'tI fE A ( � ) ) .

Proof . The s et MO

i s weak * -compact . Use Theorem XI : 3 to

decompose Jl=fdv+n and use Theorem X : 3 to prove that n .l A ( � ) .

Corollary XI : 2 . Every c losed measure i s absolutely con t i nuous

w i t h r espect to a measure in MO

.

Proo f . We know f rom Theorem X I : 4 that i f u i s c losed then

1 u = f d v+ n , v E MO

' f E L ( v ) , n .l A ( � )

wher e the decompo s i t i o n i s u n i que . I f � E C ( a � ) then �dJl i s

clos ed s o �d n A ( � ) by u n i quene s s . Hence I�dn=o for ev�ry

con t i nuous funct i on � so n � O wh i ch proves the corol lary .

- 1 3 2 -

Corol l ary XI: 3 . I f � i s a p o s i t i ve A-mea s u r e t h e n � i s

a b s o l u t e l y con t i nuous w i th r e s p e c t t o some mea s u r e s i n MO '

Proof . By The o r em X I : 3 , ]J = fd v + n , n .L A ( S1 ) a n d s i nc e � i s

pos i t i v e so i s n . Thus 0= J l o d n s o n = O .

The examp l e

and � = o l ® o O - o 1 ®0 1 ( wh e r e 0 1 i s t h e norma l i z ed L e b e s g u e

me a s u r e o n I z 2 1 = 1 ) s hows a t t h e s ame t i me t h a t A-me a s u r e s

n e e d n o t t o be c l o s ed o r a b s o l u t e l y c o n t i nuous w i th r e s p e c t t o

a ny mea s u r e i n MO ' T o s e e t h i s , t ake wEC� ( I z 2 1 < 1 ) , w ( O ) � O ,

a n d c on s i de r

Theorem XI: 5 . I f S1 i s s t r i ct l y p s e ud o convex t h e n every

A-me a s u r e i s a c lo s ed m e a s u r e .

Thus t h e equ i v a l e n c e o f A-me a s u r e s a n d c l o s ed mea s u r e s

depe nds o f t h e s hape o f a � . T h e nex t t heorem s hows t h a t a

r e l eva n t p rope r t y o f a � i s t h e ex i s t e n c e o f a � EMO s uc h

t h a t i t s a s s oc i a ted 2 . .

L - p r O ] e c t l on o n

f u n c t i on s o n smooth f u nc t i on s .

2 H ( � , a � ) maps smooth

Theorem XI: 6 . As s ume that � h a s p r o p e r t y ( * * ) ( c f . pg 1 2 8 )

and t h a t t h e r e i s a mea s u r e � E MO w i th p r op e r t y ( * ) ( c f .

pg 1 1 8 ) such t h a t � h a s n o p o i n t ma s s . F u r t he rmor e , a s s ume t hat

1 ) � 1 S closed ;

- 1 3 3 -

2 ) S [ z , � ] i s cont i nuous on TIx 8 Q \ { z=� } ;

3 ) ( ( � ( z ) - � ( � ) S ( z ' � ) ) z E Q i s a � - u n i formly i ntegrable f ami ly

for every �ECoo ( �n )

Then every A-measure i s a c l osed measure .

00 Proof . Let �EC ( � ) . Property 2 ) and 3 ) show that

for every zETI . Furthermore , by 3 )

and

i s cont i nuous for every 00 f EL ( � , 8 Q ) . As sume that 00 ( f . ) . 1 1 1 =

a u n i formly bounded sequence o f funct ions i n A ( Q ) with

l im f . ( z ) = O , � z E Q , and l e t �EC ( 8 Q ) be g i ven . We want to . 1 1-++00 prove that 1 im If . �dv=O . 1 1-++00

i s dense i n C ( 8 Q )

Put

and

for every A-mea sure v and s i nce

we can a s s ume that 00 n �EC ( � ) .

i s

- 1 3 4 -

We have a l r eady seen that I . ( z ) 1 extends to a cont i n uous func-

t i on on TI and s i nce

f . ( z ) ep ( z ) = I . ( z ) +J . ( z ) 1 1 1

so does J . ( z ) , and moreove r , 1

s up I 1 1. ( z ) I + I J 1· ( z ) I < +00 . i ElN z E n

We have a s sumed that � i s a cl osed mea s ure , s o s i nce , for

z ETI , i t fol l ows that l im i -++00

1 . ( z ) =O 1

and domi nated convergence g i ves that l im J I . d v=O i -++00 1 for every

measure on a n . Furthermore , and l im J . ( z ) =O , � z E n , . 1 1-++00 aga i n s i nce ]..I i s c l osed . Thus l i m J J . d v = O

i -++00 1 for every

A-measure v wh i ch means that

l im i -++oo

Jepf . dv = l im Jl . dv + l im JJ . dv = 0 1 . 1 . 1 1 -+ +00 1 -+ +00

wh ich proves the theorem .

