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
<
�
Given
>
-
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
(
<
E M
for
+
is
M (
lJ . )
J
. 1. J=
00
J(CP-X K)dlJ
j E ill
fCPdW
For every
O.
>
Since
E.
so that
there is a
+
E
2E
<
E
so that
<
W(K)
+
3s
- 3 -
Corollary
on
n
UCR ,
s up IJ * ( E ) IJEM Proof.
1:1. U
If
O<�
i s a l ower s em i c o n t i nuous f u n c t i o n
c ompac t , and i f
i s a capac i ty .
( � * dlJ
M= { IJ� O ;
O .s. � * d lJ� 1 }
then
s tand s f o r c o n volut i o n . )
It i s c l ea r f r om L emma 1:1
t hat
M
i s weak * - c ompact .
II
Capacitability
Let
c
be a capacity.
A set
De f i n i t i o n .
if
E
is
is said to be
sup{c(K);
c(E)
A set
Def i n i t i on .
E
E�K,
c-capacitable if
compact}.
is said to be universally capacitable
E
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
G o
consists of
a-compact Hausdorff
space.
the sets that can be wrjtten as a denumerable
intersection of open sets. Fa
consists of union of
the sets that can be written as a denumerable
compact sets.
consists of the sets that can be written as a denumerabl e Ko o intersection of sets from
A set
E
in a Hausdorff space is called F
there is a
Ko -set o
map
such that
f: F�E
Fa.
in a compact space E
=
f( F) .
W
A complete,
K-analytic if and a continuous separable and
metric space is called POlIS h.
Finally, analytic set.
a continuous image of a Polish space is called an
-
Proposition 11:1. space
U,
E
If
Proof.
is a
then there is a
and a continuous map
Put
r
=
K
=
rn(FxU)
is a
f. K
oo
WxU
-set so is
E.
f: F�E
where
K-analytic.
Since
=
f(F)
with
r
Then
E
�
proj
r
r
Thus
and
FxU
is closed in
(�rnFxU).
W�F
and
is compact so is r
W
in a compact set
F
WxU) since
(closure in
FxU
K-analytic set in a compact
{(x,f(x)), x E F}
the continuity of since
-
-set oo
W�U
f:
is as in the definition of r
5
by
and
proj:
r�E
is the required function.
Theorem 11:1. a)
Every Hausdorff space that is a continuous image of K-analytic set is
b)
K-analytic.
Denumerable unions and intersections of are
K-analytic sets
K-anal ytic.
Theorem 11:2 a)
Every Polish space is homeomorphic to a
G -set contained o
in a compact metric space. b)
Every Borel set in a Polish space is
Proof (of
K-analytic.
11:1).
Theorem
a)
is clear because of the transitivity of continuity.
b)
Let
F �B n
space let
F
F
n
f n
�
E
and
where f
n
(B
n
B
)
=
n
is a X cE n ·
K
o �u
-set in the compact
Consider
F a
be a compact space containing
f or a moment the topological sum
E F
n
=
F a (E
=
F
L
F n
n
uF x{n}). n
and
denotes Let
- 6 -
B
� B n
=
f
Then more
B
=
n
=
u
l
is a
K
oo
B-+E
where =
u
F
so
UX n n
It remains to prove that
C
the product space and let b
section of the cylinders where
each
i,
is
n
K
=
=
is
of
o
K-analytic.
C
The set where
C.
hence so is
oo
Further-
n
Therefore
K-analytic.
F
of
n
uX
n(uB .x{n}). n,l i n
is
nX n n ITB . n
K
=
b
F
Let
n
B x IT F p n prfn
n
Furthermore, we Every
C.
is then defined and continuous on
ITB
=
is the inter-
the canonical extension of
f n
denote by f n
b
feB)
is a
n B .x{n} n,l i
n
of
B . n, l
if
f (x) n
f(x) =
by
is continuous with
. nB . n, l
B x{n} n
n B
f:
and define
Then for every pair
j E C:
{z
f.(z) J
is a closed subset of
B
is a
.
=
n
.
l, J
{z E C:
K -set in oo Define now
=
f. (z)} l
C.
C
Since
f.(z) l
=
n
for every point
n
(x ) n
f
to be the restriction to
there is an
E
B
it follows that
oo
F.
x
n
and therefore
which proves the theorem.
E
B n
B
of
f . n
f (B)cX , �n E ill n n
On the other hand, nX cf(B). n
.
K
f.(z)} J
restriction is continuous and since f(B)c nX
is
For let
such that
f(x ) n
=
y.
y
E
f (x ) n n
Therefore
This
we have n
X , n y.
A
=
The f(B)
- 7 Theorem 11:2).
Proof (of
rID
Then metric
D(a,b)
ro =
j =1
h
X � Y
2)
Y
X.
0 <
be a Polish d < 1).
Y = h(X)
We define
X,
then
We
and claim
d{y,x n
dry
then
J,y)�O,
It remains to show that
Y
Q n
n
> 0
< E:.
h
E:.
E:
for given
j�+oo
o
lim d{y. ,y) < ] j�+ro
X,
h
is
is one-to-one.
d(y.,x )�d{y,x ), j�+oo, tin E ID. n J n
Then
E:
which means that
is dense in
Since
h(Y )�h(y) E Y , j
Let
d(y.,x )�d(y,x ), j�+oo, tin E ID n J n
D(h(y.))�D(h(y)), j�+ro J
This means that
> 0,
choose
is a z
Go-subset of ID r
in
�
d(x1,x2)
is open. Assume that
Zo
CD
E Yn n Q n=l
n
E:
-
ID l .
such that there
such that D(h(x2),z) < r n
n
< d(y . ,y) < d(y . ,x ) + d(y,x ) n ] J n E: E:
be the set of points
D(h(x ),z) < r ; 1 n n
(We can assume that
in
continuous.
r
x
Let now
.
Go-set.
so therefore
is an
ID
is a homeomorphism.
is a
so that
I ID = X [ O , l ]
and
h,
For if
Q
d.
is dense in
where
so
I a J.-b·1 J
L
consider the map
1)
1 =[0,1]
is a compact, separable and metrizable space with
space with metric
If
a) Let
and take
xP
< n
E X
Then each
so that
- 8 -
D(h(x
P ) , z 0 )-+- 0 , p-+-+oo. r
corresponding
and then
n
< r . n
� D(h(xP) , z 0 ) p > p n
d(xP,xq) < - < n oo (xP) p=1
Therefore lim xP = p-+-+oo n
u
j= 1 zEY
b)
rlN
U
that d
U
, ("
p,q > P n
Hence, if
1n
<
X
so
Finally,
J..-}. J
is compact every open subset of a)
choose the
so that
Pn
is a Cauchy sequence
Therefore, by
Suppose now
h >
Take
•
{y E rlN; D(y, z)
Since
F a -set.
€
> O.
and therefore
00
Y =
€
Given
rlN
every Polish space is
is a
K-analytic.
is an open subset of a Polish space; we claim
is Polish.
is a metric on
It is clear that
�E
is Polish and if
E,
v = {(t,x ) E mxE; t'd(x,E \ U) = 1} is closed. Therefore is a homeomorphism so IT
Let now
V
is a Polish space and U
is a Polish space.
be the family of subset
and its complement are
E
K-analytic. We have just shown that
IT
IT
b)
is a
a)
IT
E
of
X
so that
contains all open sets and by 11:1
V 3 (t,x)-+-x E U
contains
a-algebra and therefore
IT
X.
By Theorem
contains all
Borel sets. Corollary
Proof.
a
U
G�set
11 : 1 .
Let
P
n o. j E lN J
Every analytic set is
K-analytic.
be a Polish space. By Theorem 11: 2 a), there is contained in a compact metric space so that
- 9 -
f(nO.)=p
for a cont inuous f . In a compact metric space , every open set i s a K a -set . Therefore P i s K-analyti c and so i s any continuous image of P . J
Every K-analytic set i n U i s universally capacitable . ( Remember , U i s assumed to be F a ) . For the proof , we need two lemmas . Theorem 1 1 : 3 .
Every Koo is universally capacitable .
Lemma 1 1 : 1 .
Assume that A=nA n where A n =uK n , p ' K n , p compact p n and increasing in p . Let c be a given capacity and A a given number < c ( A ) . Since ACA 1 there i s a Pl so that c ( AnK 1 , p »A . Put a 1 =AnK 1 ' P and def ine ( a n ) n=l induc1 1 tively : I f a n- ' chosen , take P n so big that c ( a n »A where a n=a n - ,nK n , p n Since a n CK 1 , P nK n 'P n l we have that C ( K 1 , Pl n nK n ' P n »A . Proo f .
<Xl
lS
.
n . . .
.
.
.
00 n
The set K= K n P is compact and contained in A since n =l ' n C K n , p An . n Furthermore , since c ( K » \ by Axiom i i i ) , the lemma i s proved . If f: Uo�U is a continuous function between two Hausdor f f spaces and if c is a capacity on U then cof is a capacity on UO'
Lemma 1 1 : 2 .
- 10 Let
Proof of Theorem 11 : 3 .
pa c t H a u s d o r f f space pact space
B
U,
then
is a
so that
cof
K- a na l yt i c s e t i n a c om-
be a
11: 1 ,
By P r opos i t i o n Ko - s et o
conta i ni ng a
O
f: B �U O
f u nc t i on on
B
U.
A
A= f( B ) .
i s a c a p a c i ty o n
B
a nd a con t i n u o u s
If
c
B
by Lemma I I : 2 . S in c e
O
11:1
Ko - s et i t fo ll ows f rom Lemma o
i s a g i v e n c a pa c i ty
that
c ( f( B ) ) = s up { c ( f ( K ) ) : K comp a c t s u b s e t o f B } , t h e p r oo f s i n ce
f
i s con t inuous o n
there i s a c om-
wh ic h c omp le t e s
B ' O
Capa c i ta b il it y c o nc e r n s i nn e r reg u l a r i t y ; approx i ma t i o n f rom t he i n s i de w i t h comp a c t s et s . O u te r regu la r i ty ; approx i ma t i on from t he out s ide w i t h ope n s et s ,
i s t h e top i c i n t he next s e c t io n .
o f me a s u r e s , t h e r e a r e
I n c o nt r a s t t o t h e c a s e
(J
F - s e t s t h a t a r e not o u t e r r e g u la r .
Not e s a n d r e f e r e n c e s Theo r em 11:3 i s d ue t o Choque t . Choquet , G . , Theo r y o f capa c i t i e s . A nn .
In s t .
Choquet , G. , Lec t u r e s o n a na l y s i s . W . A .
Be n jami n.
( 1 95 3-54 ) .
5
Fou r ie r
Inc .
1 969. S ee a ls o chapter t wo i n :
Federer , H . , Geomet r i c mea s u r e
theory . S pr i n g e r - Ve r la g , Be r l i n- H e i de lb e r g - New Yor k . And the append i x i n :
1 969 .
Treve s , F. , Topo l og ic a l v e c t o r s p a ce s ,
d i s t r ibut i on s a nd k e rne l s . Academic P r e s s I nc .
1 967 .
There a r e a na ly t ic s e t s w i t h comp leme n t s t h a t a r e not u ni ve r s a l l y capac i ta b le , s e e :
Del lacher i e , D . ,
capa c it es , me s u r e s de H a u s do r ff .
En semb l e s a n a l yt ique s ,
S pr i n ge r LNM.
2 95 , 1 972 ,
pg 2 8.
III a Outer Regularity
I n thi s section , we assume space.
S
to be a compact and metric
Let c be a capacity 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 capacity . Def i n i t i on .
