Capacities in Complex Analysis (Aspects of Mathematics)

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. All rights reserved

© Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig

1988

No par t of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mech anical, photo­ copying, recording or otherwise, without prior permission of the copyright holder.

Produced by Lengericher Handelsdruckerei, Lengerich Printed in Germany

ISSN

0179-2156

ISBN

3-528-06335-1

Contents VII

Introduction

XI

List of notations

I.

Capacities

II.

Capacitability

III . a

Outer regularity

II I .b

Outer regularity

IV.

Subharmonic functions in

V.

Plurisubharmonic functions

4 11 22

(cont.) n JR in

30 �n _

the Monge-Ampere capacity VI.

32

Further properties of the Monge-Ampere operator

56

VII.

Green's function

66

VIII.

The global extremal function

73

IX .

Gamma capacity

81

X.

Capacities on the boundary

99

XI .

Szego kernels

1 16

XII .

Complex homomorphisms

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

m

n

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

m

n

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 operator is linear. n [

differ

(it is n-linear)

while the Laplace

This explains why the theories in

considerably.

functions is harmonic,

For example,

m

n

and

the sum of two 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. give an example of similarities and differences, following statements.

Assume first that

�

To

consider the

is an open subset

VIII

of

ffin

and that

K

i s a c l os ed s u b s et of

f ol l ow i ng p r op er t i es that F or ev ery

(i )

on

�

z O EK

K

�.

C on s i d er t h e

m a y or may n ot h a v e.

t h er e i s a s u b h a r m on i c f u n c t i on

�

s u ch t h a t l i m � ( z ) < �(z O ) , z-+ z 0

(i i )

wh er e

z E K, z 1z 0 .

Th er e i s a s u bharmon ic f u n c t i on

on

�

�,

�t- oo ,

s uch t h a t Kc{z E � i �(z ) =-oo} . (i i i )

T h er e i s a l oca l l y upp er b ou n d ed f a m i l y

s u bharmon i c f u n c t ion s on �(z ) =s up �.(z) iEI 1

wh er e

(i v)

If

�

-+

th en on

�

a nd

s uc h t h a t

of

Kc { z E � ; � ( z ) < �*(z ) } ,

�* ( z ) =li m �(z ' ) . z'-+z

i s s ubha rmon i c out s i d e

l i m �(z ' ) < + 00 , z z '

�

( �i ) i E I

wh er e

z '�K ,

K

and i f

�z E � ,

ext en d s t o a u n i qu el y d et er m i n ed s ubharmon i c f u n c t i on

�. In c l a s s i ca l p ot ent i a l t h eor y , i t i s a t h eor em t h a t 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 c t s et s t h a t h a v e t h e pr op ert i es a re exa c t l y t hos e w i t h v a n i s h i ng N ewt on c a pa c i t y . T o s t udy t h e c or r es p ond i ng p r op er t 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 ub h a r m on i c fu n ct i on s b y t h e p l u r i subharmon i c f un c t ion s . C ond i t i on s (i ) - ( i v ) a r e t h en t r a n s f ormed

IX

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

(i') -;;

(cf.

(iii)



(iii') -:; (iii);; (iv').

the last reference in Section X). Section I and II are concerne,d with general capacity

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, Sections IX, functions.

while

X and XI are devoted to capacities and analytic

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 Universite Paul Sabatier,

in the fall

Toulouse in December

1 98 5 and at 1 98 6 .

x

General references

Capacity theory

Gustave Choquet,

1-3.

Lectures on analysis,

W.A.

Benjamin,

1969. 0.0. Kellog, Verlag,

Foundations

of potential

theory. Springer­

19 29.

N.S.

Landkof,

Springer-Verlag,

Foundations of

modern potential theory.

197 2.

Complex analysis in several variables

L.

Hormander,

several varibles.

Steven G. variables. P.

An introduction to complex analysis in North Holland,

Krantz,

John Wiley

Lelong,

19 73.

Function theory of several complex &

Sons,

198 2.

Plurisubharmonic functions and positive

differential forms.

Gordon and Breach,

1969.

List of Notations

Notat i o n

Meaning

IN

t he n a tu r a l n umb e r s

JR

the real

P ( U)

numbers

t he c omplex numbe r s a ll t h e

subsets

of

t h e p r oduct s p a c e

U Ux . . . x U

t h e cha ra c t e r i st i c f u nc t i on of

U

t h e c l a s s of r e a l or c ompl ex v a l ued f u nc t i on s on or order

n

w i t h c on t i nu ou s d e ri va t i ve s or

�P

t h e d i f f e r e nt i a l ope r a t or

r

a

the d if fe r e nt i a l ope r a t or

r

au

t h e b ou nd a r y of t h e s e t

U

d d Zj

-

d -

dZ.

dz

J .

zj

J

t h e exte r i or p r od u c t LP ( ]J , U)

]J

is a mea s u r e on

c l a s s of

U

and

L P ( ]J , U )

]J-mea s u ra b le f u n c t i on s on

i s the U

w i th

I

Capacities

De f i n i t i on . c

U,

on

U

U

Let

be a a-compact Hausdorff-space.

is a set function defined on

P (U) ,

A capacity

the subsets of

with the following properties:

i)

P(U)

3 E

ii )

P(U)

3

iii) If

of

E

�

s

K , s

s E ill

U,

K

=

c(¢)

=

0 c(E)

is a decreasing sequence of compact subsets

n K

s=1

De f i n i t i on .

,

� sup c(E ) = s sEN

s-++oo,

-E,

inf c(K ) s sEN

E JR+

c( E )

s'

,

then

= c(K).

A set function satisfying property

i)

and

ii)

above is called a precapacity.

Examp l e 1:1.

If

w*

measure

W

is a positive Radon measure then the outer

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

Def i n i t i on .

Let

tends to

f

w

lim �dW S s-+oo

w

s

,

s E ill

and

weakly and write

=

f

�dw

,

w

c.

be measures. We say that

�� E C (U). O

if

-

Lemma

If

I: 1.

such that

2 -

is a sequence of

positive measures

is an upper semicontinuous func-

and if

tion with compact support then

By monotone convergence,

Theorem l�'. es.

Then

choose

J

compact, There

=

be a weak*-compact set of positive measur-

M sup

w· E M J

iii)

with

is a continuous




-

w.(K.) J

J

function

is a weak*-neighborhood of

J

is clear.

c(K.) J

-

there is an accumulation point

and since

XK.

is a capacity.

w*(E)

wEM

Everything but

Proof. K.,

c(E)

Let

W,

and

Therefore,

c(K)

�

C(Kj)

which proves the theorem.

�

f�d� j

�

X

>

K

W

(




Since

E.

so that

there is a

+

E

2E


m,

K �

is

wh ic h c omple t e s t h e p r oo f.