Corollary XI : 4 . Assume that there i s a measure ]..I on a n

with the propert i e s i n Theorem X I : 6 a nd that every point has

van i sh i ng Q-capa c i ty . Then

a l gebra o f 00 L ( ]..I , a n ) .

Proof . As sume f i rst that 00 f E H ( ]..I , a n ) and

i s a c l osed sub-

00 n ep E e ( 0: ) . Then

- 1 3 5 -

and by the proof o f Theorem X I : 6 i t f o l l ows that the f i rst term

exter.ds con t i nuous l y to TI whi l e the second i s i n co

H ( jJ , a n ) .

By approx i mat i on , i t rema i n s t o prove that co H ( jJ , a n ) +C ( jJn )

c losed and th i s fol l ows f rom i i ) i n the next theorem .

Theorem XI : 7 . Assume that n has prope rty ( * * ) and that

every po i nt i n an has van i sh i ng Q-capac i ty and that every

A-measure i s a c l os ed meas ur e . Then

i ) i f JJ E MO w i th property ( * ) . Then

j �

i s an i s omet ry for every A-mea sure \! • ,

i s

i i ) i f with property ( * ) then i s a

closed subset of co L ( jJ , a n ) .

Proo f . i ) I t i s c lear that j i s a conti nuous map f rom a

Banach a lgebra i nto a Banach a lgebra so i t i s enough to prove

that j i s b i j ect i ve . F i r s t s upp jJ = a n for otherw i se , by ( * ) ,

the Choquet boundary o f a n ( wi th respect t o A ( n ) ) would be

st r i ct l y sma l l e r than a n a nd there wou ld then be a pos i t i ve

measure e on s o that for a poi nt

� O E a n and s i nce every A-measure i s a s sumed to be a c losed

measure th i s would i mp l y that o � i s a c l osed measure . D

Coro l lary XI : 2 shows that the Q-capac ity o f { � b } i s str i ctly

pos i t i ve , a contrad i ct i on .

- 1 3 6 -

Hence , the sup norm and e s s sup ( � ) are equal for con-

t i nuous funct i on s on a n . S o i f ( f s ) := l i s a sequence o f

funct i ons i n A ( n ) OD conve rg i ng weak* i n L ( � , a n ) then the

fam i l y is un i formly bounded on n and it fol lows now f rom

Lemma X I : 3 and property ( * ) that OD ( f s ) s= 1 converges weak * i n

i s c l osed by

assumpt i on . I t fol lows that j i s a b i j ect ion wh i ch comp l etes

the proo f o f i ) .

i i \ By the above part o f the proo f we can appl y Theorem 2

and Propos i t i on 1 i n Aytuna and Cho l l et [ 1 ] , wh i ch comp l etes

the proof o f the theorem .

We f i n i sh th i s section w i th two examples where Theorem

X I : 6 and Coro l lary X I : 4 can be appl i ed .

Example 1 . We choose n=B , the u n i t ba l l i n �n and � = o , the norma l i z ed Lebesgue mea s ure on a B o Then n i s str i c t l y

pseudoconvex and Theorem X I : 5 shows that every A-mea sure i s a

c losed measure . However , we want t o show that the cond i t i on s i n

Theorem XI : 6 are sat i s f i ed . ( Hence , Corol lary X I : 4 appl i e s i n

th i s case . ) We know that

! l - < z , f,; > ! n

Thus property 2 ) i n Theorem X I : 6 i s sat i s f i ed and by P ropo-

s i t ion XI : 1 we know that a i s a c losed measure , so 1 ) holds

t rue .