To veri fy that a given capaci ty is outer regular, it i s enough t o check property i i ) . Observe also that i f c i s outer regular , then it follows from Theorem 1 1:3 that c c* on all K-analytic sets , s i nce they agree on all compacts . =
It i s clear that every pos i t ive measure def i nes an outer regular capac ity . The following example shows that there exists a compact denumerable set M of probabi l ity-measures such that i ) there is a F -set F with c ( F ) < c * ( F ) i i ) c ( E ) = 0 c* ( E ) = 0 , where c ( E ) = sup �* ( E ) . �EM �
a
and 0 n , n Elli, be Dirac measures Let respectively . Let m be the Lebesgue with mass at zero and measure on the unit interval [ 0 ,1] and denote by M the denumerable compact set o f measures { o i®m} 7 = o and let E be the F -set Exampl e I11 : 1 .
a
{( O ,y)
2 1 E R ; -2
If we , as usual , def i ne c ( E ) but c* ( E ) 1 . =
00
1=
, 1 1 sup � ( E ) we have c ( E ) �EM
1
== "2
- 12 Def i n i t ion .
Let c be a capacity. Define c by
c ( E ) = i n f {c ( F )
I;
ECF , F a G o -set } .
It i s clear that c ( E ) < c( E ) < c * ( E ) and we now use Example 111 : 1 to construct a capacity ( of the type cons idered i n Theorem 1: 1 ) such that c � c . Let M be the compact set of probabi lity me asures , M = {o x ®m}xE [ O , l ] where 0 x i s the Dirac measure with mass at x and where m i s the Lebesgue measure on [0 , 1 ]. We define the capacity c by c ( E ) sup �* ( E ) . �EM We f i rst define E n ' n Ern, by i nduction . Let E 1 be the set in Example 111 : 1 . To construct E n , n>l , divide [ 0 , 1 ] in n equal i ntervals and do the construct ion o f Example 111: 1 in each interval so that the set so obtai ned does not i ntersect E n- l ' Put E E n=l n ' Then c ( E ) -2-� hut i f F i s any G o -set conta ining E we claim that c ( F ) = 1 . For i f n 0 s = F , EC O s E lN are s=l open sets then the l-dimens ional sets Ss = { x E [ 0 , 1 ] , {x} x [ O ,l]CO S } ' s E IN are open and dense i n [ 0 , 1 ] by the construction of E . Hence , n Ss is dense and i n particular s= 1 non-empty which means that c ( n O s ) = c ( F) = 1 . s=l Exampl e 111:3. Let c be the capacity def ined i n Example I I I : 2 . Then there exists a G o -set A contained in [ O ,l ] x[ O , l ] such that c(A) = 0 i) Example III:2.
=
=
00
u
=
00
s'
00
00
- 13 i i ) c ( O)::;;
for every open set
contai n i ng A .
0
Thi s follows directly from the exi stence o f a Go -set A contained in [ O , l ] x [ O,l ] such that A i s the graph of a lower semicontinuous function . 1) 2) I f K i s a compact set i n [0 , 1 ]x [O,l 1 with projlK = [ 0 , 1 ] then AnK � ¢. Def i n i t i on . A set function c i s called strongly subadditive i f for all compacts K 1 , K 2 • Every strongly subadditive capacity on Theorem 111 : 1 . outer regular .
S
is
Assume that c is strongly subaddi tive and that < < h are open subset of V . Then n n n n c ( 1 U.) < c( U V. ) E c ( U . ) .1::;; , 1 i::;E ;' c ( V 1. ) . i= 1 i =l
Lemma 1 11 : 1 .
U 1. �V 1. ,
1
U
1
The proof i s by i nduction . Assume that U 1. �V 1. , , _< j < n , are open sets . We want to prove that n n n n c ( UJ. ) + . E c ( V . ) < c ( V . ) + . E c ( U . ) . J= 1 J. = 1 J= , J'=l Thi s i s true i f n = 1 . If n = 2 we put U = U 1 and V = v,UU2. Then c ( U1UU 2 ) + c ( V 1 ) < c ( UuV) + c ( U n V ) < c ( U ) + c ( V) = c ( V , UU2 ) + c ( U , ) . Proo f o f the l emma .
U
1
U
1
1
+
- 14 we On the other hand , i f we put U = U 2 get c ( U 2 UV 1 , + c ( V 2 ) � c ( UuV ) + c ( Unv ) < c ( U ) + c ( V ) = c ( U 2 ) + + c ( V 1 uV 2 ) · Addi ng the i nequal ities gives which proves the lemma for n = 2 . Assume now that the formula has been proved for n . We then prove i t for n + 1 . Put n U 1 = u U.J , U 2 = U n+ 1 j= 1 n u V.J , j= 1 The case n = 2 then g ives c ( U 1 U U 2 ) + c ( V 1 ) + c ( V 2 ) < c ( V 1 UV 2 ) + c ( U 1 ) + c ( U 2 ) and the induction assumption n n n n c ( U U J. ) + E c ( V. ) < c ( V . ) + E c ( U J. ) . j= 1 J j=1 j=1 J j = 1 Hence n+1 n+ 1 n+ 1 n+ 1 c ( u U J. ) - E c ( U J. ) - c ( u V . ) + E c ( V . ) j= 1 j= 1 j= 1 J j=1 J u
n n n+ 1 U. ) + c ( U n+ 1 ) - ( U V.J ) - c ( Vn+ 1 ) + c ( u V J. ) + j= 1 J j= 1 j=1 n+1 n+ 1 n 1 n E c (U . ) - c ( +u V . ) + E c ( V. ) = c ( u U J. ) + J J j= 1 j=1 J j = 1 j= 1 n n n + E c ( V. ) - c ( u V J. ) - E c ( U J. ) < 0 j=l j= 1 j= 1 J C
U
- 15 b y t h e i nd u c t i on ass ump t i on . I t is e no u g h t o p r o v e t ha t
Proof o f the theorem .
true f o r
c* .
fo r eve r y that
S o ass ume t h a t
E """ E , s-++oo . s
U. 1
t h e r e is a n o p e n s et
i
c( U . ) - c*( E . ) 1 1
IS
<
ii) IS
G i ve n
h o l ds
> 0,
E. 1
c o n t a i n i ng
s uc h
2i
-
By Lemma 1 1 1:1 we h a v e n c( u U . ) . 1 1 1=
so
c(
n u
i=1
U. ) 1
+
-
n
1: c*( E . )
1
. 1=1
c*(
n n < c*( u E. ) + 1: c( U . ) 1 1 ' 1 1=1 1= .
n u
i= 1
E. ) 1
n 1: c( U . ) 1 i =1
<
- c*( E . ) 1
<
E: .
H e nce n
00
0 < c( u U.) - c*( E ) < l i m ( c(u U . ) - c*( E ) ) = n 1 i=1 1 i n-++oo n = lim ( c( u U.) 1 . n-++oo 1= 1 H ence
c*( E )
Let
M
< l i m c*( E ) n n-+ + oo
c*(
n u
. 1=1
1
E. ) )
<
IS.
wh i c h p r oves t h e t h e o r em .
b e a s et o f p os i t i v e measu r es o n
l ess or eq u a l to
1.
su p �( E ) ]J E M
De f i n i t i on .
o us f u n c t i o ns
L et
w i th mass
W e f i n is h t h is s e c t i o n by p r ov i ng
theor em t h a t gi v es s u f f i c i e n t c o nd i t i on o n f u nc t io n
S
M
s o that the s e t
is o u t e r r e g u l a r o n i ts z e r os ets . N
b e t h e s et of pos i t i ve ,
l owe r s em i c o n t i nu -
on
S
w i th t h e p r o p e r t y t h a t t o e v e r y
there is a n o p e n s e t
A
w i th
�
a
s up �(A) � EM
<
E:
IS > 0
a n d s uc h t h a t t h e
r e s t r ic t i o n o f
�
to
Let
Lemma 1 11 : 2 .
Q
f u n c t i on s s u c h t h a t Then
i n f s UE � EQ u EM
f � dU
=
\ A
a > sup i n f ]..l E M � EQ < a
J � d]..l .
so
A
�
f � du i n f f �du . � EQ
M 3 � -+ s uE. u EM
i s c o n t i n u ou s f o r e v e r y
s UE i n f ]..I E M � EQ �
G iven =
i s c on t i n u o u s .
be a d ownwa r d d i r e c t e d f am i l y o f po s i t i v e
It i s c lear that
Proof .
f �d�
S
E M
f CP d �
{u E M ;
f � dU
f
< i nE s u£ �d]..l . � EQ ]..I E M
there i s a
< a}, � E Q
� E Q
� E Q.
Let
w i th
i s then an open
T c o v e ri ng o f M a n d s i n ce M is c ompac t we c an c hoo s e ( �. ) . 1 1 1= T B u t s i nc e Q s o that :J M . u A i s downwa r d d i r ec t ed t h e r e i s �. an
f�d �
� E Q
< a,
i=1
1
wh i c h i s dom i n a t e d b y a l l \:f� E M
Theorem 111 : 2 .
so
i n f sUE �EQ ]..I E M
A s s ume t h a t
se m ic on t i n u o u s . T h e n
� E N
f cp d]..l �
�. , 1
< i < T. 1 -
T he r e f o r e
< a wh i c h p r ov e s t h e l emma .
i s b o u n d e d , po s i t i ve a nd l ower
i f a n d o n ly i f
f
M :3 u f+ �d � i s c o n t i nu ou s . � ) A s s ume t h a t
Proof o f Theorem 1 1 1 : 2 .
G i ven to
€ > 0
c hoo s e
� ' cont i nuous on €
O€ U.
cp E
N
and that
a s i n t h e d e f i n i t i on a n d e x t e n d Then
lJ �]..I . s �
-
so �)
l im s .... O
l
f � dU s - f �du l M
A s s ume t h a t
= O.
-
( No te t h a t
f
u
3
17
� � dU
E:
].)(0
i s c o n t i n u o u s a n d le t
po s i t i ve c on t i n u o u s f u nc t i o n s dom i n a t e d b y we h a ve
�
If',
c hoose
then
Then
uEM
If' E N
'
i E
E N
1
� uEM f
t i on s w i t h
Then
sup
l
IN,
<
1
]
1
2
c on t i n u o u s o n
co
111:2
we put
< -
2K 22K+2
,2, . .. , OK <
and w i th l i m i t � .
{ x:
==
1
co L If' j+l-If'j > -'K j=K co 2 == OK· Let
2K co j==LKIf' ]'+l-If',
1
K=m
<
c on v e r g e s u n i f o r m l y o n
If',
u
E
2K' CE;
> 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
<
a nd o n
m=T
B y Le mma
We c a n
co == If'1+ L If',+l-If',] a nd i f j==l J co ]J(OK) .s. f2K(,L If',] +,-If',)dlJ ] J=K
c (E )
b e the
l
t o b e a n i ncr ea s i ng s eq ue nc e of func-
If',d]J
_
�.
N
==
�
wh ic h s hows t h a t
N
f ( � -If' ) dU
o = s UE i n f
by Lemma I: 1) .
E:
< -
)
and
c on t a i n s a c on ve x c o n e
N R
b e de f i ne d a s a bove . A s s ume t h a t o f f u nc t i o n s w i th the f o l l ow i ng
prope r t ie s . i)
1 E R.
i i)
If
(� ,) ,
�n
R,
co
] J== 1
then
�u E M
whe re
m i no r a n t of iii)
If
� , If' E R
iv)
If
(A
co
,J ),J==1
lim
j....
i s a u n i f o r m l y b ou n d e d a n d monot o n e s eq ue nce �
( l im �, )
j....+co
O �
O
R
and
�, d]J == � dU f fO ' .... co ] + j l im
i s t h e l a r g e s t l owe r s em ic o n t i n u o u s
+co ]
then
E
*
l im � , . j ....
i n f ( � , If' )
E
R.
i s a d ec re a s i ng s eq ue nc e o f o pen se t s w i th
s u p ]J ( A , )
+co ]JEM
]
J
==
0
then
- 18 -
in f { s up l im lJEM j-++co v}
If
K
f CPdW i
cP E R ,
cP
� 1
is a comp a c t s u b s et of co
(A J. ).J =1
there i s a s eq u e n c e
A.} = J
on S
w it h
o.
s up lJ ( K } lJEM
=
0
o f ope n s e t s c o n t a i n ing
then K
s uc h that l im s u p j-++co lJEM
G(E) =
Then
l a r capa c i ty .
in f { sup lJEM
Corol lary 1 1 1 : 1 . 111:3.
lJ(A.) = J fCPd lJi
O. cP E R ,
A s s ume t ha t
M,
1 on
E}
i s a n o u t e r r e g u
a nd
R
a r e a s in T h e o r em
N E
T h e n , t o every Bore l s e t
in co
S
( A J ) J. =1
there i s a dec rea s i ng s eq u e n c e
E
cP >
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 ( � ) <
,
'tj
continuous � .