CE

M

Let

Theorem 1 11 : 3 .

j=l

we h a v e

CE

J+ 1 - If' ]'

L

j= l

,

22j+l




with

s up lJ E M

lJ(E) =

0

o f open s e t s c o n t a in i ng

w it h l im sup lJ ( A . ) J j-++oo lJEM

Proo f .

=

O.

S i n c e a l l f u nc t io n s i n

N

a n d h e n c e in

R

a r e l ower

s e m icont inuou s , the set f u nc t io n

G(E) =

i n f { sup lJEM

fCPdUi

cP E R ,

cP > 1 o n

E}

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

G(E) = inf{G(A)i ECA Th i s p r o v e s that

G

open } .

s a t i s f ie s ax i om

iii )

and a l so that the

c o r o l la r y f o l l ow s f rom the t h e o r em . Let n ow 1

n

R,

< I J -

cp.

co

(CPj)j=l a nd l e t

be a d e c r e a s ing s eq u e n c e e o f f u n c t i o n s b e the l a r g e s t l owe r s e m i c o n tinuou s

- 19 -

minorant of l im � J. . We claim that to every £ > 0 there is a -+ +oo � £ E R such j that � £ = on { l im � j > � O} and such that sup f � d w < £ . Let £ > 0 be given . Si n ce all the funct ions wEM �O ' ( � j ) j = l belongs to N , there is a decreasing sequence of open sets ( A.)J J. = 1 with lj imoo sup w ( A J. ) 0 and such that all -++ wEM the functions are continuous on CA J. , j E m. By iv) , there is a �,.. E R such that sup f� £d w < £/3, � £ -> 1 on A J. for £ EM some A J. . Then {x E CA J. � J. -> �O + l} v = K�J is a decreasing £ £ sequence of compact sets and sup w( n K�) = 0 by i i ) . Hence WEM j=l J by v ) and iv ) there i s a sequence ( �v ) oov= l of functions in v K� and such that R with � > on j=1 J E:

00

00

=

c.

W

;

00

00

n

Then and T so � £ proves

T

00

= inf ( L �v , l) E R by ( i i ) for i ncreasing sequences , v=l > on {ql 0 < l im ql.} nCA J. . Furthermore sup f d w < -} J wEM £ and sup fql£ < £ which + T > 1 on wEM the claim . +

T

T

00

Let now ( E J. ) J. = 1 be an i ncreasing sequence of subsets of S with E We want to prove that lim G ( E J. ) G( E) . E .. j-++oo j=l J Choose ql� E R , ql� � on E.J so that SUPfql�dW---"G(E.), K-++oo , J wEM �j E where we can assume that � Kj+ l < qljK ' j, K E m. We denote by �� the largest lower semicontinuous mi norant of l im �Kj . K-++oo =

m

00

u

=

- 20 We can assume that min {cp ml i I -> j , I + m < cp j as above so that

CP oj+1 ' j E ill ( for we can replace CP� by + K}) . Let € > 0 be given and choose where lim and so that K-+ +oo

j

E

00

Then'!' J. = Cp� + inf ( � cp€s , l ) E R , '1'J. < '1"J + 1 and'!' J. > s= 1 on E J. . Hence G( E ) � sup I l im 'I' J.d� = sup lim I'!' J. d� = �EM j-++CXl �EM j-+ oo = sup �im I ( CP6 i nf( � cp� , 1 ) ) d� � l im sup I CP 6d� + E . But s=1 �EM J-++oo j-++oo � EM since all functions ( CP Oj ) j=1 and are lower semicontinuous we have by i i ) -

+

+

00

sup ICP�d� = sup inf I CP�d� � inf sup I CP�d� � G ( E J. ) . K �EM �EM �EM K which proves the theorem . Hence G( E ) -< l- im G ( E J. ) j ++oo

+

E

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

Remark.

Notes and references Example 111 : 1 i s due to B . Fuglede, Capacity as a subl i near functional general i zing an i ntegral. Der Kongel ige Danske Vi denskabernes Selskab. Matematisk-fysiske Meddelelser . 3 8. 7 ( 1 97 1 ) . The exi stence of a G o -set A with properties 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 this context was observed by C . Del l acher i e , Ensembles

- 21 a na l y t i q u e s , c a pa c i t es , me s u re s d e H a u s d o r f f . Spr i ng e r LNM . 1

972 .

pg.

1 06 Ex.

T h e o r em

4.

1 1 1 : 1 i s a v a r i a nt o f a t h e o r em due to Choquet .

S e e t h e r e f e r e nc e s i n S e c t i on I I . Repre s e nt a t i o n o f s t r o ng l y s u badd i t i ve capac i t i e s b y mea s u r e s h a s b e e n s t ud i ed by Ber nd Anger , Repr e s e nta t i on o f c a pa c i t i e s . Math . A nn .

2 2 9 ( 19 7 7 ) , 2 4 5- 2 5 8 .

2 95 ,

III b

Outer Regularity

(Co nt.)

In this section , we continue our study of outer regularity but in a more special situation . Many problems in complex function theory are related to outer regular capac ities - in par ticular outer regularity of zero sets . We therefore proceed as follows . Let in what follows F be tive and lower semicontinuous funct ions compact and metric space U . h g inf { cp E F; g < cp}. H g sup{0i 0 continuous , 0 < g} E F for every where we assume that function g and that continuous Assumpt i ons .

a convex cone of posi( l.s . c . ) defined on a

==

-

==

LS

bounded and pos itive if g is .

Let 6 be a given probabi l i ty measure on U such that fh g d6 for all bounded positive functions gi we also assume that f cpd6 < �cp E F . +00,

Furthermore , we assume that i f {m.}� is an increasing 1 '1" 1 sequence of functions in F with lim < +00 then J ep.d6 . 1 1-*+00 lim ep.1 E F . . 1==

1-*+00

Observe that H h h g for all l . s . c . g and that g hep ep for al l ep E F . Note also that ep l , ep 2 E F implies =

Hcp

==

==

- 23 -

For i f 9 i s l.s.c . , choose g,"'g , g J. c ontinuous . J Then h g . H h < H h < hg . g J. 9 J Now h g . E F· h g . � so l im h g . E F and since lim hg . J J J J we get that h 9 l im h g . E F and that h 9 H h . =

,

>

9

=

=

9

J

The "f i ne" problem is now to decide i f E f+h X ( z ) i s a E capacity for every f ixed z E U ( X E i s the characteristic function for E ) . The "coarse" problem is a capacity.