- 1 3 7 -

I t rema i ns to understand that 3 ) holds trup . Choose

�E ] l , n-7/2 [ and cons i de r

I f sup l ( z ) <+oo then i t i s c l ear that 3 ) holds . But z E B

=

< 2 1 / 2

By Rud i n [ 7 , Propos i t ion 1 . 4 . 1 0 ] sup l ( z ) <oo . z E B

2 1 1 - < Z , c:: > l n - ( 1 /2 )

Example 2 . Let be n number s i n the i nterva l

] 0 , 1 [ and put

Then i s a pseudoconvex set w i th 2 C -bounda ry , but a s soon

a s at least one a . i s l e s s than 1 , 0 i s not str i ct l y J a

pseudoconvex .

We sha l l now see that the cond i t i on s i n Theorem X l : 6 are

s at i s f i ed for a certa i n measure u that we de f i ne i nduct i ve l y .

For n= l we def i ne u =o l = t he norma l i zed Lebesgue mea sure on

I z 1 = 1 .

As sume now that i s de f i ned for a l l a ' = ( a 1 , · · · , an- 1 ) ·

We then de f ine for

- 1 3 8 -

a= ( a ' a ) , n a s

f f ( z ) dWa = a D

2 / a f f ( ( l - l z 1 n ) 1 /2 z ' , z ) . n n

a a D . x { l z 1 < 1 } a n

2/a 1 • 1 1 . ( 1 - 1 z 1

n ) a - dw d z n a ' n '

where d Z n denotes the Lebesgue measure on { l z n i < l } . I t i s

c lear that w aEMO and that wa i s c l osed i s eas i ly seen f rom

the fact that G 1 i s c losed .

For Bonami and Lohoue have c a l cul ated the corre-

spond ing S z ego kernel S Wa a nd they prove i n P ropo s i t i on 2 : 1

that i t extends to a n i n f i n i te l y d i f f erent i able funct i on on

D xD out s ide the d i a gonal o f a D x a D . ( Bonami and Lohoue [ 2 ] . ) a a a a

Fur thermore , they prove i n Propos i t i on 5 . 3 that

1 S un i formly i ntegrable in LP ( Wa ) for ever y

p< wh i ch shows that 1 1 . f - 2 n 1 n ak

i s a W -un i formly i ntegrable f am i ly f or every L ipsch i t z a func t i on � . I t fol lows that the cond i t i on s of Theorem X I : 6

are sat i s f i ed and we can a l s o apply Coro l l ary X I : 4 .

- 1 3 9 -

5 . The Cauchy trans form o f measures

We con s i der aga i n the u n it bal l B i n �n and a , the

norma l i z ed Lebesgne mea sure on a B . We wr i te C and P for

the correspond i ng Cauchy and P o i sson kerne l , and we say that a

funct i o n f EH ( B ) belongs to HP , i f r ( f ) E HP ( a , a B ) . I f

g ( z ) = f P ( z , � ) d lJ ( � ) E H ( B ) a B

for a comp l ex measure lJ then lJ=fdlJ where f EH ( o , o B ) and

supp lJ = a B ( or empty ) . Th i s i s so because

so and ( g ( r U ) O < r < l converges i n 1 L ( a , a B ) to g * ( Pr opos i t i on X I : l ) . O n the other hand ,

converges weakly to lJ so lJ=g * do .

Reca l l now that the Cauchy kernel relat i ve l y B and a

i s

C [ z , � ] = n ( 1 - < z , � » = L:

0. > 0

z E B , � E a B . I f lJ i s any mea sure o n a B for wh ich the func

t ions ( � o. ) are pa i rw i se orthogonal the n

fC [ z , � ] d]..t{ � ) = lJ ( l ) .

by

- 1 4 0 -

For i nstance , let w be de f i ned on the un i t sphere i n �n

I � ( � ) dw ( � ) = 2TI d B

2 n

I i G � ( e , 0 ) dG o

( i . e . w=o®oO

) . Then W i s s i ngular relat ively 0 but the Cauchy trans form of W � l . But there are restrictions on � i n order to have

Theorem XI : 8 . Let � be a real measure so that g ( Z ) = IC [ z , � ] dW ( � ) E H 1 . Then W i s a c losed measure .