Thus i f s s+ -s is the decomposit ion of s in positive and negative parts , it follows from ii i ) that f �ds+ � L ( � ) , 'tj continuous � . =
- 25 -
A s s ume that
{ � i } 7=
� E F : choo s e
and i nc r ea s i ng
1
f
sequencp o f cont i nuous f unction s w i th l i m i t == � . Then �d s + l im � i d s + � l i m L ( � i } < L ( � ) we have p r oved that s + E M so c ( g ) = L ( g } , f o r a l l c on t i nuous g . I f 9 i s uppe r
f
{g.1 } � l= 1
sem i co n t i n uous , choo se
to be a decrea s i ng sequence
of c o nt i nuous f unct i on w i t h l i m i t = g . Then 1 im . 1
that
1
< c ( 91) .
)..l i �)..l .
. 1
im
)..l. 1
f h gd O
= l i. m 1
Hh
g i--"g i
f h g . do 1
=
g 1.
�.
8
�.
1
< inf i EN
s i nc e
f Hi n f
i EN
Hh �
1
9
do
E F, i E
8
then < h =
f
< h .
3)
I n othe r words :
so
f hgd O
l im
j-++oo
proof . Propo s i t i oD III: 1. Proof .
2)
�
3})
or c l osed . S o i f
=
2)
h
9
f
�.
1
.
==
9
If
h9 = H h
8 Hh
is =
g
we g e t �.
i s l ower
so
9
l.s.c.
H lD . f
i EN
�.
1
w i th so
a . e . ( do ) .
1
i n f � 1. do l
wh i ch comp l et e s the
3).
c(E)
=
i s out e r r e g u l a r t h e n
inf L ( XO } = i n f c ( O ) EC O ECO o open
whe r e we c a n a s s ume
u s e Choquet's l emma a nd
= inf i EN
By Lemma 111 : 3 , c
..A
1
s o t hat i f
. f � . = in f < h lD 9 1 i EN i EN .
hg
I n p a r t i cu l a r
9
i n f � 1 do i EN
( gk )
==
-
� 1. � g
m,
1
L ( g i ) = c ( g i } < c ( g ) , i -++00.
=
F i na l l y , i n order t o pr ove choos e
)J
( gk ) <
L ( g } � l im L ( g 1. } .
L ( g ) � )..l ( g } � c ( g } .
Th i s g i ve s
semi cont i n u ou s then L(g ) =
)..l l'
1 im
( g 1. ) <
=
c ( E) .
L ( XE }
if
E
i s open
c ( E ) � L ( XE } �
- 26 -
O n the other hand ,
3)
111: 3 3)
together w i t h Lemma
p r ov e s
that 2 ) holds . 3)
Corol lary 111 : 1 .
Then
but ¢:)
Proo f .
�).
c*( E ) =O
We have that
Lemma 111: 4 .
f cpdo<+oo
�
cp
I E = +Cfl
�
t h e r e is a
�EF
w i th
0 E:= { z E U ; E:cp ( z » l } , E: > O .
Put
c(0E:) � E: f cpdo -+ O ,
0
i s open , conta i n s E a nd E: �) Choos e f o r every s EThl , O s open w i th hO =H ' wh i c h we denote by f s ' i s i n s h°s
1
C ( Os ) <
2s
F.
E: + - O. •
Then
00
Then
c ( Os )
so
Propos i t i on 111 : 2 .
4)
s
+ 5)
I t i s enough to prove that C(K)=
E /'I E s Lemma
f hX
do
�
3).
is a capac i ty s i nce
f o r a l l compact s e t s
K
by Lemma
K
111: 3.
be a g i ve n i nc r ea s i ng sequence o f s e t s a nd 111: 4
Ii' s E F
there i s a
have
Hence
f hX
+ EF
lim
E
f Hh
d6 = s
d6� XE
s
f hX
E
do
f Hh
<.
.�. Put
2s
so
�
d6.s. l i m XE
s
f Hh
Hh
XE
>H s
00
Let
(> 0 .
XE
s
F= l: Ii' s' s= l
•
By
} c {1i' s =+=} we then
� 1 im
Hh + E: F . + s -+ oo X E XE s
d6 + E: XE
wh i ch m e a n s that
and the proposit i o n f o l l ows .
{h
such that
f Ii' s do
where we can a s sume that
lim s-+ oo
i s the requ i red f u nctio n .
cp= L: f s =1
f FdO .
Thus
s i s a capac i ty
-
Fol l ows f r om Theorem 1 1 1 : 3 .
Proo f . a
I n t h i s s i tu a t i o n , t h e r e i s
s i mp l e r p r oo f . I t f o l l ow s f rom t h e p r o o f o f P r opos i t i on 1 1 1 : 2
that i f
c* ( {h -
> Hh } )
9
9
=
w i th
I im � . = H
wi t h
c( Os ) < s
i EN
U\ Os .
1
Then
t: N = { x � 0 c"s
:
h
a.e.
s uc
{h
h
that
-
9
1
� . EF
( do ) .
CP i '
t o be a decrea s i ng sequence
G i ve n i EJN
s > 0,
and
co
9 > Hh g } c O
h (x) > H
Each 9
9
h
9
9
f o r a l l pos i t i ve a nd bounded
0
then 3 ) holds true . Choose
{h
-
� 3) .
1) + 5)
Propo s i t i on 1 1 1 : 3 .
27
t: U U N s t: s = '
H
h
choose 9
0 c-
open
"-
i s con t i nuous o n
whe r e
( x ) + l} . s
S C * ( N ) =O s
i s c l os e d s o
wh i ch mea n s t ha t
c a n be cove r ed by open s e t s o f a r b i t r a r y sma l l
> Hh } 9
capa c i ty wh i c h proves t h e p r opo s i t i o n .
Then
cp E N
ii )
Proo f .
i s a bounded f un c t i on i n
+
F,
00
L Ut ) � cp ,
t=p
and
p�+oo
out s i d e a s e t 00
cp E F n N n L .
A s s ume f i r s t t h a t
s uch t h a t
E
with
con t i n u o u s f u n c t i o n s i n
\lEM
f(
'cp - If' . ) < -
J
2J .
•
F
w i th
Lemma 1 1 1 : 3
2 )
c * ( E ) =O .
A s i n the p r o o f o f
The o r em 1 1 1 : 2 w e c a n choo s e a n i nc r e a s i ng s equence
sup
F.
i f a nd o n l y i f t h e r e a r e two seque n c e s o f con-
t i n u o u s f u nc t i o n s i n
F 3 ( uP
cP
A s s ume t h a t
Theorem 1 1 1 : 4 .
l i m If' . = cp j �+oo J g i ve s
and
00
( If' . ) .
J J= 1
of
- 28 u
D efi ne
=
P
u
T hen al l
p
an d
p
= H
h ( 'I'
'1'
p+ 1 - p
)
the fu nctions ar e c ontinu ou s and in
< H '1' J. + 1 - 'I' J. - h '1' .
Sinc e
J+ 1
00
uP +
'I'
uP + 1 - u P - u
= 00
E u t t =p
{ l im u
+
=
'I'
we g e t J .
- 'I'
p+ 1
P - H h'l' p + l - 'I' p
is a d e c r e as i ng s eq u e n c e .
uP
P
P
- 'I'
<
Cjl < u
P
00
+
00
E
t=p
u
E
u E u > Cjl } c { t p t=p t= l
=
so that
< 0
Furthermore
so
t
00
F.
wh i ch comp l e t es the p r oo f i n
+ oo }
00
th i s
di re c ti on s i nce
f E UJ
. do
j= 1
On th e oth er hand ,
<
+00.
if
p r ope r t i es a bove , w e w i s h t o p r ove t h a t 0E
choos e
wi th
l im
S-4- + OO
E ) <E
f Cjl d ]Js
U
<
xEU
<
E
+
l im
S-4- +OO
f Cjl d ]J
caE
su p Cjl { x ) xEU
Cjl
f Cjl d ]J
then
E sup Cjl ( x )
Cjl E N .
an d s u c h t hat
If
outs i d e <
c(O
h ave t h e
and
+
S
G i ve n l i m (u p-+ +oo
� l im S-4-+OO
< E
f
E>O , P
00
+
Cjl d]J
U
sup Cjl ( x ) xEu
I u ) t
t=p
s +
<
- 29 -
+
lim s ___ + oo 00
00
Si nce the the
f
L: u t E F , L: u t d 8 < +00 t=p t=p U
r i ght h a nd s i de t e n d s t o t h eo r em
Not e s
,
so l et t i n g €
sup xEU
�(x) +
p
tend t o
f �dw
+00
wh i ch p r ove s
U
u s i ng Theorem 1 1 1 : 2 .
and r e f e r e n c e s A p r oo f o f " Choquet ' s l emma " c a n b e f ound i n Doob , J . L . ,
Cla s s i cal pote n t i a l t h e o r y a n d i t s p r ob a b i l i s t i c c ou n t e r p a r t , Spr i n g e r -Ver l ag
( 1 984 ) .
IV Subharmonic Functions in IR" Le t t i on t o whe r e
B B
R
n F ,
be the u n i t ba l l i n
l et
be t h e r e s t r i c -
F
o f a l l p o s i t i ve s up e r h a r mon i c f un c t i o n s o n
>1 .
i s a f i xed numb e r
L e b e s g u e mea s u r e on
B,
I f we t a k e
i t i s we l l k nown t h a t
0
RB ,
to be the U=B
0
and
s at i s f i e s a l l the a s s umpt i o n s made i n S e c t i o n I II ; t h e " co ar s e " p r o b l em h a s a p o s i t i ve s o l u t i on . B u t much more c a n b e s a i d : t he f i ne p r o b l em h a s a pos i t i ve s o l ut i o n . Fix K B,
xEU
a nd de f i ne
d
x
( K ) = i n f { cp ( x ) E F ; cp � 1 o n K }
compa c t . Con s i de r t h e c l a s s m = { )J � O ; x
Theorem 1 : 1 ,
S cpd p � cp ( x ) , M
x
M
o f p o s i t i ve mea s u r e s o n
x
Ii cp E F } .
for
The n ,
by Lemma 1: 1 a n d
g i ve s r i s e t o a capac i ty :
c ( E ) = s u p )J ( E ) x \.l E M x a nd by a p r oo f , s i m i l a r t o t h a t o f Lemma 1 1 1 : 3 we have t h a t c (E) < d ( x x
E)
E
w i th equa l i ty i f
F u r thermo r e ,
i s c ompa c t or ope n .
i t i s a c o n s e q u e n c e o f t h e max i mum p r i nc i p l e
+ d ( K 2 ) s o b y Theorem x capa c i ty a n d t h e r e f o r e
I
II : 1 ,
d
x
e x t e n d s t o a n oute r r eg u l a r
c =d x x
Moreove r , t h e f o l l ow i ng s t r on g e r v e r s i on o f T h e o r em 1 1 1 : 4 ho l d s t r u e i n t h i s ca s e . I f s u p e r h a rmon i c f un c t i on on
f RB
i s a bounded a n d p o s i t i ve then
- 31 00
E
f
whe r e
j=1 00
(f . ) . J J= 1
-
f .
J
i s a s eq ue n c e o f pos i t i ve a nd cont i nuous s up e r -
harmo n i c f u nc t i o n s o n
B.