1S

to dec1de i f E

f+

fh X ( z ) d o ( z ) E

Assumi ng all this about F , we defi ne a class of positive measures M , M = {w � 0; f�dW � f�dO , �� E F } . It i s clear that M is convex and since every function i n F is l . s . c . , M i s compact by Lemma 1:1. We now define c , c ( E ) sup w ( E ) which i s a capacity by Theorem 1:1, and the lJ E M connection with outer regulari ty i s that c outer regular i f and only i f E f h X d o i s a capacity ( cf . Propos ition 111:1 E below ) . =

f+

1)

2) 3)

4) 5)

We now turn to the study of the following statements . Every bounded function i n F i s a member of N . c i s outer regular . c ( E ) fh X d o for every Borel set E . E I f E i s a Borel set with c { E ) 0 then c* { E ) O . c { {h 9 > H h } ) = 0 for every pos itive and bounded function g . 9 =

=

=

- 24 Define for bounded functions g : sup J9dIJ. and L ( g ) J h g d o . IJEM

Lemma 111 : 3 .

c(g)

=

=

Then 1)

2) 3)

c(g) � L(g) . Equal ity holds in 1 ) i f 9 is upper or lower semicontinuous. L { g ) = i n f { L ( � ) ; 9 < � E l . s. c . } inf { L ( � ) ; 9 < � E F } . =

1 ) Assume 9 � o . Since J h g dO J H h d o , there is, by 9 Choquet 's lemma , � . > h 9 , � . E F, i E ill, a decreasing sequence of functions such that f � i d o J h g dO, i� oo . =

Proo f .

1

1

-

�

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

�

J � i dIJ � J � i dO which gives

IJEM

2 ) It is clear that the functional L has the following propertiel i ) L ( ag ) a L ( 9 ) , a > O. i i ) L ( gl+g 2 ) i L { g 1 ) + L ( g 2 ) · i i i ) If 0 � g 1 � g 2 then L { g 1 ) < L ( g 2 ) ' From i ) , ii ) and the Hahn-Banach theorem it follows that to every continuous function g there is a measure s such that =

J gds

= L

(g)

J �ds L ( � )


wher e

J

f n U j dd c

then

E:� O

and u s e t he

quas i cont i nu i ty we get the d e s i red c o n c l us i on .

e.

Compa r i son theorems Let

U

Lemma V : 2 .

f U Proo f . K,

u=v

f

CX)

MA ( U ,

. .

. ,U)

K,

G i ven

u \

on

and i f

u , vEPSHnL ( U )

If

udd c

�n .

be and open and bounded s u b s e t of

dd c u

=

f

U

MA ( v ,

compact i n K.

J

Then . A dd c u

U

x

•

•

•

u=v

near

au

then

,v)

U,

CX)

choo s e

X E C O ( U ) , X == c X dd u 1\ MA ( u , . . . , u ) =

J

1

uX " b y S tokes f o r mu l a . But u=v A .. supp dd C X s o the r i gh t h and s i d e equal vdd c X 1\ dd c v A c = X ( dd V ) n = x MA ( v , . . . , v ) .

=

f

f

Lemma V : 3 .

\ixE a U

f

If

then

f U

MA (

CX)

u , vEPSHnL ( U ) , u > 1 2 J dd c [ maX { log l z l , -109 4 ) ] 2 = J { ddCU K ) 2 = d ( U K ) . ( 10g 4 ) Note that i n view of the results i n Section V ,

=

'

and are outer regular capaciti es . We now turn to the problem of estimati ng the Monge-Ampere mass of plur i subharmonic functions defined outside a compact set . Let � be an open and bounded pseudoconvex set in [ n , n> 2 and K a compact subset of n so that n \ K i s connected . By Hartogs extension theorem , every analytic function on � \ K extends to � . When it comes to pluri subharmonic fun c tions the situation i s d i fferent . Assume that � E P S H n Looloc ( n \ K ) where K is a removable singularity set for the pluri subharmonic functions . Then J ( dd c � ) n < + oo when Kcc n ' c c � . r.l '\K Propos i t i on VI : 3 .

- 60 -

From Section V we know that ( dd c � ) n i s a well-defined positive measure on �\K. Since � extends to a pluri subharmonic function on � , we can choose a sequence � ]. E PSHnC near TI' such that Proo f .

00

Le t W= { zEn l i d ( z , K ) > �d ( a n , K ) } . Then 8=inf � ( z » - oo zEW and i f we put ¢ ].=sup ( � ]. , 8 ) then ¢ ].=� ] on W. I

.

So f ( dd c ¢ j ) n = f ( dd c � j ) n by Stokes theorem . nl nl I

= f ( dd c sup ( � , c ) ) n < + 00 since sup( � , 8 ) EPSHnL l oc ( � ) . nl Theorem VI : l . Let ¢EPSHnC 2 ( n ) and assume that n ' = { ¢< l } and that K= { ¢�s } for an s , O <s< l . I f �EPSHnLooloc ( �\K ) then 00

f I � I ( dd c � ) n- 1 ,\ dd c ¢
.;. n 2' n + "2 2 = so 2 1 w i 8 > 21 - 1 z 1 2 > 21 ( 1 - ( 1 +n ) - 2 ) = 21 2( n+n } 1 +n ) 2 ( 1 +n ) 2 Z,

- 63 -

Assume now that � < l z I 2 + 1 I w I 8 « ��) 2 and that 2 } In + �2 < w I 4 < 2 3 9 In + i . I 15 1 + n I } 40 1 + n

which is a well def ined domai n i f 2 r-2 - }2 I w l 8 ( l +n ) 2 4 0 Y �3 - 8 } 2 t- ) - ( 1 + n ) I w i 4 ) 3 +

+

d 1 /4 .!L 2 )

---=2---- � +oo , n� O

(n

where

/� :

n =

Furthermore , l im W( z ,w ) � l im W( z , p ) � l im z-+p z-+p z�p the proof of Proposition VII : l . =1

_

Z

so therefore v f ( z , w ) =O which completes p

- 69 Let

f : D�D ' be a ho l omorph i c map between two Dc� n , D ' c� n . Then

Theorem VI I : l .

open s e t s

WD ' ( f ( z ) , f ( w ) ) �W D ( z , w ) . Proo f .

I t i s c l ear that

2 -P S H ( DxD )

.

Fur the rmo r e

WD ' ( f ( z ) , f ( w »

15

nega t i ve and

WD , ( f ( z ) , f ( w ) ) �l og l f ( z ) - f ( w ) 1 -

- l og max [ d ( f ( z ) , C D ' ) , d ( f ( w ) , CD ' ) ] = l og I z -w 1 + l og

1 f ( z ) -f ( w ) I -

I z -w l - l og max [ d ( f ( z ) , CD ' ) , d ( f ( w ) , C D ' ) ] s o Wp , ( f { z ) , f ( w ) ) - l og { z -w ) i s l oca l l y bounded above near 6 . The r e fore WD , ( f ( z ) , f ( w ) �

� WD { z , w l . K l i mek a nd Dema i l l y have stud i ed the f o l l owi ng funct i on . u ( z , w ) =u D ( z , w l =sup { u ( z l E PSH ( D ) ; u�O , u ( z ) - l og l z -w l above near

bounded

w} .