Proof . Let f . EA { B ) , i Eill , be a uni formly bounded sequence 1 such that l im f . ( z ) =O , �z E B . Then . 1 1-++00

� im If i { U dW { � ) = � im 1-++00 1-++00

l im I ( Jc [ rz , � ] f i { Z ) dO ( Z ) ) dW ( � ) = r-+ 1

= 1 im If . ( z ) g * ( z ) do ( z ) = 0 . 1 1-++00

by Propos i t ion XI : l where g * i s the L 1 ( o ) - l imi t o f g { r z ) , wh i ch ex i st s s i nce 1 g E H .

We have proved that � i s an A-measure and we know that every A-measure is a c losed measure .

Corollary XI : 4 . I f JC [ z , � ] d � { � ) EH l for a real measure w { tO )

then the Q-capac i ty o f the support of w i s pos i t i ve .

- 1 4 1 -

Proo f . By Theorem X I : 8 and Corol l ary X I : 2 , w = f d v where vEMa and fEL 1 ( v , d � ) . Thus supp w= ( supp v ) n ( f� O ) .

Remark . There i s no converse of Theorem X I : 8 . Choose n= l and 1 a non-negative funct ion T E L ( a , d B ) \ L log L . Then

closed measure but g ( z ) = fC [ z , � l Tda�H l because i f Tda I S a

1 gEH , Re g�O then Re g * EL log L . Now , s i nce 2 Re C- l =P we get that

2 Re g * - f T ( U da ( � ) = T ,

a contrad i c t i on , s i nce the l e ft-hand s ide i s i n L log L .

For the rest o f thi s sect i on , let w denote a pos i t i ve measure so that the funct ions 0: ( � ) 0: > 0 are pa i rwi se orthogonal and remember that C denotes the Cauchy kernel relat i ve to B and a .

By construct ion , i f 2 <p EL ( a , d B ) then

. h 2 . . f I S t e L -pro J ect I on 0 <p i nto H 2 . The fol lowing theorem shows what happens when we r eplace a by w .

Theorem XI : 9 . The map L2 ( W , d B l 3 T � fC [ z , � l T ( � ) dW ( � ) i s i nto H2 i f and only i f sup

do: < +00 where do: = f ' �o: , 2dlJ ( � l and

0: > 0 Co: Furthermore , i f L: d = +00 for a sequence 0: . J

00 ( a . ) . 1 J J =

- 1 4 2 -

c wi th � ( d

a j ) 1 /2 < +00 then there i s a cont i nuous j= 1 a . J

funct ion T on d B such that

( Remember that we have as sumed that fC [ z , � ] d U ( � ) = U ( 1 } > o . )

Proof .

where

Hence

I f 2 cpEL ( u , d B )

cp ( z ) = 2: a > O

then

a- a fC [ z , U CP ( S ) d U ( S ) = f 2: ;}- cp ( � ) d ]J ( S ) =

a

a z dU ( � ) 172 c a

so i f sup :a < +00 then fC [ z ' � ] CP ( S ) dU ( � ) E H 2 . As sume now that a a

da sup - = +00 and choose a sequence a ca

c 00 a . 2: (�_J_ ) 1 /2 < +00 and cons ider

. 1 d J= a j k

00 ( a . ) . 1 J J = such that

- 1 4 3 -

1 / 4 c 1 /4 a . c co a . a . co a . J

J � +00

J = l a . J

a . J

d 00 a . 1 / 4 a .

z J

a . J

fe [ z , � ] <p ( (,; ) d w ( t,; ) = L ( _J ) . 1

c J = a . 172 c

a .

wh i ch i s not i n s i nce

J

d a .

l im _J c

j�+oo a . J

J

= +co •

F i nal l y , assume that

Then

00 c a . 1 / 2 L ( � ) < +co but

. 1

d J = a . J

c 00 a . 1 / 2 a . 1.jJ ( z ) = E (�) Z 1

j = l a . J

i s con t i nuous on a B but

00 L d = +00 .

j = 1 aj

1 / 2 c a . J

d1 / 2 a . J

a . z J c a .

J

d co a . 1 / 2 a . E ( --.J. ) z J

. 1 c J = a . .

J

so

- 1 4 4 -

. 00 I I Ie [ z , s ]\(J ( s ) d w ( s ) I 2 da ( U = L: d = +00 j = l a j

wh i ch completes the proof o f Theorem X I : 9 .