Not e s a n d r e f e r e n c e s Brelot , M . , E l ement s de l a t heor i e c l a s s i qu e du pote n t i e l .
C e n t r e documen t a t i o n U n i v e r s i ta i r I e S o r bonne , Choquet , G . ,
Theory o f c a p a c i t i e s . A n n .
1 965 .
I n s t . Four i e r 5
( 1 953-54 ) . Choquet , G . , L e c t u r e s o n a n a l y s i s . W. A.
B e n j am i n . New York
a nd Ams t e r dam 1 9 6 9 . Landkof , N . S . , Foun d a t i on o f mod e r n p o t e n t i a l theor y ,
Ve r l ag ,
1 972 .
Spr i ng e r -
Plurisubharmonic Functions in en The Monge-Ampere Capacity
V
Let
B
be the u n i t b a l l i n
[
n
F
and l e t
be t h e
rest r i ct ion to
B
o f a l l p o s i t i ve p l ur i s up e r h a r mo n i c f u n c t i o n s
on
R
i s a f i xe d n u mb e r
RB ,
where
the Lebesgue mea s u r e on
B
I t i s t h e n t r u e that
and
F
>1 .
a nd f o r m 0
c
We t a k e
t o be
as i n Se c t i on I I I : b .
meet a l l t h e r e q u i r emen t s i n
S e c t i o n I I I : b a n d we a r e go i n g t o s e e t h a t and
0
h a s p r ope r t y
F
1)
h a s p r ope r t y 5 ) ; t h u s 2 ) a n d 3 ) h o l d t r u e by P r opo-
c
s i t i on s 1 1 1 : 3 a nd 1 1 1 : 1 .
a
The d i f f e r en t i a l ope r a t o r s
a
a � d Z J' J J=1 n
,E
=
a.
1
E
j=l
ai , J
do , J
a r e d e f i ned by
so that
d= a + a
a nd
De f i n i t i on o f t h e Mo n g e - Ampe r e o p e r a t o r Let
V
a
n
and
a
and
,
and
•
•
•
U
n [ o
be a n ope n a n d bounded s u b s et o f
n , v E c 2 \, U ) ,
n 1 MA ( v , . . . , v )
we d e f i ne
If
to b e t h e s ymme t r i c
n - l i n e a r ope r a t o r
1
MA ( v , I f moreove r
. .
. , v
n
)
c
1 n v , . . . , v E P SH ( U )
pos i t i ve mea s u r e .
1 . . . A dd c v n .
= dd v A
then
c 1, dd v fl
•
•
•
c n dd v
is a
- 33 Theorem V : 1 .
If
1 n MA ( v . , . . . , v . )
J
PSHnC
2 ( U ) 3v i � v i E L00 ( U ) , . J
j -+ + oo ,
l < i
then
i s a weak l y c o n v e r g e n t s eq u e n c e o f pos i t i ve
J
mea s u r e s . Mor e ov e r , the l i m i t i s i ndepe n d e n t o f the p a r t i cu l a r l
cho i ce o f t h e d e c r ea s i ng s eq u e n c e s
Proo f .
i 1 V 1 �V j '
J
T h e s t a t eme n t i s pure l y l o c a l , 1 �i�n ,
j aN
c
jEN.
s o we c a n a s s ume that
out s i de a f i x c omp a c t s u b s e t
G i ve n
f 8dd
1 < i
K
of
U�B ;
n � .
the u n i t ba l l i n
lim j -++oo
v . ,
we wan t t o p r ov e t ha t 1 v . /1.
J
•
•
•
C n /lA dd v .
ex i s t . Take
J
ll � l
and c h o o s e
n e a r t h e s uppo r t o f
8,
union
K.
T h e n by S t o k e s f o r mu l a ,
f 8dd C v 1.
J
We co n s i de r
1\
A
1\
"
•
•
•
c n dd v . J
f ll v 1. dd c 8 1 1\A dd c v 2. A . . . dd c v n. J
J
f ll v j1 + 1 A dd c 8 1 /\ " A
<
J
c n . dd v + j 1
fv 2j + 1 dd c 8 1 " dd c ( n v j1 )
A /I
•
<
•
•
a nd o b s er ve t h a t
f n V j1 dd c 8 1 A 1\
c n dd V + l j
c n . . . dd v + 1 j
<
- 34 c n dd v j+1 '
we g e t
f ( v 2j + 1 -V 2j
) dd
c
81
A /1
C jA
dd v 1
f n V j1 A dd c 8 1 /\ dd cV 2j
<
A /I
II
•
dd
•
•
A /1
n= l
s i nc e
near
c n dd v
j+1 +
A
C 3
Vj + 1 /\
•
•
•
c n A dd v • j+1
Repea t i ng t h i s , we get t h a t
i s dec re a s i ng i n
j
wh i ch p r o v e s tha t t he l i m i t e x i s t .
We now p rove t h a t t h e l i m i t does n o t depend o n t h e pa r t i c u l a r cho i c e o f the approx i ma t i ng s e q u e n c e s . W e f i r s t n o t e t h a t i t f o l l ows f rom the proo f a b o v e t h a t i f s u b s equences o f
i v ., J
t o t h e s ame l i m i t a s
t he n
v
n J .
•
c l dd v . " . J
But i f
A . . II
•
•
•
c 1 /\ 1\ c n dd v . . . . dd v . . J J
i t l S e nough to show t h a t l im i t a s
1
c A dd v . II Jp
c n dd v . J
c dd v if
;A
•
�n v . J
we h a v e t h a t
•
•
A
i v . , Jp
C n dd v . Jp
S i nc e c dd v� J
l < i
j Em p
are
t e n d s wea k l y MA
is
n-l i near ,
t e n d s t o t he s ame
h a s t h e s ame prope r t i e s a s
- 35 n �n v -v -+ 0 , k-++oo k k
and s i nc e whe n
k-++oo
t he r i ght h a n d s i de t e n d s t o z e r o
f o r every f i xed
oo 1 v EPSHnL ( U ) ,
If
Def i n it ion .
1 n c 1 r, MA ( v , . . . , v ) = dd V 1\
E
If
Def i n i t i on .
f
j
•
•
•
A 1\
wh i ch p r ove s t h e theorem .
l < i
c n dd v
2
we de f i n e
to b e the weak 1 i m i t o f
i i P S H n C ( U ) 3v . � v ,
i s a Borel subset o f
d ( E ) = s u p { MA ( U , . : . , u ) : u E PS H ( U ) , n t 1. m e s
j-++ oo ,
J
O
E
U,
l
we put
It i s e a s y t o s e e
that t h e f o l l ow i ng propos i t i o n ho l d s . Propos i t i o n V : l .
00
ii )
d(
iii )
If
u
i=l
E. ) < 1.
E J.
./I
00
E
- i=l d ( E 1.. )
E:
j-++ oo
( We w i l l s e e l at e r t h a t If
Theorem V : 2 .
l im d ( E . ) = d ( E ) . J
then d 1.
i s a c a pa c i t y i n Choquet ' s s e n s e ) . 1.
00
P S H 3v . ,; v E PS H n L (U) , l oc J
o c 1 C n 0 c 1. v . dd v · A . . . I\A d d v . z. v dd V fl J
Proo f . o i v =v ' 1 j
J
J
•
•
The s t ateme n t i s p u r e l y l oc a l , O�i�n ;
j Elli
As s ume f i r s t t h a t we f i nd a s
•
j-++oo ,
i
2
If
s o we c a n a s s ume that 00
JO
c 1 dd v . J1
K
of
O,S,n E C ( B ) , n = l O
i n t h e p r oo f o f T h e o r em I V : 1 , t h a t
f n v 0.
then
c n dd v .
out s i d e a f i xe d c ompa c t s e t v . E PSHnC ( B ) . J
O < i
U=B . near
K,
- 36 in
i s d e c r e a s i ng
( jo ' · . . , jn ) .
O n t h e other hand , 1 1m
.
-
j-++oo
�
1\
•
•
.
•
J
j -++ oo
.
1\
/A\
C n 1' dd V . � 1m
J
c n dd V
k-++ co
f
°
we g e t c
n V dd v K
1 1\
.
•
.
1\
c n dd v
•
c n dd v . =
J
J
f n v ° dd c v 1 A
1\
•
•
•
1\ dd c v n
i s a n y ac cumu l a t i on po i n t f or o c v . dd v . /I.
1
J
•
•
•
c n dd v . , J
<
u
o
c dd V
1
A 1\
•
•
•
j ElN
o C v dd v
1 1. .
00
/\
.
.
c n dd v
a nd s i n c e
Gu
o
1S
o � GEC ( 8 ) O
upper s em i co n t i nuous i f
Z
•
A
h a s t h e s ame ma s s a s
Thus
1 S d e c r ea s i ng ,
f n v °. dd c v 1. 1\ . . . 1\
l im
J
�
A /\
J
0
The r e f o r e so i f
J
f n v dd C v 1 A
<
o J
nv .
s i nce
f n v °. dd c v 1.
The r e f o r e
C n dd v
a nd t h e y have t h e s ame ma s s s o
they have to b e equa l .
Now ,
if
i v . J
r e g u l a r i z ed f u n c t i on s
Then ,
if
PSHnL
are i n
00
( eJ ) J. = 1
i
v .
J
,
1S
E:
00
l oc
E PSHnC
00
(8)
o n l y , w e c on s i de r t h e
i n a sma l l e r ba l l .
a n y s eq u e n c e d e c r e a s i ng t o z e r o w e
<
-
37
-
get f rom above that o v 1. , ( . dd c v 1l. , ( . 1\ 1
1
•
•
•
0 dd c v n1. , ( . Z, v dd c v l 1\ 1
Pj
•
•
we c a n t o every
On the other hand , g i ve n (
.
dd c v n . j
f i nd
s o that
f 8v 0. dd c v 1.
-
...
1\
J
J
•
•
•
A /\
dd C v n. l < 1.
J
J
beca u s e the theorem i s a l r eady proved f o r C 2 - funct i on s . o A c n wh i ch The r e f o r e v . dd c v l. (\A . I\A dd c v n. �v 0 dd c v n \ / \ dd v J
J
.
proves the t heorem .
b.
I
J
.
Quas i co n t i nu i ty w i t h re spect t o
Theorem V : 3 . o
open s e t Proo f .
(
To every s u c h that
d(O
A s s ume f i r s t that
compact s u bset o f
U.
•
vE P SH ( U ) v l u- o
a nd
( ) <£
•
d.
and every
£>0
•
(
there i s an
i s cont i nuou s .
O
Choo s e
a n d that K i s a g iven 2 near v . E PSHnC K , V "'lI. V 1 , j -++oo . j J
We c l a i m that 1
i rn j -+ +oo
Assum i ng t h i s f o r a mome n t , c hoose sequence o f c ompact s u b s e t s o f
U
k.
J
t o be a f undame n t a l
a n d l et
£>0
be g i ven .
- 38 -
Choose for every j , v J' such that ( v j -v ) ( dd u ) n sup f uEPSH ( U ) k O
u
1
co
I
,
�
==
-
fd { v j -v k ) A d ( u-u ) /\ ( dd c u ) n- l c
�
sin c e fd { v j -vk ) /\ d u A ( dd c u ) n - l = 0 . s ince { dd cu ) n- l i s a positive ( n- l , n- l ) -form we get I fd ( V j -V k ) /\ d C ( u-u } /\ { dd c u } n- l I � 1 2 nC A ( dd u ) l ] < [ f d ( v j -v k ) A d ( v j -v k ) 1 2 nc l ] . [ f d( U -� } /\ dC ( u-� } /\ ( dd u ) 1 C C nl '2 ] . [ f ( v j -v k ) dd ( v j -v K ) ( dd u ) f ( U-� ) dd c ( u- � ) /\ ( dd c U ) n- l ) 2 < 1 1 < ( f ( v j -vk ) ddC ( v j +Vk ) A ( dd c u ) n- l ) '2 ( 2d ( L ) ) c�
1\
C
f\
•
=
1
•
.