I t i s c l ea r that Propos i t i on VII : 2 .

W O whi ch completes the proof of Proposition VII : 2 . Let D be a domain in � n . I f ( z , w ) EDxD we De f i n i t i on . define the Caratheodory d i stance ....

C D ( Z , W ) =Sup { P ( F ( z ) , F ( W ) ) ; F : D F

....

U,

F holomorphi c }

- 71 and t h e Kobayas h i d i stance m K ( z ,w)=inf{ L 0 ( 2 , , 2 ' - 1 ) ; D j=l D J J whe r e

0

D

( z , w ) = i n f { p ( � , n ) ; 3 f : U4 D

f h o l omorph i c }

whe r e

U

z = z ' z =w } m 0

with

f ( � ) =z ,

i s the u n i t d i s c i n

i s t h e hyp e r bo l i c d i s ta nc e o n

�

f ( n ) =w , and where

U.

p

Propo s i t i on VI I : 3 . l og t a n h C ( 2 , w ) �W ( z , w ) �u ( 2 , w ) �1 0 g t a nh o ( z , w ) .

If

D

i s con vex , then equa l i t y h o l d s . We have that

Proo f .

l og tanh C ( Z ' W ) = s U P { l Og

l f1 -( Zf )( -z f) �f W( w) ) I ;

f : D4 U ,

f

h o l omorph i c }

s o by Cor o l l a r y V I I : 2 t he f i r s t i n equa l i t y f o l l ows . The s econd i s c l ea r a nd the t h i rd f o l l ows f r om Theorem VI I : 1 s i n ce u = l og t a nh ( p ) . U that

If

D

i s convex ,

i t i s a r e s u l t o f Lemp e r t

C=o . Wh en

n= 1 ,

we have f o r e v e r y compa ct s e t

KCU ,

the u n i t

d i sc

whe r e

u

When

i s a u n i qu e l y d e t e rm i ned p o s i t i ve mea s ur e on n) 1 ,

t h i s i s no l on g e r t r u e s i n ce t h e r e a r e comp a c t

p l u r i po l a r s e t s s uppor t i ng p o s i t i ve mea s u r e s s u ch that

fW ( z , � ) d U ( � )

K.

i s bounded o n

D.

- 72 -

Note s and r e f e re n c e s The f u n c t i on

u

i s i nt r oduced by

M.

Kl i mek i n t h e a r t i c l e

Ext r ema l p l u r i s ubha r mon i c f u n c t i on s a nd p s e udod i s t an c e s . Bu l l . Soc . Math . F r a n c e

1 1 3 ( 1 98 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 ha rmon i qu e by J . -P . Dema i l l y , Ma t h .

c o n t i nued ,

Z .

1 94 ( 1 9 8 7 ) , 5 1 9-56 4 ,

t h i s s t udy i s

i n c l ud i ng a r e p r e s e n t a t i on formu l a and expl i c i t

examp l e s . Th a t

CO= K O

f o r convex s e t s

in A n a l y s i s Mathemat i c a

8 ( 1 9 82 ) .

0

wa s p r oved by L . Lempert

VIII The Global Extremal Function L

D e n o t e by on

er

n

t h e c l a s s o f p l u r i s u b h ar mo n i c f u n c t i on s

s uc h that

where

i s a c o n s t a n t ( de p e n d i n g o n V ( z ) = s up { f ( z ) ; E

and

V* E

fEL ;

f� O

on

V* E L E

If

VE E L

�

E

If

Ecer

n

we de f i n e

E}

i s not p l u r i po l a r .

t h e n the f u n c t i on i s c a l l ed t h e g l oba l extremal

p l u r i s u bh a r mo n i c f u nc t i on r e l a t i ve l y to A s s ume t h a t

Theor em VI I I : l .

E

a b s o l u t e l y c on t i n u ou s . Let

uEL

n c ( dd h * ) E

a r e mu t u a 1 1 y

( S ee V I f o r t h e de f i n i t i on o f

h ' ) E

and de f i ne

Y (u) = r a

E.

i s a r e l a t i ve l y compa c t s u b s et

o f the u n i t b a l l . Then

The n

f) .

t o be the s ma l l e s t upper s em i c o n t i nuous ma j or a n t o f

Propos i t i on V I I I : l .

whe r e

f

f

u ( rw ) d a ( w ) - l og r

I w l =l

i s the norma l i z ed L e b e s gue mea s u r e o n the un i t sphere . i s convex a n d b o u nded a t

c r ea s i n g a n d we d e n o t e by

y(u)

+ 00

the l i m i t

so

Y

r

(u)

lim Y ( u ) . r r-+ +oo

i s de If

- 74 furthermore u�log+ l z l

then

There i s a c o n s t a n t

Lemma VI I I : l .

Propos i t i on VI I I : 2 .

If

I uddClog + I z I II I z I 0

be g iven .

- 82 If

Theorem IX : l .

f

is

{Xi

p-c apac i ta b l e t h e n

f(x»

s}

is

p - c apa c i t a b l e . . A s s ume t ha t

Proof .

f

is

p - c apac i t a bl e . T h e n t h e r e i s a n i n ­

{ f n } �= l

c r ea s i n g s equence

o f upper sem i - c o n t i nuous f u n c t i o n s

wh i ch a r e sma l l e r o r equa l t o

f

with

I f n dp = I f dp .

l im n-++oo

I t i s n o r e s t r i ct i on t o a s s ume t h a t e v e r y

f

n

h a s comp a c t

suppo r t . I t i s e a s y t o see t h a t p( {Xi

P ut of

E

m,n

{Xi

={x;

f(x» s} )

f ( x ) >s n

f(x» s}

and

+

= p( {Xi

1}. m

Every

00

u

l n= l

so i t f o l l ows f rom Lemma I X : 1

that

c i t a b l e a nd

n

{Xi

Theorem IX : 2 .

f (x» s}

A s s ume t h a t

c i ty . D e note by

B 1 ( f ) =inf (

E

n

(x»

min

s} )

\:1 s > O .

i s a comp a c t s u b s e t

00

l im f ( x » s } = u n -+ +oo m=

{Xi

f

l im n -+ +oo

h a s to be c

E

min

f

(x» s} l im n n-++oo p - c apac i ta b l e .

{Xi

i s a s t ro ng l y s u ba dd i t i ve capa-

t he c h a r a c t e r i s t i c f u n c t i on of n

A ) l: a 1. c ( 1· i i=l

i s p - c apa-

n

f) . l: a i X A . � .= 1 l 1

A.