Example . I f v=a , x o O then vEMO and I�a� SdV=O , a � S . But

d ( m , O ) = l , �mElli , so Theorem XI : 9 shows that the cont i nuous

funct ions do not operate on the c l a s s T of mea sures

Propos i t ion XI : 1 says that i f ( f ) 00 s= l

A ( B ) , bounded i n

then f da :4 0 , s-++oo. S

2 H ( a , a B ) and such that

is a s equence i n

l im s-++oo

f ( z )=O , �zEB s

We do not know i f th i s can be extended to arbi trary

measures in MO but we have the fol l owi ng theorem .

Theorem XI : l 0 . Suppose that

t i ons i n A ( B ) such that

I I 1 2n+€ s up f . da < +00 . J J

00 ( f . ) . , J J= i s a s equence o f func-

and that l im f . ( z ) ex i st s for every z E B . I f �EMO then j-++oo J

f . d w i s convergent i n the sense o f d i st r i bu t i on theory . I f J

furthermore sup I I f . I dw < +00 then f . dw i s weak l y c onvergent . j J J

Proof . Suppose that 00 ( f . ) . , J J = sat i s f i es the assumpt ions l n the

- 1 4 5 -

theorem and that l im f . ( z ) = O , � z E B . We then have to prove j-++co J

that

l im f f j epdlJ = O , 'ti<pECoo ( 3 B ) j-++oo

where lJ i s f ixed i n MO '

Assume that epEC 1 ( �n ) and let O < r < l . Then

where we have used that lJ EMO ' I t fol lows from Propos i t ion X I : l that l im ff j ( s ) ep ( r j s ) dO ( S ) =O for every sequence r . .A 1 , j-++co •

J

I t rema i ns to prove to take care o f the f i rst term . Note f i rst that i t follows f rom the calculat ion i n Example 1 , that

where a = 2 n + E and a + a' = 1 .

There fore , for z EB f ixed ,

1 im f f . ( s ) ( ep ( r . z ) - ep ( r . U S ( r . z , U d o ( U = 0 j-++ro J J J J

for every sequence

- 1 4 6 -

r . .A 1 , j-++oo . J

Thus , by dominated con-

vergence , the f i rst term at the r i ght hand s i de tends to z ero

for every sequence r . .->1 1 , 1 -++00 . J

To f i n i sh the proof o f Theorem XI : l 0 , we need only observe

that every cont i nuous funct i on on a B can be un i formly approx i

mated by funct i ons i n Coo ( �n ) .


General references for t h i s sect i on are Bungart [ 3 ] and

Rud i n [ 7 ] .

Theorem XI : 3 was proved by a comb i nat i on o f arguments o f

Gl i cksberg , Kon i g , Seever and Ra i nwater . See Rud i n [ 7 , pg . 1 9 4 ]

for deta i l s .

Theorem XI : 5 i s due to Henk i n , see Henk i n and C i rka [ 4 ] .

We have no exact descr i p t i on o f the sets that sat i s f i e s

cond i t i on ( * * ) , ( pg 9 4 ) .

By tak i ng d i l atat i on s , i t i s c l ear that every star shaped

doma i n sat i s f i es ( * * ) . I t i s known that th i s i s true a l s o for

smooth s t r i ctly pseudoconvex doma i n s , c f . : N . Kerzman ; Holder

and LP e s t i mates for solut i on s o f aU= f i n strongl y pseudo-

convex doma i ns , Comm . Pure Appl . Math . 2 4 ( 1 9 7 1 ) , 3 0 1 - 3 7 9 , and

Brian Cole and R . Michael Range ; A-measures on comp l ex man i -

folds and some appl icat i ons , J . Func . An . 1 1 ( 1 9 7 2 ) , 3 9 3 - 4 0 0 .

That Hoo+C i s a c losed subalgebra o f Loo was f i rst

proved by Sara son [ 8 ] on the u n i t c i r c l e , by Rudi n [ 6 ] on the

- 1 4 7 -

un i t spher e i n �n , by Aytuna and Cho l l et [ 1 ] on the boundary

of str i c t l y pseudoconvex doma i ns .

Jewel l a nd Krantz [ 5 ] proved the results for convex sets

in [ 2 w i t h r eal anal yt i c boundary and they a l so remarked that

th h d t h It f "' 0 cfT"n h ( ) 1 ElN ey a e r esu or 0 \L, w ere a= a 1 , . . . , a , - , a n ak 1 <k < n . The methods i n the above paper s are d i f ferent from our s .