2"
- 39 Cont i nu i ng i n th i s manner , we get a n e s t i ma t e o f the f orm
where
i s a constant and whe r e Theor em V : 2 ( v j - V k ' dd c ( v j +v k ' n � j � +oo .
c
J
Then aga i n by Theorem V : 2 ,
J(
v
. -v
J
c ) dd ( v . +V ) n � J
Let k�+oo , then by ( V -V l dd c ( V +V ) n . Let now j j
a> O .
J
0
wh i c h p r ov e s the c l a i m . c.
D i r i ch l et pro b l em
Theorem V : 4 .
doma i n i n
�n .
Let If
be a s t r i c t l y ps eudoconvex a nd bounded
U
a nd
hEC ( a U l
un i que p l ur i s ubharmo n i c f un c t i on
fEC ( Ul ,
then there ex i st s a
such t hat
uEC ( U)
au
on
d.
U
Cont i nu i ty o n i nc r ea s i ng sequ e n c e s
Theorem V : 5 .
A s s ume that
where
i s a non-dec r ea s i ng sequence such that j� +oo .
Then
MA ( u j , . . . , u j l n t imes
�
l im u j =u O
MA ( u 0 ' . . . , u 0 ) .
(u
00
J. ) J. = 1
( a . e . ) when
- 40 -
Proo f .
�
Let 1 im
j-+ +oo
J 8dd c u j A
Take
If
be g i ve n . We have t o p r ove that
8EC ( U )
•
•
•
1l = 1
w i th
c
!\ dd u
on
J 8dd c u O !\ . . . " dd c u O ·
=
j
s upp 8
a nd
we k new that C dd u j
A 1\
•
•
•
A 1\
dd C uj
by monoton i c i ty , so s i nce
J ll u k dd c u o /\ . C
A
J J TlU O dd C u j A llu . dd ]
where the
U . /\ ]
last
dd C 8 1 4dd C U o
A /\
. . .
A /\
dd C U o
. .
d dC 8 1 (
*
)
uk
a r e qua s i co n t i nuous :
A dd c u O A dd c 8 � 1
" C A C I\dd u . dd 8 1 � 1 1 m
'
•
•
.
C . . A dd u
•
]
c j l\ dd 8 1
=
j -+ +00
J
J
C ll u . dd u . A ]
]
"
. . . /ldd
c u ' l\dd " C8 < 1
-
J
Tl u O dd c u o A . . . A dd c u O /\ dd c 8 1
i nequa l i ty f o l l ows f r om t h e f a c t that
qua s i cont i nuous .
A /\
k�j ,
we would have for
Thus
A 1\
Uo
is
-
-
41
F r om Lemma V : l be l ow we g e t tha t C c c c Tj u k dd u o !\ . . " dd u O !\ dd e 1 = Tj u o dd u o /I.
lim
f
and
fu J. dd c u J. 1\
k-t- + oo
J
.
.
•
•
/\ dd c u .
J
c 1\ dd e
'¥EC
00
•
•
•
c c !\ dd u o /\ dd e 1
can be handled i n exactly the
2
same way prov i ded we know t h a t C C c c A dd U j 1\ dd ,¥ � dd u O /I. dd U j 1\ •
•
•
•
•
•
c
c
/\ dd u o 1\ dd ,¥
f o r eve ry
(U) .
Th i s we c o u l d do a s above i f we knew that c C C c c c c c dd u A A dd u . /I. dd ,¥ 1\ dd y � dd u . 1\ 1\ dd u II dd ,¥ /I. dd y , .
•
J
.
•
J
00
J
•
•
.
.
J
\i'¥ , y E C ( U ) . I f we repeat th i s true . Thu s
(*)
n- l
t i me s we get a s tatement wh i c h i s
and Theorem V : 5 i s pr oved by the f o l l ow i ng
Lemma . Lemma V : l .
If
J u . J < cons t . J
-
u . E PSH ( U ) J
one h a s u j .?l u O E P SH ( U ) , then c n � u dd c v l 1"\ A 1\ dd v u j dd c v l /I. O •
•
.
i oo v EPSH ( U ) nL ,
Proof .
•
•
l
•
A 1\
\:i j E lN ,
dd C v n
a nd i f a . e . on
U
for all
Fatou s l emma g i v e s t h a t i f u i s t h e weak l im i t o f -then l i m u . d c v l /\A . IIA dd c v n > u . But J
u O> l i m u 1.
.
.
w i th equa l i ty a lmo st everywhere w i th r e spect to the
Lebe sque mea s u r e s o i t i s e nough to p r ove that u a nd c C C u O dd v 1\A /lA dd v n have ( l o c a l l y ) the s ame mas s . We can a s s ume J. J. 2 that v =v E E C out s i de some f i x compact subset K o f U whe re v '::l. v j , E-t- O on U . •
�
•
•
- 42 CX)
If
Ju . dd c v A 1
J
nEC ( U ) , n= l o •
•
•
-- E: + f n u ' dd c J
f
A dd c ( v n - v n ) = E:
u n i formly i n
l"
fI . . . 1\
(v
s i nce
j
f
such that v
K
near
n - v n ) dd c V 1 1\,. E:
vn J
.
•
1\
•
•
1
"
c
n
f
A C v n - 1 1\A I\dd
•
•
A cu . � o , l\dd
•
J
1"\ dd c v n E: l n u . dd c v 1 . . . /\A dd c v n�
c 1 A dd c v 2 A dd c v nE: > - nE: + nu . dd v 1\ E En J n- 1
.
j�+CX)
dd c v nE - E: n .
f
•
•
•
J
I f we now l e t
1
n E:
. . . /\ dd ( v -v ) =
E: 1
1 1m
1 n U O dd c v A . E: n
•
v
i s quas i cont i nuous . Thus n u . dd c v 1 1'1 . . . 1\A dd C v n >
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
v, . . . ,v) <
f U
MA (
on
u, . . . ,u) .
U
and
if
near A ddeu on •
.
•
dd c v
=
l i m u ( z ) -v ( z ) =O z�x z EU
- 43 -
Proo f .
G i ve n
put
v =sup ( v- £ , u ) . £
, V£ ) =
f MA ( u , . . . , u ) .
£>0 ,
V: 2
so b y Lemma
f MA ( v£ '
•
U
S i nc e
as MA ( v £ '
.
•
•
) 4
near
v £ =u
au
U
we have by Theorem
£ ":11 0 ,
. . , V£
Then
V: 5
MA ( v , . . . , v )
wh i ch proves the c l a i m . Theorem V : 6 .
If
00
u , vEPSHnL ( U )
a nd
l im ( u ( z ) -v ( z » O
Z4 a U zEU
f
f
then
MA ( u , . . . , u ) . MA ( v , . . . , v ) < { u< v } { u
l i m u ( z ) -v ( z » o
f o r s ome
Z4 d U
0 >0
the n
zEU
f
f
MA ( u , . . . , u ) . MA ( v , . . . , v ) < { u< v } { u< v } Proo f .
We f i r s t note that t h e s econd statement f o l l ows f rom the
f i r s t by c on s i der i ng
u
and
zero . W e a l so n o t e that i f
v+ £ u
and l et t i ng
a nd
f i r s t s tateme nt f o l l ows f r om Lemma
v V: 3.
u+ £
dec r e a s e t o
a r e a l s o cont i nuou s , the To prove the theorem ,
i t i s n o l o s s o f gener a l i ty t o a s s ume that othe rw i s e
£
{u
( con s i der
i n stead of u ) . 00
( w . ) . = 1 i s a decreas i ng sequence J J of p l ur i subha rmon i c funct i on s on U w i th l im w . =wE P S H n L ( U ) J We f i r s t c l a i m that i f
00
- 44 -
then
f
f
c n ( dd c w . ) n ( dd w ) < l i m J j-+ +oo { u < v } {u
0>0
cw ) n
> l im
j-++oo
00
( dd c w ) n . j
u
and
{u
be g i ven . S i nc e
there i s a n open s e t
f
w i th
two con t i nuous func t i ons
u
The r e f o r e
and
s up . J
v
v
a r e qua s i cont i nuous
f( dd cw J. ) n < o
and the r e a r e
such that
{ u�u } u { v�v } c O o .
0o
f
( dd c w ) n < l im { u
f
_
f
_
f
wh i ch
g i ve s the s econd pa r t o f the c l a i m .
and
We can now f i n i sh the p r o o f o f the theo r em . Choose
Uo
v E:
and
t o the d e c r ea s i ng s equences o f cont i nuous pl u r i sub
ha rmon i c f u nc t i ons de f i ne d i n a n e i ghborhood o f s uch that
{u < v 1 } c u
.
{u
- 45 -
the ) < l im f ( dd c v ) n < f ( dd c v ) n < (claim ( (-+ 0 {u
<
Hence
=
-
E:
f ( dd c v ) n �
for all small but pos itive n .
{ u + n
with l im u ( z ) -v ( z » O and z-+(lU zEU if MA ( u , . . . , u )�MA ( v, . . . , v ) then u>v on U .
Corol l ary V : 1 .
Proo f .
If
u , vEPSHnL
Let O >pEPSHnL
00
(U)
(U)
such that f ( dd c p ) n > O for every V
Borelset V in U with pos itive Lebesgue measure . If there is z O EU with u ( z O )
O so small that u( z O )
=
- 46 Corol lary V: 2 .
I f u , vEPSHnL o
00
then
(U) ,
u
l im
u ( z ) -v ( z » O
z-+ a u zEU
> v on
and i f
U.
I f {u O so that Proof .
{u
=
O.
f ( dd c p ) n = O whi ch i s a contradiction . { u
Agai n
- 47 '-1
I f K J K , jEm i s a decreas i ng sequence of compact subsets of B then u*K . u*K so J
Remark .
.
by Theorem
V:6
Theorem
/f
and Theorem
V:5
V:7.
now g ives that
This together with Proposition f.
Q uasicontinuity
1n
V: l
proves that d i s a capacity .
the system F , c
I f K i s compact i n B then d ( K ) = f MA ( u K , · · . , uK ) where u K =sup { uEPSH ( RB ) ; - l �u� O ; u I K = - l } . RB Proo f . Given K compact i n B we know from Theorem V : 7 that Propos i t i o n V : 2 .
f MA ( uK ' . . . , uK ) = f MA ( uK ' . . . , uK ) . K RB Let �EPSH ( RB ) ; - l <�
.
.
- 48 A s s ume f i r s t that
f MA (
have
i s cont i nu ou s . By Theorem V : 6 we
f MA ( u K , · . . , u K )
<
=
{ u <
K
f MA ( u K ' . . . , u K ) .
f MA ( u K ' . . . , u K )
=
K
B
If
to be a decrea s i ng
i s not cont i nuou s , c hoose
sequence o f compact s e t s w i th i nt e r sect i on equal to a . e . s0
f
f
f
Then
MA (
f MA (
K.
.
MA ( u , . . . , u K ) = < l im Ks s-+ +oo B s RB equa l i ty c ome s f r om Theorem V : 5 . If
Corol lary V : 3 .
d(0)
=
f MA ( u O '
•
RB
•
•
IS
0
, uO )
a n open s u b s et o f
B
then
•
Choose a n i ncrea s i ng s equence o f c ompact s e t s
Proo f . 00
w i th Then
u
j= 1
{ k s } oos= l
k . =O . J
d ( ks ) =
f MA ( U k s , . . . , uk s )
RB
f
by P r opos i t i o n V : 2 . Thus
f
d ( O ) =l im d ( k ) = 1 i m MA ( u k . . . , u k ) ::=; MA ( u O ' . . . , u O ) b y s s + s-+ +oo s s RB -+ oo RB ,
Propos i t i on V : 1 and Theorem V : 5 . We a r e now i n pos i t i o n to p r ove t h e r e s u l t s s tated i n the i ntroduct i on t o th i s sect i on . We f i r s t p r ove that Re ca l l that RB ,
R> l .