Put

- 83 -

Then Assume that c is a capacity as in Theorem I X : 2 . Then the Choquet i ntegral is subadditive ( and therefore a seminorm on the non-negative functions ) .

Corol lary IX : l .

Corollary I X : 2 .

IX : 2 .

Assume that c i s a capacity as i n Theorem

Then

f fd c

= inf

f gd c ; f s } ) ds = E v-+ + oo V 00 U

v= l

fC ( { xE U ;

L 00

( x » s } ) ds

U

0

v= l

E

=

v

E) .

K ' v E lli , v

be a d e c r ea s i ng s eq u e n c e o f c omp a c t s u b s et s

Then 00 inf C ( K ) = l im v v-+ +00 v E IN

fC ( {XEU 0

L

K

00 = l im v-++ oo

v

( x ) > s } ) ds =

00

fC ( {xEU ; LK 0

v

( x ) ::. s } ) d s =

( x ) ::.s } ) d s = i m c ( { xE U ; L f vl -++oo K V

0

00 =

fC ( {X EU ; o

00

(x»

L 00 n K

v= l

v

s } ) ds = C (

n K

v= l

v

) .

The l a s t s ta t em e n t i n t h e t h e o r em f o l l ows f rom Coro l l a r y I X : 1 . Theorem IX : 4 .

Let

(L ) E E CV

be a swa r m . T h e n

L

E

i s a u n i ve r ­

s a l l y capa c i t a b l e f u nc t i on f o r e v e r y u n i ve r s a l l y c a pa c i t a b l e set

EE P ( V ) .

- 85 -

S i nc e ,

Proo f .

eve r y capa c i ty set

c,

S i nc e

L

is

E

f

L d E c

i s a capa c i ty f o r

w e h ave f o r a n y u n i ve r s a l l y capa c i t a b l e

=

C( E) =

s up C(K) = s up KCE KCE K compa c t K c ompa c t

i s uppe r s em i - c on t i n u o u s a n d l e s s o r equa l t o

K

c - c a p a c i ta b l e . But

c i ty s o i t f o l l ows that Theorem IX : 5 .

L { x ) =c ( { yE V ; E

in

L

c

wa s a n a r b i t r a r i l y c h o s e n capa­

c

i s a c apa c i ty o n

( x , y ) E E } ) , ECU x V L (X) E

L , E

i s u n i ve r s a l l y capac i ta b l e .

E

A s s ume t h a t

i s s ubadd i t i ve t h e n

V.

Then

i s a swarm . Furthermore ,

if

c

i s s u b a dd i t i ve f o r eve r y f i xed

x

U. a ) c l ea r .

Proo f . 8)

C(E)=

E

f L Ed c L

IX : 3 ,

by Theorem

L e t a c omp a c t s u b s e t

that

L

K

of

Ux v

be g i ve n .

I t i s c lear

h a s comp a c t s uppo r t so i t rema i n s to p rove t h a t

K

1 S upper s em i con t i nuou s .

We have t o p r ove t h a t

L ( x ) �a . K O

x � x ' n�+oo . O n

such that

G i ve n

Put

and o

n

= { yEV ;

( x , y ) EK} . n

I t i s e a s i l y v e r i f i ed t h a t

a>O

and

Choose

L

x E { x E U ; L ( x ) �a } . O K x

n

with

L ( x ) �a K n

K

- 86 -

co

co D :> n ( O

u D . ) .

i= , j = i J

S i nce

c

i s a capa c i ty we h ave co

co

co = l im c ( u D . ) i�+co j=i J

> u.

m

L (X ) > l im c ( D . ) = l i K j j�+ co j�+co

J

The l a s t s t a t eme n t i n the t he o r em i s obv i ou s and the p r o o f i s comp l ete . De f i n i t i on ( Pr oduct Capa c i ty ) .

on

U

UxV

a nd

V

Let

c

and

d

be c a pa c i t i e s

r e spect i ve l y . The p r od u c t c a pa c i ty

cxd

on

i s de f i ned by

where L ( X ) =d ( { yE V : E By Theorems if

c

IX : 5

( x , y ) EE } ) .

IX : 3 ,

and

c xd

i s a c a p a c i ty . F u r t h e rmor e ,

i s s t r on g l y s ubadd i t i ve and

IX : 2

f o l l ows f rom Theorem Exampl e I .

Let

c i ty . Co n s i d e r i n

Im z , = O } .

U=V=� �

2

that

cxd

the s e t

i s s ubadd i t i ve ,

it

i s s ubadd i t i ve .

a n d d e n ot e by

c

t h e Newton i an c ap a ­

E={ ( z " z 2 ) ; I z , I + l z 2 1 = ' ,

I t i s eas i l y seen that

cha nge t h e v a r i a b l e s

d

a nd

E ' = { ( Z l , z 2 ) ; I z , I + l z 2 1 = 1 : Im z 2 = O } L , ( z } = O , \:f z l E� , so cx c ( E ' } =O . E 1

cxc ( E » O .

But i f we i n te r -

i . e . con s i der the set i t i s c l ear that

- 87 -

Let now on

U.

U

be a n ope n s u b s et o f

on

c =c 1

and

c

a c a pa c i ty

b y i nd u c t i on

on

We c o n s t r u c t

�

U

c =c x c n- 1 n I t i s c l ea r t h a t s u b a dd i t i ve , t h e n

By T h e o r em I X : 5 ,

n U ,

i s a c ap a c i t y o n c

(L

n

if

i s strongly

n n-p EcU , x E U

i s s ubadd i t i ve . Put f o r

n-p ) E n Ec U

c

i s a swarm .

I t f o l l ow s f rom T h e o r em I X : 4 a n d Theor em I X : l t h a t

Remark .

�

n p -P { xE U - ; L (x» s}

i s u n i ve r s a l l y c a pa c i t a b l e f or e v e r y

a n d e v e r y u n i ve r s a l l y c a pa c i t a b l e s e t Let

c

E.

b e a capa c i ty o n a n ope n s u b s e t

d e f i n e a p r e c a pa c i ty =c

on

U

of

on ( x , y ) EE } »

n O } ) , ECU .

1 ) P ( E ) =O � c ( E ) = O , n n 1 L (X» O} ) , E n i s a p r e c a pa c i ty o n U ,

Theorem IX : 6 .

2) 3)

P

4)

e v e r y u n i ve r s a l l y c a pa c i t a b l e i s

5)

if

n

c

�.

b y i nd u c t i on

. n l P ( E ) =c ( { xE U ; P _ ( { y E U - ; n 1 n

i s s u b a dd i t i ve , t h e

P

n

s>O

P - c a p a c i ta b l e , n i s s ubadd i t i ve .