[ 1 ] Aytuna , A and Choll et , A . -M . , Une Exten s i on d ' un resultat de W . Rud i n . Bul l . Soc . Math . f rance 1 0 4 ( 1 9 7 6 ) , 3 8 3 - 3 8 8 .

[ 2 ] Bonami , A . and Lohoue , N . , P r o j ecteurs de Bergman et S z ego pour une c l a s se de doma i ns fa i b l ement pseudo-convexes et e s t imat i ons LP . Compos . Math . 4 6 ( 1 9 8 2 ) , 1 5 9 - 2 2 6 .

[ 3 ] Bungart , L . , Boundary kernel funct i ons for doma i ns on compl ex man i folds . Pac i f ic J . Math 1 4 ( 1 9 6 4 ) , 1 1 5 1 - 1 1 6 4 .

[ 4 ] Henk i n , G . M . and Ci rka , E . M . , Boundary proper t i es o f holomorph i c funct ions o f several complex var i able s . J . Sov iet Math . 5 ( 1 9 7 6 ) , 6 1 2 - 6 8 7 .

( 5 ] Jewe l l , N . P . and Krantz , S . G . , Toep l i t z operators and related f unction algebras on certa i n pseudoconvex doma i ns . Tran s . Amer . Math . Soc . 2 5 2 ( 1 9 7 9 ) , 2 9 7 - 3 1 2 .

( 6 ] Rud i n , W . , Space s o f type ( 1 9 7 5 ) , 9 9 - 1 2 5 .

00 H +C . Ann . I nst . Four i er 2 5

[ 7 ] Rud i n , W . , Funct i o n theory i n the u n i t bal l o f [n o Spr i nger-ve r l ag . New Yor k , He i del berg , Ber l i n 1 9 8 0 .

[ 8 J Sarason , D . , Genera l i z ed i nterpolat i on i n Ame r . Math . Soc . 1 2 7 ( 1 9 6 7 ) , 1 7 9 - 2 0 3 .

. 00 H . Trans .

XII Complex Homomorphisms

I n the last sect i on , we saw that one could a s s i gn

" boundary val ue s " to certa i n analyt i c funct ions by con s i de r i ng

closed extens ions o f the rest r i c t i on operator .

Here we use the a lgebr a i c structure o f Hoo ( � } a nd con

s i der e l ements i n Hoo ( � } a s conti nuous funct ions on a compact

set " conta i n i ng " � . We start with a very b r i e f rev i ew of the

elements of Ge l fand representat i on .

Let A be a uni form and commutat i ve Banach a lgebra ( wi th

i dent i ty ) . A complex homomorphi sm ( or l i near mU l t i p l i cat i ve

funct iona l ) on A i s an el ement O�mEA ' such that

m ( f g } =m ( f } m ( g ) , � f , g E A .

We wr i t e MA for the compl ex homomorph i sms o n A . Note that

MA is contai ned in the u n i t sphere in A ' .

There fore , by the Banach-Al aog l u theorem , MA i s compact

i f we g i ve it the weak* -topo l ogy , wh i ch i s gene rated by nei gh-

borhoods of the f orm

One can show that every max ima l i de a l i n A i s the kernel

of an element in MA , there f ore MA is somet i me s cal l ed the

max i mal i deal space of A .

- 1 4 9 -

W i th every e lement f EA , we assoc i ate f EC ( MA ) by f ( m ) =m ( f ) , mEMA and i t fol l ows from the de f i n i t ion of the weak * -topology that f i s a lways conti nuous on the compact space MA o

I f we denote by A the set o f f , f E A we have an algebra A

homomorph i sm f4f , sup I f ( m ) I =sup I m ( f ) I � " f l l and f i s mEMA mEMA

cal l ed the Ge l fand trans form o f f .

A c l osed subset K o f MA i s cal led a boundary for A

i f

I I f l ! = sup I m ( f ) l , 'tf f EA . mEK

One can prove that there is a smal lest boundary ; thi s boundary i s cal l ed the Sh i l ov boundary o f A .