F
1 )
holds .
denot e s t h e p l u r i s uperharmo n i c f unct i o n s o n
-
G i ve n
uEF .
{ o s }'X>s= l
o f open s e t s u l co
s s�+oo .
By Theorem
-
49
V:3
t h e r e i s a decreas i ng sequence
such t h a t
l i m d ( O s ) = 0 and s uch that s�+ oo i s cont i nuou s . By Co r o l l a r y V : 3 , MA ( U O , . . . , u O ) � o , s s
I t f o l l ows f rom Coro l l a r y
V: l
that
Uo ( z ) = O s
l im
a . e . so ,
00
i f n e ce s sa r y , by pa s s i ng t o a s u b s equence The r e s tr i ct i on o f
* f � d O� O ,
C ( CK T ) i
and
u
to
T�+oo .
K T = { ��-T }
Uo
s
E P S H ( RB ) .
i s a l ways con t i nuous uEF
B
1S
c.
Cond i t i on 5 ) i s f u l f i l L ed Let
0ig
be a bounded f un c t i on . We w i sh to prove that
c ( { h > Hh } ) =O . 9 9
(8)
E
s= l
Th i s p r oves that eve r y
qua s i co n t i nuous w i t h r e spect to
g.
�=
,
i�+ oo
Choose
and put
h 1. E F : h 1. >g
so that
A r . = n. { h 1. > r . > H h } J 9 J 1
rat i on a l number s i n
( 0, 1 ) .
Then
hi
9
enough to prove that
Hh
9
{ r . } �= 1 J J
wher e
{ h9 > Hh } c
�
a.e. a re the
00
u
j= 1
A
rj
so i t i s
c ( A r . } = O , tf j E lN . F i x j and l e t k be J a g i ve n compact subset o f A . ; i t i s enough to prove that r J Hh h. -2 c ( k ) = O . S i n c e �> r .- 1 on k but r . < 1 on A r . i t 1 S c l ear J J J 00
that
Hh < 1 9
on
i s compact and
k Hh
so k
s
kc
1 < 1 -s
u
s= 1 on
1 }. { Hh <1-s 9 ks
Now
A s s ume that
rp E F ;
l i m rp ( z ) =O , z� �
- 50 'tf [, E a RB
and that
Then
Hh
k
s
2.
( 1 - s1 ) cp
on
ks .
Further-
, . . . , H h ) =0 o n RB \k s by Theorem V : 7 . H e n c e , ks s by Coro l l ar y V : 2 H h � ( 1 - s1 ) cp o n RB s o we have t hat k s 1 « l - -1 ) h < h = H a . e . ( do ) . « l -) hk and so H h Hh s s k s- k s h k ks ks s s
more , MA ( H h
k
I t f o l l ows that
hk = 0 s
wh i ch proves the c l a i m .
a . e . ( do )
S ome consequences
h.
We have now seen that i n the p l u r i s uperharmon i c c a s e , F
and
c
sat i s f i e s cond i t i o n s
1)
and
5)
r e spect i v e l y ;
here we l i s t some conseque n c e s o f th i s . Theorem V : 8 .
Let
{ uj } j E I
be a f a m i l y o f p l u r i subharmon i c
f unct i on s , l o ca l l y bounded a bove . Then there i s a p l u r i subharmon i c f unct i on
�
such that
{ sup u . « sup u . ) * } c { � = -oo } . jEI J jEI J Proof .
P ropo s i t i on
f r om Corol l a r y Theorem V : 9 .
111 : 1
111 : 3
g i ves
a n d Lemma
The capac i t i e s
3) 111 : 4 .
c
and
have the same z ero sets . F u r t hermo r e
f o r every Bor e l s e t
E.
s o the s t ateme n t f o l l ow s
d
a r e outer r eg u l a r and
- 51
-
That c i s outer regular follows from Propos ition 1 1 1 : 3 and Proposition 1 1 1 : 1 . By Lemma 1 1 1 : 4 c ( E ) = 0 � 3uEPSH such that u=- oo on E . Therefore c ( E ) = O � d ( E ) = O . Assume for a moment that we have proved that
Proo f .
d ( E } = f( dd c -h X ) n , for al l Borel sets E . E I f d ( E ) = O then by Corollary V : I -h X =0 a . e . so c ( E ) = O . E c ) n for all Borel I t remains to prove that d ( E ) = f ( dd -h XE sets E and we know this for E compact or open by Proposition V : 2 and Corollary V : 3 . I f E i s a given Borelset then , since c is outer regular , we can fi nd a decreasing sequence ( 0 J. ) 00J = 1 at open sets containing E such that ( -h X } * = ( inf h ) * . E j X O ]. Hence d ( E } -
c ( E ) = Sh X d o E is a capacity , we can find an increasing sequence ( K J ) J = 1 of compact sets contained in E with l im c ( K ]. ) =c ( E ) so 00
.
( -h X ) * K ]. Then Theorem
-�
.
( -h X ) * . E by
V:2.
- 52 -
d ( E ) = f ( dd C ( -h X
Thus
Theorem V : 9 .
n
wh i ch comp l e t e s the p r o o f o f
The set f u nc t i o n
Propos i t i on V : 3 .
G(E) =
E
)*)
inf [ s up - l
f u ( ddc v ) n ]
E
is
a n outer r eg u l a r capac i ty . The r e s u l t s i n th i s s e c t i on s how that Theorem 1 1 1 : 3
Proo f .
app l i e s . A s s ume that
Theorem V : l 0 .
00
O > u E P S H n L ( RB )
and p l ur i s ubharmo n i c f unct i o n s
i)
on uP
ii )
+
L
In
outside
U t /'f u
a s e t o f c - capa c i ty z ero .
III ) .
The p o i n tw i s e ( " f i n e " ) behav i ou r o f
E f+ H X
E
(z) ,
z
fixed
i s not c l ea r ; we do n o t k n ow i f t h i s i s a capac i ty i n
Choquet ' s s e n s e . The m i s s i ng part s�+oo
such t h a t
B
th i s s ec t i on , we h ave s e e n the s o l u t i on o f the " coar s e "
probl em ( c f . Sect i o n
B
on
Theorem 1 1 1 : 4 .
Proo f .
in
Then
B
00
t= p
R> l .
of n eg a t i ve ,
and
t h e r e are two sequences
continuous
whe n
we woul d l i ke to k now i f
we have i s
the
h
f o l l ow i ng t h eo r em .
is
s ?l' E , s�+oo . What
proper ty i i ) : I f
XEs ( z )� hXE ( z ) ,
E
-
53 -
Assume that K s /f K , s-++oo are compact sets of the unit bal l . Then for every f ixed z E B , l im h X ( z ) =h X ( z ) . s-++oo K s K
Theorem V : l l .
Follows from Lemma city i n Choquet 's sense .
Proo f .
111 : 1 ,
since E� c { X E ) is a c apa
Notes and references General references : Hormander , L . , An introduction to complex analysis I n several variables . Van Nostrand, 1 9 6 6 . Function theory of several complex variables . Wi ley-Interscience series , 1 9 8 2 . Krantz , S . G . ,
Plurisubharmoni c functions and positive di f ferential Gordon and Breach , 1 9 6 9 .
Lel ong , P . , f o rm s .
Theorem V : 8 as well as many other results in this section was f irst proved by E . Bed ford and B . A . Taylor i n : A new capacity for plurisubharmonic functions . Acta Math . Vol . 1 4 9 ( 1 982 ) .
Theorem V : 4 is proved by the same authors in : The Dirichlet problem for the complex Monge-Ampere operator . Invent . Math . 37 ( 1 9 76 ) .
J.
See also L . Caf farel l i , J . J . Kohn , L . N i renberg and Spruck . , The Dirichlet problem for non-l inear second-order
- 54 -
e l l i pt i c equat i on s I I . Comp l ex Monge -Ampe r e , and u n i f o r m l y e l l i pt i c equat i o n s . Commu n i c a t i on s o n P u r e and Appl i ed Mathema t i c s ( 1 9 8 5 ) . S . -Y . Cheng and S . T . Yau , O n t h e e x i s t e n c e o f a comp l e t e
Kah l er met r i c on n o n -comp a c t comp l ex man i f o l d s a n d t h e r e g u l a r i ty o f F e f f e rman ' s equat i on . Commu n i c a t i on s on P u r e a nd App l i ed Ma t hema t i c s 3 3 ( 1 9 8 0 ) . U.
Cegre l l , O n the D i r i c h l et prob l em f o r t h e Mo n g e - Ampere
op erator . Math .
Z.
1 85 ( 1 984 ) .
The qua s i c ont i nu i ty h a s a l s o been p r oved by A . Sadu l l aev , Ra t i on a l approx i ma t i o n a n d p l u r i p o l a r s e t s . Ma th . U S S R S b o r n i k . Vo l .
4 7 ( 1 9 8 4 ) . No .
1 .
A s t r o n g e r ve r s i on o f Theorem V : l 0 i s p r o v e d i n : U . Cegrel l , S ums o f cont i nuous p l ur i s ub h a rmon i c f u n c t i o n s and t h e c omp l ex Mon g e - Ampe r e ope r a to r . Mat h . In
Z.
1 93 ( 1 986 ) .
c o n t r a d i s t i nc t i on t o t h e s ubha rmon i c c a s e , E � h
XE
(x)
not s t rong l y subadd i t i ve . Th i s i s s hown by Johan Thorbi6rnson i n : A counte r ex amp l e t o the s t r o n g s u badd i t i v i ty of e x t r ema l p l u r i s u bha rmon i c f u n c t i on s . To appear i n Mh . Math . Theorem V : I I was f i r s t p r oved i n : U . Cegre l l , Capac i t i e s a n d extrema l p l u r i s u b h a rmon i c f un c t i o n s o n s u b s e t s o f A r k . Mat .
1 8 ( 1 980 ) .
n [ o
is
- 55 A set a ba l l
E
i s ca l l ed
,
B( zO r )
a nd
EnB ( z O , r ) c { �=-oo } .
n � - p o l a r i f t o every
� E P SH ( B ( z O , r ) )
zOEE
s u c h that
� j - oo
there i s and
It wa s s hown by B. Jos e fson : O n the
equ i va l e n c e b e tween l oc a l l y p o l a r a n d g l oba l l y pol a r s e t s f o r p l u r i s u bh a r mo n i c f u n ct i on s i n o n e a lway s c a n t ake
n [ o
Ark . Mat .
1 6 ( 1 9 7 8 ) that
n
�EPSH ( � ) .
F o r a s tocha s t i c po i n t o f v i ew s e e
M.
Fukush ima and M . Okada ,
On D i r i ch l e t f o r ms f o r p l u r i s u bharmo n i c f u n c t i on s . Acta Math .
1 59 : 3-4 , 1 987 .
VI
Further Properties of the Monge-Ampere Operator
Let B be the unit ball i n [ n and let P be the restri ction to B of all negative pluri subharmonic functions on RB where R> l . We saw in Section V that F=-P and 0 , the Lebesgue measure on B , give ri se to two natural capacities d ( E ) =Sup{ f ( ddc U ) n , - l
where f is any bounded function on B . We do not know i f { ( dd c u ) n ; - l �u�O , uEP} i s compact or convex but i f we def ine N to be the positive measures on B that are dominated by d ( i . e . O 0 and K compact , choose f O i f � J. EN , � J. an open subset A containing K such that d ( A )
:A �
J
-
-
57
-
a ) Every measure in M is absolutely continuous with respect to a measure i n N . b ) There i s a positive constant K such that KNCM .
Propo s i t i on VI : l .
a ) Since N i s weak*-compact and convex every Borel measure � has a unique decomposition U = � l + � L. ' where iJ 1 is absolutely continuous with respect to a measure in N and iJ 2 is carried by an F a -set E such that sup { V ( E ) , vEN} = O . I f uEM it follows from Theorem V : 9 that � ( E ) = O so � 2 = O which proves the claim . b ) Follows from Proposition V I : 2 below. Proo f .