We

- 88 1 ) , 2 ) i nd u c t i o n . n = l c l ea r . A s s ume t h a t 1 ) a n d 2 )

Proo f .

ho l d for n - 1 .

Prove 1 ) a n d 2 )

P ( E ) =C ( { x E U ; n

n.

for

P n _ l ( {yEU n- l ;

= c ( { xEU ; c _ ( { yEU n 1

n- l

;

( x , y ) EE } » O }

=

( x , y ) EE} » O } ) =

= c ( { xE U Thus

and i t i s c l e a r t h a t

1 c { xEU ; L ( x » E

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

4)

L�

i s a swarm .

i s a c apac i ty a nd

A s s ume that

E

i s u n i v e r s a l l y c a p a c i ta b l e . By Theorem

IX : 4 ,

i s u n i ve r s a l l y capac i t a b l e a n d

wher e of

c

E .

K v ' vErn,

i s a n i nc r e a s i ng s e q u e n c e o f compact s u b s e t s

Hence

�

c ( { xEU ; L ( X » =

S } ) = c ( { xE U ;

l i m c { x E U ; LK ( x » V v-++ro

s} )

l im

L�

v-++oo v

(x»

s} )

=

- 89 for a l l

s�O ,

so

P ( E ) = l i m P ( K ) = s u p { P ( K ) i K compa c t , Ke E } n n \I

wh i c h mean s t h a t

5 ) I nduc t i on .

E

n= l

v� +oo

is

P - c ap a c i t a b l e . n

c l e a r . A s s ume t h a t

P

i s s ubadd i t i ve .

n- 1

n- l P ( E l u E ) =c ( { x E U i P _ ( { y E U ; ( x , y ) E E UE2 } » 0 ) = 2 n 1 n 1 n- l n- 1 = C ( { XE U i P _ ( { y E U i ( X , y ) E E } U { yE U i ( X , y ) EE } >0 ) } < 2 1 n 1 n- l < C ( { XE U i P _ ( { yEU ; ( x , y ) EE } » 0 } U n 1 1 Then

U

{ xE U ; P _ ( { yEU n 1

f o l l ows t h a t

P

prope r t y .

�

( x , y ) EE } » 2

i

A s s ume t h a t 2 c ( u E ) =0 , \I \1= 1

c

i s a capac i ty o n

then

c

A s t h e p r o o f o f T h e o r em I X : 6 ,

Proo f .

Theorem I X : 7 .

and i t

0 } ) �P ( E 1 ) + P ( E 2 ) n n

i s s ubadd i t i ve .

n

Corol lary I X : 3 .

C ( E ) = O , \1 = 1 , 2 \I

n- 1

Let

and

n

P

If

h a s the same

n

2.

be a ( pr e ) c ap a c i ty o n

c

U.

V

and

a comp l et e n o r ma l f am i l y o f c o n t i n u o u s f u n c t i o n s ,

(a ) i iEI

a . : U-+V . 1

The n C ( E ) =SUp c ( a . ( E ) ) 1 iEI i s a ( p r e ) c a p a c i ty o n

U.

If

c

1 S s ubadd i t i ve , then

C

is

subadd i t i ve .

i ) , i i ) c l ear .

Proo f .

i i i ) Let G i ve n then

E ' \l Eill , \I

£>0 .

Choose

such that

b e a n i nc r e a s i ng s e q u e n c e o f s u b s e t s o f 1

a.

such that

£

c(a.

1(

1(

( E ) ) n ( u a n ( K n ) ) . . n= 1 1. = 1 n = 1

G i ve n 00

z E

n

. 1= 1

00

u a (K ) ) . n . n n=1

U.

We c l a im that

- 91 Then 00

z E

Choose Then

z � E a. z l = a. 00

x Q,�x E

so 00

n

1. = 1

U

0.

n n = l.

(K ) n

(K � ) n( �) n( )

( x� } n(l)

n K

v v= 1

'

z =o. ( x ) O

\:l i E N .

� � + oo .

whe r e

where

n( �»

x EK � l n( }

�

s u ch that

z �� z ,

�� +oo .

and we c a n a s s ume t h a t

Now

and s i nce

z

w a s a n a r b i t r a ry e l ement i n

00

u a. n ( K

n = 1.

n

o. o (

} )

we have proved t h a t

00

00

00

n K } :::;) u 0. ( K ) ) . n n n 1 n = i i= n= 1 (

n

To f i n i s h t h e p r o o f , we c a n n ow a r gue a s i n the end o f t he proo f o f T h e o r em

IX : 5 .

Ron k i n ' s g amma - ca pa c i t y a n d Favorov ' s c apa c i ty Denote by

c aP

2

and

loga r i t hm i c c apac i ty o n a s f o l l ows

the i n t e r i or and ext e r i or �,

r e spect i ve l y .

i s t he n de f i ned

- 92 -

Ronk i n ' s g amma - capa c i t y i s t h e n b y de f i n i t i on r

n

( E ) = s up { y

n

(a(E) ) ;

y

Propos i t ion IX : l . E,

sets Proo f .

whe r e

P

n

n

c omp l e x u n i t a r y t r a n s f o rmat i on } .

a

f o r a l l u n i ve r s a l l y capa c i ta b l e

( E ) =P ( E )

n

I nd u c t i on . C l ea r l y P r opo s i t i on

A s s ume i t � s t r u e f o r A s s ume t h a t

E

I t i s clear that

caP 2 .

i s f o rmed w i th r e s p e c t t o

n- 1 .

I X : 1 i s true for

P r ov e i t f or

n= l .

n.

i s u n i ve r s a l l y c a pa c i t a b l e .

{ z E cr

n- 1

;{z

l

� s u n i ve r s a l l y c a pa c i t a b l e .

, z ) EE}

Hence

for a l l

z E cr . l

Then

= ( Theorem I X : 6 , ( by d e f i n i t i on )

= ( Rema r k p . 6 6 )

1))

= c a P { { z E cr ; c _

2

l

n 1

( { z E cr

n- 1

;(z

l

, z ) EE } »

O} ) =

- 93 -

a n d t h e p r opos i t i on i s prove d . i s u n i ve r s a l l y c a pa c i ta b l e t h e n

Coro l l ary IX : 3 .

If

E

Coro l l ar y IX : 4 .