Let now n be an open and bounded subset o f n a: , n� l . Let A be a uni form Banach a lgebra o f conti nuous funct ion s on n ( wi th supnorm ) . Then every z O E n g i ves r i se to an element

We denote by TI ( m ) = ( m ( z 1 ) ' " . , m ( z n ) ) ' mEMA , and TI* = the weak* -c lo sure of

- 1 5 0 -

Propos it ion XII : 1 . The Shi l ov boundary o f Hoo ( n ) i s contai ned

i n IT* .

Proof .

then

I f sUE I f ( m ) I < I f ( rna ) I mE n *

and l / f - f ( mo )

wh i ch i s a contrad i c t i on .

are i n 00 H ( B ) . There fore

We wi l l need the fol l ow i ng r e f i nements o f Lemma I I I : 2 a nd

Theorem X : 3 .

Theorem XI I : 1 . Let G be a convex subset o f some vectorspace ,

K a convex subset o f some t opol og i ca l vector space and

F : G x K�m a funct i on such that G 3 x � F ( x , y ) i s convex on G

for every yEK . K3y � F ( x , y ) i s concave and con t i nuous on K

for every xEG . Then sup i n f F ( x , y ) = i n f sup F ( x , y ) . yEK xEG xEG yEK

To formulate the next t heorem , we need s ome more notat i on .

Let X be a compact Hausdor f f -space , M ( X ) i s the set o f a l l

regular Bore l measures o n X . A funct i on a l gebra A o n X i s

a c losed s ubalgebra of C ( X ) ( wi th supnorm ) . We a s sum� that

A conta i n s a l l the constants and separates po i nt s on X .

I f h i s a mUl t ipl i ca t i ve l i near funct i onal on A , then i t fol l ows f r om the Hahn-Banach theorem and R i e s z repr esentat i on

theorem that there ex i st s a probab i l i ty measure p EM ( X ) such

that

h ( f ) = f fdp , tf f E A

- 1 5 1 -

and we then say that p represents h . We de f i ne Mh to be

a l l the probab i l i ty measures that r epresents h .

Theorem X I I : 2 . With notat i on a s above , suppose E i s an

F - set i n X such that a . sup p ( E ) =O . p EMh ( X )

{ f } oo o f func t i ons i n A such that m m= 1

I I f II < 1 , tfmElN m -

l i m m-++oo

f ( x ) =O , m tfxE E

l im f ( x ) = 1 a . e . ( p ) , m m--)o+oo

Let now B be the un i t bal l i n

Then there i s a sequence

n 00 [ , n � 1 , A=H ( B ) . Then

Hoo ( B ) i s a c losed subalgebra of C ( MA ) and by the Stone

We i erstrass theorem , ( fg ) f , gE Hoo ( B ) generates C ( MA ) . There

fore , i f W EMO then

def i ne s a con t i nuous l i near operator on e ( MA ) . Here f *

denotes the weak * - l i mi t o f ( f ( r � ) ) O < r < 1 relat i ve l y w . See

Theorem X : 8 , X I : 5 and Lemma X I : 3 .

So by the R i e s z representat i on theorem , there i s a regular

Borel mea sure 0 on MA so that

- 1 5 2 -

In part i cular , L ( f ) =f f *d�= f ( O ) =O ( f ) so OEMO .

On the other hand , i f DEMO ' cons i der for f , gEA ( B )

fgdD . Then , th i s determi nes a cont i nuous l i near funct ional on C ( S ) , S= 3 B . Aga i n , s i nce f -+ ffdD=f ( O ) =O ( f ) , R ies z representat ion theorem g ives a measure �EMO such that

When

f (gd� = ff�dD , b' f , gEA ( B ) .

n= 1 , and M� o contai ns only one e lement . When n > 1 , th i s i s not so and the fol low i ng two quest ions are natural to ask .

1 •

2 •

Given that

a . e .

D EMO ' determine � EMO as above . Is it then true f f *g *d � == ff�dD , 00 b' f , g EH ( B ) ?

I s i t then true that l im f ( � ) ==f ( � ) , r-+ 1 r

( 0 ) for a l l D EMO? ( Here fr ( � ) = f ( r � ) ; th i s would be an analogue o f Fatou ' s theorem extended to MA ) .

I n view o f Henk i n ' s theorem ( Theorem XI : 5 ) 2 ) woul d imply 1 ) . But 1 ) i s false - assume for a moment that 1 ) i s true .