There i s a constant c such that f - CP ( dd c U ) n � C f - cpd o , 'dcpEP ; 'duE P , - l u on RB\ rB and v r-
1 - cp ( dd c u ) n � f - cp ( dd c U 1 ) n = B
rB
= f - cpdd C V r II ( dd C u 1 ) n - l + f - cpdd C ( u 1 -v r ) /I ( dd C u 1 ) n- 1 rB = f - cpdd C v r 1\ ( dd C u 1 ) n - 1 - f ( u 1 -v r ) dd c cp ,\ ( dd C u 1 ) n- 1 < rB � f - cpdd c v r /\ ( dd c U 1 ) n- 1 � ( repeat n- l times ) < rB � f - cp ( ddC v r ) n c r J - cp ( w ) da ( w ) rB I w l =r =
:=;:
- 58 -
whe r e [
n
a
i s the s u r f a c e mea s u r e on the bo unda r y o f c
a n d where
Now ,
�
s i n ce
i s a con s t a n t , o n l y depend i ng on
r
rB r
in and
n.
i s s u bharmon i c we k n ow
f �d o
< 2n r
B
f
< 2n r
�d o
rB
f
�d a .
The r e f o r e
c
r
f
-
< r rr �d o -
2n
f
-
B
cp d o
wh i ch proves P r opos i t i on V I : 2 . Note t h a t i f d ( X ) �d ( U ) K K
K
i s a c ompact s u b s e t o f
a n d we c a n have s t r i ct i n equa l i ty .
For l et
and
L
=
{ z E a:
2
;
I z I � "43 } .
Then u
u and
K
L
B
=
=
max {
l og l z l l og 4
max {
1 02 1 z 1 l og 4 / 3
' -1 } ,
'
-1
}
then o f c ou r s e
- 59 dd c u K ::::; = log 4 i og 4/3 J u L dd c [ max ( log I z I , -log 4 ) ] dd c [ max ( log I z I , -log 4 / 3 ) } c ::::; 10g 4 i09 4 / 3 J u Ldd [ max ( log I z I - 1 0g 4 / 3 ] 2 = = 10g 4 1�9 4/3 Jdd c [ maX { log l z l , -109 4 ) ] 2 > > 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 ¢ <
{ s<¢< l }
+ 00
and ( ¢-s ) ( dd c � ) n < J { s<¢< l } Since � there is to every f ( z } ff ( z O ) ' �zEK. apply the maximum bounded above on Proof .
+ 00 .
is pseudoconvex and ¢ plur i subharmoni c , z O E { s<¢< l } an analytic functions such that I f we restrict � to { f ( z ) =f ( z O ) } and principle , we c onclude that � i s uni formly TI'\K . It i s therefore no restriction to
- 61
-
assume that �iO . Let s
f - � ( dd c � ) A ( dd c � ) n- 1 + f ( � _ r ) ( dd c � ) n <
{ r<�< l }
{ r<�< l } � l im - �dd c max ( � , t ) " ( ddc � ) n- l + f ( �-r ) ( dd c � ) n = f t r { r<�< l } { r<�< l }
+
f d max ( � , t ) 1\ d C � /I ( dd c � ) n- l + f ( IV - r ) ( dd c � ) nJ =
{ r<�< l }
{ r
+ f ( ma x ( IV , t ) -r ) d c � " ( dd c � ) n- 1 { �= 1 }
f ( max ( 1jJ , t ) -r ) d C � 1\ ( dd C � ) n- l +
{ IV =r } +
f ( r-max ( � , t ) ) ( dd c � ) nJ � f �d C � 1\ ( dd c � ) n- l + -
{ r<�< l }
{ �=1 }
+ f ( �-r ) d c � A ( dd c � ) n - l . { �= 1 } Since the r ight hand side i s uni formly bounded in r , s
- 62 -
f I qJ l dd c 1jJ J\ ( dd c qJ ) n- 1 +
f ( 1jJ-s ) ( dd c qJ ) n < +oo
{ s<1jJ< l } { s<1jJ< l } where each number is non - negative . An example
The following example shows that the convergence factor 1jJ-s really is needed in the theorem . There are funct ions qJ and 1jJ such that
f ( ddc qJ ) n =
+00
{s<1jJ< l } where qJ can be taken to be pluri subharmon ic and bounded . Defi ne u ( z ) =- li z I 2 � then u i s subharmonic on 1
-
_
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 + i2 ) ] . 2 Then qJ n i s plurisubharmonic on { .l2 < 1jJ < (�) 2 1 } for i f ( l ':- n ) 2 1 z 1 2 .s. 1 we have , when 1jJ ( w ) > .;. 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 <6 ( n + r ) -8 /32" ( n +
2
r ) ( l + n ) I w l 4 + 4 ( l +n ) 2 I w l 8
n+
2 2 o < 5 ( n + i ) + � 4 ( 1 + ) 2 1 w 1 8 - a ll in i ( 1 + Tl ) I w 1 4 +
n
which holds with I w l 4 i n the interal above - the right hand side is then strictly larger than n22 2 + 2 n 39 - - a -( n !l- ) 1 +n ) 2 1 ( .! 5 ( n + !l2 - ) + J... 3 40 2 O . 5 ( 11n)2 •
Therefore
+
=
dz �w l ( dd c CP n ) 2 � c f « 2 ) 3/2 1 +n ) I z I A 2 1 8 2 0 2 2 2 n 2 < l z l +} I w l « 21 )
f
_
- 64 -
where
2 2 l 23 ( n + t- ) - ( 1 + n ) 1 w 1 4 ) n ) 2 < ( 1 + n ) 2 1 z 1 2 < 21 + 4 (/ 2 An = ; �2 ; n � n + 2 +2 4 w l < < B + l o. 1 n +n
1(1
- 32 1 w i 8 )
rn -�
(1+
'--
3 9 Y0.1 and where c i s a and where we have chosen Y135 < o. < 8 < 4O" strictly positive constant . The right hand s ide i s not smaller than 2 (n .!L)dw 3 9 12. 2 c { ( a 2 - -35 ) + > 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 ,
and
+
( d positive constant )
=
00
-
2 1
<
,, 2 1 j =20 j
f ( dd c
2 so
if
we put
f }.
- 65 It i s possible to mod i fy the cp n : S to get the function cp of class C 00 on We do not know i f one but sti l l have f ( dd c cp ) 2 = +00 . can mod i fy to be C00 on Q\K Notes and references
Remark .
For the de c omposition of measures used i n the proof of Proposition VI : 1 a ) see Secti on XI . Proposition V : 2 was f irst proved by Dema i l ly , J . -P . , Mesures de Monge-Ampere et caracteri z ation geometrique des vari etes algebriques affines . Bulletin de la So ciete Mathematiques de France , Memoire 1 9 , 1 1 3 ( 1 9 85 ) , 1 - 1 2 5 . Propos ition VI : 3 i n the case K equal to one point was proved by Gr i f f i th , P . , Two theorems on extension of holomorphic mappings . Inv . Math . 1 4 ( 1 97 1 ) , 2 7-6 2 . Theorem VI : 1 i n the pluri subharmonic and c 2 case was proved by For naes s , J . E . and S i bony , N . , Pluri subharmonic functions on r ing domains . Krantz , S . G . , Ed . , Complex analysi s . Seminar University Park PA , 1 9 8 6 . Springer Lecture Notes i n Mathematics . Vol . 1 2 68 . Proposition VI : 1 and Theorem VI : 1 wil l appear in Cegre l l , Plur i subharmonic functions outs ide compact sets . Proc . AMS . The above example shows that there are functions that can be subextended and sti ll have unbounded Monge- Ampere mass near a compact set . In smooth domains , there are pluri subharmonic functions that cannot be subextended : Bed ford , E . and Taylor , B . A. , Smooth plurisubharmonic functi ons with no subextensions . Manuscript 1 986 .
U. ,
VII
Green's Function
An open subset D of [ n i s cal led pseudoconvex i f -log d ( z , CD ) i s plurisubharmonic . ( d ( z , CD ) =inf j z-w l ) . wECD I f there i s a continuous and pluri subharmonic function � c l w l 2 ' �PE a D , �wE[ n i , j == l a Z 1. dZ J.
D,
for some c>O then D i s said to be strict ly pseudoconvex . De f i n i t i on . Let D be an open subset of [ n , n> l and assume that �ESH ( DxD ) . Then � i s called 2 -pluri subharmonic ( 2 -PSH ) on DxD if �Bz � � ( z ,w ) EPSH ( D ) , 'dwED n
3w � � ( z , w ) E P S H ( D ) , 'd zED.
De f in i t i on .
Green 's function relatively to D is
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 ) ] } .
- 67 -
u E P SH ( D ) , u < O , wO E D i s bounded a bove near t he n
Lemma VI I : l l .
If
and i f
u ( z ) - l og ! z -w o l
u ( z ) �l og l z -w o l - l og d ( z , CD ) , � z E D . Proo f .
Then
Con s i de r f o r
l E Vz
TE[
a nd
zED
a n d s i nce
l S s u b h a rmon i c on
Vz
we g e t
g ( l ) � s u p g ( T ) �SUp - l og I T ( Z -w o ) 1 = T EV T E av z z
Ther e fo r e
u ( z ) �l o g ! z -w o l - log d ( w O , CD ) , z E D .
Corol l ary V I I : l .
near the d i ag o n a l
If
u�O , u ( z , w ) - l og l z -w l
�cDxD
and i f
loca l l y bounded
u E 2 - P SH ( Dx D )
then
u ( z , w ) �l og l z -w l - l og max [ d ( z , CD ) , d ( w , CD ) ] .
Corol l a ry V I I : 2 .
If
v f ( z , w ) =l og
f E H oo ( D ) , I f l � l , ftcon s t . then
f ( Z ) -f ( W ) I � W( z , w). I l-f( z )£ (w) _
- 68 -
It is clear that v f E 2 -PSH ( DXD ) ( but v f i s not PSH ( DxD ) ! ) , vf�O . Furthermore , v f ( z ,w ) -log l z-w l i s locally bounded above on so by Corollary VII : 1 , v f�W' Proo f .
6
Propos i t i on VI I : l .
2 ) WE 2 -PSH 3) I f furthermore
1 )
0
W( z , w ) =W ( w , z ) is strictly pseudoconvex then
l im W( z , w ) =O , �wED . z-+ aD Since log l z-w l -log max [ d ( z , CD ) , d ( w, CD ) ] is symmetric so i s w. 2 ) Denote by W* the smallest upper semicontinuous ma j orant of W. Then W i s subharmonic on DxD and we claim that W* 2 -PSH ( DxD ) . I f we take � to be a function only depending z --) w-n on I z I and consider W E , c ( z ,w ) = f W( � , n ) � (-E � � (-C-) then W w* when 0 0 and W i s 2-PSH . Furthermore W* ( z ,w) < log l z-w l -Iog max [ d ( z , CD ) , d ( w, CD ) ] since the right hand side is upper semiconti nuous . Therefore W = W* which proves 2 ) . 3 ) If is strictly pseudoconvex it i s well k nown that to every pE aD there is a function f P . analyti c on D and continuous on D such that f ( p ) = l and I f 1 < 1 on D . Proof .
1 )
IS
1: � E , v
E ,
':.
1: ( , v
0
By Corollary VII : 2 v f p ( z , w ) og I f ( Z ) -f ( W) I .s. W ( , w ) l -f ( z ) f ( w ) o > 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
As s ume that
D
i s s t r i ct l y pseudoconvex .
The f o l l ow i ng cond i t i ons a r e equ i va l e nt . i) lim ( dd max ( W ( z , � ) , t » n = ( 2 TI ) n , � z E D t�-oo D
f �
ii )
W=u
iii )
u ( z , w ) =u ( w , z ) , � ( z , w ) E DxD
iv l
D3w
�
u ( z , w ) E PS H ( D l , � z E D .