By C o r o l l a r y I X : 3 ,

r

n

{E

1

) =r

n

{ E ) =0 2

i mp l i e s

that

Eve ry u n i ve r s a l l y c a p a c i ta b l e s e t i s

Propos i t ion IX : 2 . r - c apa c i ta b l e .

n

Proo f . A s s ume t h a t

i s u n i ve r s a l l y capa c i t a b l e and l et

E

be g i v e n . Choo s e a c omp l ex u n i t a r y t r an s f ormat i on r

n

(E) < y

n

a

(>0

such that

( a { E ) ) + (/2 .

i s u n i ve r s a l l y capa c i ta b l e s o b y Theorem I X : 6 t h e r e i s a

a{E)

comp a c t s u b s e t

r

Thus

n

set o f

{ E ) -00

wh i ch p r ove s t h e p r o po s i t i o n .

n The f o l l ow i n g examp l e s hows t h a t t h e r e a r e n o n - � -po l a r s e t s o f z e r o gamma - c ap ac i ty . Exampl e 2 .

r (H» O. n

Then

Put Denote by

It i s c lear that g

t h e b i h o l omo r p h i c m a p

- 95 -

Any complex l ine cuts g ( H ) i n at most four points so Now , by Propos ition IX : 2 , H i s not � 2 -polar and s i nce g i s biholomorphic , g ( H ) cannot be � 2 -polar . Hence , there are non-� 2 -polar subsets of � 2 with vanishing r 2 -capac ity . To get a capacity with null sets i nvariant under holomorphic mappings we proceed as fol lows . Let B denote an open and bounded subset of by An the holomorphi c mappi ngs f , f : B n�B n . h n ( E ) =sup r�( f ( E » , Def i n i ti o n . fEA n

[.

Denote

Theorem I X : 8 .

i ) ECB n ==> h n ( E » -h n ( f ( E » , fEA n , vani shes on � n -polar subsets of follows from the definition of h n . Observe that thi s means that h n i s i nvariant under biholomorphic mappings of B n onto i tself . It follows i i } Assume that N is a � n - polar subset of Let now fEA n be given . from Proposition IX : 2 that We have to prove that Proo f .

i)

r� ( f ( N } ) = O . Denote by T ( f ) the Jacobian of f . It i s clear. that

- 96 -

s o by Coro l l a r y I X : 5 i t r ema i n s t o p r ov e that r ( f ( Nn { l ( f ) =O} ) ) =O . n

Th i s f o l l ows f rom Coro l l ar y I X : 6 be l ow .

A s ub s e t

Def i n it ion .

of

E

w

ana l yt i c s e t i f f o r every °

w

w

of

Let

Theorem IX : 9 . F=

( f1

'

•

•

•

Eno

s u c h that

,f ) n

U

w

[

n

in

i s c a l l ed a ( proper ) l oc a l l y

th e re

E

i s a n e i g h bo rhood

i s a ( proper ) a n a l yt i c s et i n

be a n open s u b s e t o f

a h o l omorph i c map

F:

U-+ [

n

[

n

w

.

and

Then

.

o

F ( { t ( F ) =0 } )

i s cont a i n ed i n a d enume r a b l e u n i on o f p r op e r l o c a l l y a n a l yt i c

sets . Put

Proof .

P

J= { Z E U i l ( F ) = O } .

We c l a i m that

d i m < m ==) F ( p n J )

a n a l yt i c o f

i s c o n t a i n ed i n a denume r a b l e u n i o n o f p r oper l oc a l l y a n a l yt i c sets . Th i s i s c l e a r l y t ru e f o r

f or an a n a l yt i c s e t o f

m= O ,

for

m- l

a s s ume that

Choo s e

G=de t

P

we have to p r ov e i t f o r

( aa fz �. p J lq

PnJ

( a f iP az

t o

.

m dim P=m.

i f the s t a t e e n t i s t r ue

z e r o d i me n s i on i s d e nume r a b l e . Now ,

w i th

We c a n

i s connected .

lq

)

,

p= 1 , . . . , s ,s q= 1

where

•

•

w i th

•

s =max r a n k

(::� J

PnJ

)

.

.

.

1 , J= 1 ,

.

.

•

,n

- 97 -

Put

Q= ( J n P )

note s b e l ow ) ,

reg

n { G� O } .

By t h e r em a r k i n Remme r t ( s e e t h e

i s c o n t a i ne d i n a d e n ume r a b l e u n i on o f

F(Q)

pr oper l oc a l ly a n a l yt i c s e t s . d i me n s i on

<m- 1

1 Q = ( Jn p )

s o , by a s s umpt i o n ,

reg

n { G= O }

F(Q' )

is of

i s con t a i ned i n a

d e n ume r ab 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 . d im ( JnP )

F u r t h e r mo r e , s i nc e for

F ( Jn P )

) . s I. ng

< m- 1 , . s l ng-

t h e s ame i s t r u e

S i nc e

F ( Jn P ) CF « JnP )

) uF ( Q ) uF ( Q ' ) s I. ng

the s t a teme n t i s prove d . To p r ove the t h e o r em , Corol lary IX : 6 .

in

a:

n

and i f

n a: - po l a r i n Proo f .

If

N

F : U-+ a:

i t i s e nough t o choo s e 2 a: - po l a r i n

is n

U,

P=U .

whe r e

i s a n a n a l yt i c map , t h e n

U

i s ope n

F(N)

is

n a: .

I t i s c l ea r t ha t

F ( Nn h ( F ) �O } )

Theorem I X : 9 i t f o l l ow s t h a t

is

F ( Nn h ( F ) = O } )

s i nc e p r o p e r l oc a l l y a n a l yt i c s e t s a r e

n a: - p o l a r . From is

n a: - po l a r ,

n a: - p o l a r .

Not e s a n d r e f e r e n c e s T h e o r em I X : 2 i s d u e t o

F . Tops¢e , On c on s t ru c t i on o f

mea s ur e s . K¢be nh avn s U n i ve r s i t e t , Mat .

I n s t . P r e p r i nt s er i e s

1 974 : 27 . Corol lary IX : 1 Ann .

i s due t o G . Choquet , Theory o f c a p ac i t i e s .

I n s t . Fou r i e r 5 ( 1 9 5 3 - 5 4 ) .

- 98 -

The notat i on o f swa r m i s c l os e l y r e l a t ed t o that o f � noyau capac i t a i r e r eg u l i e r � as d e f i ned i n C . D e l l acher i e , E n s emb l e s a n a l y t i que s . Capac i te s . Me s u r e s d e H a u s d o r f f . S pr i ng e r LNM .

295

( 1 972 ) .

i s d u e to V . S e i now ( s e e Ronk i n s book b e l ow ) . v

Ex amp l e 1

Examp l e 2 i s due t o C . O . K i selman . Ma n u s c r i pt . Upp s a l a 1 9 7 3 . The g amma capa c i t y was i nt r oduced i n L . I . Rank i n ,

I n tro­

duct i on to t h e t h e o r y o f e n t i r e f u nc t i o n s o f s eve r a l v a r i a b l e s . Ame r . Math . Soc . P r ov i de n c e . R . I .