- 1 5 3 -

Then , by T heorem XI : 5

a � i n f f I g - f * 1 2d ].l � i n f f I ;-f 1 2

d O g EA ( B ) g EA ( B )

s o Theorem X I I : 1 woul d g i ve

f A A 2 f A A 2 a ::: s up i n f I g - f I d O = i n f sup I g - f I dO =

O EMo g EA ( B ) g E A ( B ) O EMO

= i n f sup f l g - f * 1 2 d ].l . g EA ( B ) ].l EMa

Note that

i s con t i nuou s s i nce I f I lS con t i nuou s on MA f or a l l

f E H<X> ( B ) .

w i t h

Th i s g ive a contrad i c t i on , s i nce t h e r e ex i s t f E H<X> ( B )

i n f sup f l g - f * l d ].l > a g EA ( B ) ].l E Ma

( c f . Theorem X : 6 a nd the examp l e prec eed i ng i t ) .

Notes a nd r e f erences

For r e s u l t s and re ferences conce r n i ng Banach a l gebras we

r e f e r to J . B . Garnett , Bounded a n a l yt i c f u nc t i on s . Academi c

P r e s s , 1 9 8 1 .

Aspects of MathEmatics E ng l i sh- language subseries ( E )

V o I . E 1 : G . Hecto r / U . H i rsch, I ntrod uct i on to the G eometry of

Fol iations, Part A

Vo l . E2 : M . Knebusch / M . Kolster , Witt ri ngs

V o l . E 3 : G . Hector I U . H i rsch , I nt rod uction t o t h e Geo metry of

F o l iat ions, Part B Vo l . E 4 : M . Laska, E l l i pt ic Cu rves over N u mber F ie lds w i t h Prescr ibed

Reduction Type

V o l . E5 : P . St i l ler , Automorph i c Forms and the Pica rd N u mber of a n

E l l i pt ic Su rface

Vol . E 6 : G . F a l t i n gs / G . Wiistholz et a I . , R at iona l Poi nts

(A P u b l i ca t i o n o f the M a x - P l a n c k - I nstitut fur Mathemati k . B o n n )

Vo l . E7 : W. Sto l l , Va l ue D istri b ut ion T heory for Meromorph ic M a ps

Vol . E 8 : W. von Wah l , The Equat ions of N avier-Stokes a n d Abstract

Pa rabo l ic Eq uations

Vo l . E 9 : A. Howard / P.- M . Wong ( Ed s . ) , Contr ibut ions t o Severa l

Complex Var iab les

Vol. E 1 0 : A . J . Tromba, Sem i na r o n N ew R esu lts i n Non l i nea r Part i a l

D i fferentia l Equations

(A Publ icat ion o f t h e Ma x-P l a n c k - I n'st i t u t f u r Mathemat i k , B o n n )

Vol . E 1 1 : M . Yosh ida, F uchsian D ifferent i a l Eq uations

( A P u b l i ca t i on of the Max-Pl a n c k - I n stitut fur Mathemati k , Bon n )

Vo l . E 1 2 : R . K u l ka r n i , U . P i n ka l l ( Ed s. ) , Conformal Geometry

(A P u b l i cation of the M a x - P lanck-I nstitut f u r Mathemat i k . B o n n )

Vo l . E 1 3 : Y. And re, G - F u nct ions and Geometry

(A Publ i cation of the Max-P l a n c k - I nstitut fur Mathemat i k , Bon n )

Vol . E 1 4 : U . Cegre l l , Capacit ies i n Complex Analysis

Aspekte der Ma1hanatik Oeutsc hsprach ige U nterre i h e ( 0 )

B a nd 0 1 : H . Kraft, G eometr ische M ethoden i n der I n va ria nte ntheorie

B a nd 02 : J . B i ngener, Loka le M od u l ra ume i n der ana lytischen G eometr ie 1 B and 03 : J . B i ngener, Loka le M od u l raume i n der ana lyt i schen Geometrie 2 B a nd 0 4 : G . B a rthe l / F . H i rzebruch / T . H ofer, G e rade nkonfigurationen

u nd A igebra ische F lachen

( E ine Veroffen t l ichung d e s Max-Planck-I nstituts fur Mathemat i k , B o n n )

Band 05 : H . Stieber, E xi stenz sem i u n i verse l le r O efo rmationen i n

der komplexen Analysis

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