1 97 4 ,

a nd the mod i f i ed g amma

c a pa c i ty i s i n S . Ju . Favorov , On c a pa c i t y c h a r a c t e r i z a t i o n s o f n sets i n � . C h a r kov 1 9 7 4 ( Ru s s i a n ) . The r ema rk by Remme r t , u s ed i n the p r o o f o f Theorem I X : 9 i s i n R . Remmert , H o l omorphe u n d me r omorphe Abb i l du n g e n komp l ex e r Raume . Ma t h . An n .

1 33

( 1 957 ) .

The s e t f u n c t i on s

Y

a nd r has been used i n connec­ n n t i o n w i th r emov a b l e s i ng u l a r i ty s e t s ; c f . U . Cegre l l , Remova b l e s i ng u l a r i t y s e t s f o r a n a l yt i c f un c t i o n s h a v i ng modu l u s w i t h bounded Lap l a ce ma s s . P r oc . Ame r . Ma t h . Soc . Vo l . P . Jarvi , Remov a b l e s i n g u l a r i t i e s f o r

P r oc . Ame r . Ma t h . Soc . Vol .

86

88

( 1 983 ) .

H P - f u n c t i on s .

( 1 982 ) .

J . R i i hentaus , An e x t e n s i o n t h e o r em for me r omo r ph i c f u n c ­

t i on s o f s e ve r a l va r i ab l e s . A n n . Acad . S c . Vo l .

4

Fenn . S e r e AI .

( 1 978/79 ) .

Some o f the ma t e r i a l o f t h i s s e c t i o n h a s b e e n p u b l i s h e d i n s em i na i r e P i e r r e Le l ong Spr i n g e r LNM 8 2 2 .

1 980 .

H e n r i Skoda ( An a l y s e )

1 9 78/7 9 .

X Capacities on the Boundary Let

be an ope n , bounded a n d c o n n e c t ed s ub s et of

U

con t a i n i ng z e r o . The n

H(U)

a n a l yt i c f u n c t i o n s o n

U

in

oo (H (U) )

and

A( U)

t i ve me a s u r e o n c l o s u r e of For

au

con s i s t s o f the f u n c t i on s

A(U) z EU

in

U.

P H ( W , aU )

we de f i n e

n

i s t h e c l a s s o f ( bounded )

t h a t extends c o n t i n uou s l y t o

H(U)

�

If

w

i s a pos i ­

( 1 �p < +00 )

t o be the

LP ( w , a U ) .

we d e f i ne t h e c l a s s e s o f proba b i l i ty me a s u r e s

M = { w>O ; z

f

f { z ) = fdW ,

�fEA( U ) ,

s upp w c a u }

and

( PSH

and

C

a r e t he p l u r i s u bharmon i c f u n c t i o n s a nd the

con t i n u o u s f u n c t i o n s r e spec t i ve l y . ) We wr i t e

f o r t h e u n i t ba l l i n

B

norma l i z ed L e b e s g u e me a s u r e o n

aB o

�

n

Then

a

a nd

i s the

f E HP ( a , a B )

i f a nd

on l y i f

sup O

If(z) I

- 1 09 �

1

im r� l

s o Coro l l a r y X : 1 proves that comp l e t e s the proo f .

1 f ( r t;; ) 1 ( 0 ) � I I f II A(B)

I n one var i ab l e , the f u nc t i on s i n

LH

wh i c h

1

Hl

are den s e I n

( take d i l atat i on s ) . Example .

We are go i ng to c o n struct a bounded a n a l yt i c f u n c t i on

f

2

on

Bc�

such that

1)

l i m f ( r t;; )

2)

f ( r t;; )

Let

t;; K =

ex i st s

� t;; E S .

do not converge to

(� ,

/1

- �2 ) ,

K E JN.

f

in

Then

LH 1 . { e i G t;; K ' G E m }

a r e c l o s ed

and d i s j o i nt sets s o we c a n choose open d i s j o i nt sets iG B where V k conta i ns { e � K ' G ElR} . For each

K

choos e

nK

at mos t one Let then

n < z , t;; > K K K= l K.

f(z)=

The n

0 -00

X:6

we c a n t o eve r y

00

out s i de a s e t ( on a B ) o f van i sh i ng

Q - c apa c i ty .

So we put f* ( z }

t: E a B ,

-

1 13 -

we get a function defined outside a set of vanishing Q-capacity . Furthermore , l im sup f l f* ( � ) -f ( r� ) I d w = O . 1 \.lEM O We now use this last property to prove a result related to inner functions . Observe we do not assume I f I to be bounded . r-+

that

( n 2 ) Assume that fEALH 1 ( B ) and that tI\.lEM O · Then f ::: constant .

Theorem X : 7 .

f I f * I dw = 1 , Proo f .

If

>

\.l = a

we get that

fQ ( Z ' � ) I f* ( S l l d a ( � ) 1 0

=

1

where Q is the classical Poi sson kernel . To prove that f ::: const . it is enough to prove that I f ( 0 ) I = 1 because forces I f I to be harmonic a nd t h e r e f o r e constant .

that

Choose for O

l im f l f* l dW r r-+ 1

l im f l f* ( S l -f ( rU l d\.l r ( � ) = r-+ 1 since the last term vanishes by remark following Theorem

X:6.

- 1 14 -

We now return to M O ; we have seen that to every function fEALH 1 there i s a "boundary value function" f* so that 1 ) f ( O ) =Jf*d\.l , 'd\.lEM O J l f* ( � ) -f ( r U l d\.l=o . 2 ) lr/1im 1 sup I.l EM O The above example shows that 2 ) need not hold i f ALH l i s replaced by H <Xl ( B ) . We do not know i f 1 ) i s true for H <Xl ( B ) . But f* i s wel l defi ned for every fixed \.lEM a ! complex measure on S i s cal led an A-measure i f for every sequence , L] EA( B ) , I f J. ( z ) I -<M , 'djEJN, 'dzEB with l im f ]. ( z ) = O tl zEB it follows that l im .ff J. d\.l = O . A

\.l

If i s an A-measure and i f f ]. EA ( B ) , I f J. ( z ) I �M , 'd z EB , jEJN such that l im f J. ( z ) 3 t1 z E B then f J.d\.l i s j -++<Xl O weakly convergent ( i . e . l imJ�f j d\.l 3 t1�EC ( S ) ) .

'rheorem

X: 8 .

In particular , it foll ows from Theorem X : 8 that i f \.lEM O ' fEH<Xl ( B ) then f ( r� ) d\.l ( � ) I S weakly convergent so there I S a function ( determined I.l-a . e . ) f *E L<Xl ( S ) such that I.l

Let be a probabi l ity-measure on S such that there i s a constant c with

Theorem X : 9 .

sup f l f ( r U I 2 d \.l ( � ) � cJ l f ( t;; ) 1 2 d \.l ( t;; ) , tl fEA ( B ) . O