- 70 -
Proo f .
ii )
�
i i i ) follows f rom Propos ition VI I : l .
i i i ) � iv) si nce z.... u ( z , w ) EPSH ( D ) , tlwED . iv) � i i ) . I t follows from Corollary VII : l that u ( z , w ) W . i i ) i ) ( dd c u ( z , w ) ) n �O outs ide w . Therefore l im f ( dd c ( maX u ( z ,w) , t ) ) n �l im f ( dd cmaX ( 109 ! Z-W ! , t ) ) n � ( 2 1T ) n . t D t D i ) � i i ) . Fix wED . Since max ( W( z , w ) , t )imax ( u ( z , w ) , t ) and since both functions have boundary values z ero , it follows from Lemma V : 3 that f ( dd c max ( W ( z , w ) , t ) ) n � f ( dd cmax ( u ( z , w ) , t ) ) n � ( 2 1T ) n �
.... - oo
.... -oo
•
o
C
0
Therefore i ) means that ( dd W( Z , W ) ) n = 0 for z �w because , since W( z , w ) -log ! z-w ! i s bounded near z=w, l im f ( dd c max ( W( z , W ) , t ) ) n = ( 2 1T ) n for every neighborhood E of t -oo E w. Then ( l -E ) W( Z , W ) I S equal to z ero at a D , larger than u ( z , w ) near w . Therefore , Corollary V : l gives that ( l -E ) W( Z ,W)�U ( Z , W) for every E> 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
8
is a
8=
+ I udClog + I z 1 ,\ I z l =R Proo f .
y ( u )�O . C
n
= ( 2 1T ) n
d - c l o s ed
so that
( n - l , n - l i - f orm t hen
S log + I z I dd c u ,\ 8 + I z I
8 - l og
R I dCu II 8 . I z l =R
S t oke s ' f ormul a .
Theorem VI I I : 2 .
If
10g+ l z l �uEL
then
y ( u ) = I u ( w ) da ( w ) - 1 n I l og + I z I dd C u j\ ( dd C lo g + I z I ) n - 1 . C l u l =l [n Proo f .
8
Take
to be
( ddClog+ l z l ) n - l
in
P r opos ition
VI I I : 2 .
Then C
n
I u ( Rw ) da ( w ) -log R I ddCu A ( ddClog+ l z l ) n- l = I z l �R Iw l =l
= C
n
I u ( w ) da ( w ) - I 10g+ l z l dd c u A ( ddCl og + l z l
I w l =l
) n- 1
I z l
a n d t h e l e f t h a n d s i de i s , by Lemma V I I I : l n o t l e s s t h a n z e r o .
- 75 -
Hen c e ,
f
o <
�n
So
l 1 0g + l z l ddcu /\ ( dd C l 09+ l z l ) n - ..s. C n
l u l =l
f
Cn I
ddcu A ( dd C l og+ l z l u ( zw ) do ( w ) - l ogR I w l �l I z l
=
cn f
f
U ( RW ) dO ( W ) - 1 09R ) + 1 09R [
[n
I w l =l
S
ddc u A ( ddC l og + I z I ) n - l ] I z I ..s.R Cn [
a nd
1 0gR
sin c e
f
�n
S
Iw l =l
dd C u 1\ ( dd C l og + I z I ) z I �R
l og + I z I
wh i c h proves
A
n- l
f
1 0g + l z l ..s.u E L
f l og + I z I /\ dd C U /\ ( dd C l og
+
) n- l _
ddcu A ( ddC l o g + l z l I z l �R
-+
0;
R -+ +00
ddcu A ( ddc l og + I z I ) n- l < +00
If
) n- l
ddcu A ( ddC l og+ l z l
the theorem .
Corol lary VI I I : l .
u ( w ) do ( w ) .
==
u ( Rw ) do ( w ) - l ogR ] + l ogR
S I
I
then
IzI)
n- l < +00 .
) n- l
- 76 -
L
u 1.
If
Lemma VI I I : 2 .
i s a d e c r ea s i ng s equence o f f u n c t i o n s i n
l i m u 1. =u o > l og + l z l
such that
then
l im y ( u , » y ( u O ) . J O n the other hand , we have f rom Theorem V I I I : 2 that + l im l og 1 z 1 dd c u j A ( dd c l og z ) n � j-+ + oo
It i s c l ea r f r om the de f i n i t i on that
S
�
S
+1 1
u O da ( w ) - c l og 1 z 1 dd c U o A ( dd c l og n z l
S
+
+ 1 z 1 ) n- l
by Theorem V : 2 . I f we l e t
R-++oo
a nd app l y Theorem V I I I : 2 aga i n we g e t the
des i red c o n c l u s i on . Def i n i t i on .
a nd + 00 ,
Let
uEL ,
we t h e n de f i n e
f ( u ) =l im f r ( u ) ( s i nce fer ( u ) r-++oo f r ( u ) I S decreas i ng i n r ) . I t i s c l ear that
In
if
n=2 ,
then
y ( u )� f ( u ) .
y ( VE ) = f ( V E ) .
IS
If
f r ( u ) = s up u ( z ) - log r z I =r convex and bounded a t
1
ECB ,
B
t he u n i t bal l
The f o l l ow i ng examp l e shows t hat
we can have s t r i ct i nequal i ty .
- 77 Fix a> l and take K= { ( z l , z 2 ) E[ 2 ; 1 z 1 I 2 +a I z 2 I 2� 1 } Then VK( z ) = 2"1 log + ( I z 1 I 2 + a I z 2 1 2 ) so y ( VK ) = � f log ( I w I 2 +a l w 2 1 2 ) d O ( w ) < � log a = f ( VK ) · I w l =1 Examp l e 1 .
Propos i t ion VI I I : 3 .
log + l z l
Proo f .
If
we fi rst take j =l , Proposition VII I : 2 gives
+ f ud C log + l z l f\ ( dd c u ) n - l _log R f d C u /\ ( dd c u ) n - l � C n r R ( u ) + I z I =R I 1 =R z
+ log R f ( dd c u ) n -+ C n f ( u ) , R-++oo since 10g R f ( dd c U ) n-+ O , R-++oo . I z I �R I z l >R Assuming the propos ition for j , we want to prove it for j+ 1 . C
fu8 i + 1 == fu ( dd u ) n- j
-
l f\
( dd C log + 1 z I ) j + l =
•
- 78 -
2.
. ) fu ( ddC u ) n-j A ( dd C log + 1 z I ) j C n f ( u ) 2. ( assumpt Ion
2.
f u ( dd C log + 1 z I ) n + C n J. f ( u ) + C n f ( u ) ::: fu ( dd C log + 1 z I ) n +
+
Corol lary V I I I : l .
I f 10g + l z l EuEL then
f u ( w ) d a < �n f 10g + l z l ( dd c u ) n + y ( u ) + ( n- l ) f ( u ) . I w l ::: l In particular i f
ECB ,
the unit
ba l l ,
then
f VE ( w ) d a ( w ) 2. y ( VE ) + ( n- l ) f ( VE ) ·
I w l ::: l
Take j=n- l in Proposition VI II : 3 . Then , VII I : 2 , we get Proo f .
by
Theorem
fU ( W ) da ( W) - y ( u ) ::: � J 1 0g + l z l dd C u ( dd C log + l z l ) n - l n n I w i ::: 1 C
<
which proves the first statement . The second statement follows from the fact that i f E c B then supp ( dd C VE ' n CB .
- 79 Remarks and references A proof of Proposition VII I : 1 is i n J . S i ci ak , Extremal plurisubharmonic funct ions i n � n , Proceedings of the f irst Finnish-Polish Summerschool in Complex Analysi s in Podlesice , 1 9 7 7 , pg . 1 2 3 - 1 2 4 . Theorem VII I : 1 i s due to N . Levenberg , Monge-Ampere Measures Associated to Extremal Plur i subharmonic Functions in � n . Trans . Am . Math . Soc . 2 89 ( 1 9 85 ) , 3 3 3 - 3 4 3 . Lemma VII I : 1 i s proved by B . A . Taylor , A n estimate for an extremal plur i subharmonic function . Seminare d ' Analyse P . Lelong , Dolbeault-H . Skoda , 1 9 8 1 / 1 9 8 3 . Spr inger Lecture Notes i n Mathematics 1 028 . Thi s paper also contains a somewhat weaker version of Corollary VI II : 1 . In S . Kolod z i e j , The logarithmic capacity i n � n ( To appear -f ( V ) in Ann . Pol . Math . ) , it was proved that C ( E ) =e E i s a - (V ) capacity in Choquets sense . That e y E i s a capacity was proved by the same author i n : Capacities associated to the Siciak extremal function , Manuscript . Cracow . 1 98 6 . The relationship between y and f has also been studied by J . S i c i ak , On logarithmic capacities and pluripolar sets i n � n . Manuscript , October 1 9 8 6 . Using Corol lary 6 . 7 i n E . Bed ford and B . A . Taylor , Pluri subharmonic functions with logarithmi c s i ngularities , -f ( V ) Manuscript 1 987 , one can prove that e E is an outer regular capacity . has studied capacities and extremal pluri subharmonic functions in connection with transf i nite diameter v. P .
Zaha r j uta
- 80 and the Bernstein-Walsh theorem : Transfinite diameter Cebychev constants and capacity for compact in � n . Math . USSR Sbornik , Vol . 2 5 ( 1 975 ) , No . 3 . Extremal plurisubharmoni c functions , orthogonal polynomials and the Bernstein-Walsh theorem for analytic functions of several complex variables . Ann . Polon . Math . 33 ( 1 9 76 ) . ,
For results and more references , see : Nguyen Thanh Van and Ahmed Z er i ah i , Fami lIes de polynomes presque partout bornees . Bul l . Sc . Math . 2 c Serie 1 07 ( 1 9 8 3 ) . OO P l e s n i ak and W . PawXucki , Markov 's i nequality and C functions on sets with polynomial cusps . Math . Ann . 2 75 ( 1 986 ) , 467-4 80 .
w.
Pl u risubharmoni c measures and capacities on complex mani folds . Russian Math . Surveys 36 ( 1 98 1 ) .
A . Sadul l aev ,
IX Gamma Capacity
( the Choquet I ntegral ) . Assume that f 1 S a non negative function and c a capacity . Then f fd c i s def ined by
Def i n i t ion
co
ffd c = fC ( {x ; f ( x » s } ) ds . o
Let p be a precapacity . f is said to be p-capacitable i f
Def i n i t i on .
A
non-negative function
f fd p sup f gd p ; g� f , g upper semi -continuous . =
A ssume that E v ' of p - capacitable sets . Then
Lemma I X : l .
00
U
We have p ( E v ) = s u p V= 1 vEN Choose v so that Proof .
p(
is an increasing sequence i s p - capacitable . E v v= l
vEN ex>
u
p( E
v
U
) '
Le t
E ) < p ( E ) + E/2 v v= 1 v and a compact subset k of E v such that
Then
i s compact in
k
p( so
00
U
v= l
E
v
co U
v= l 1S
co
U
v= l
E
v
Ev ) < p ( k ) + £ p-capacitable .
and
£>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
Swarms . Product capacities . De f i n i t i on .
A c lass of functions
(L
' F EcV
is called a swarm i f a) for every fixed xEU , E�L E ( x ) is a capacity on v , for every compact subset K of LK ( X ) i s a bounded , s) upper semi -continuous function with compact support . v,
Choose U=v=� n and denote by X E the character istic function of E . Then ( X E ) E c � n i s a swarm .
Example .
c
Assume that i s a capacity on U and ( L E ) ECV a swarm. Then C ( E ) = fL E d C i s a capacity on V . Furthermore , i f L E ( x ) is subadditive for all x ln U and i f c i s strongly subadditive then C is subadditive .
Theorem I X : 3 .
- 84 Proo f .
i i ) Let
i ) c lear . E ' v
vEJN
f
be a n o n -d e c r e a s i n g s eq u e n c e o f s u b s e t s o f
V.
Then
s up C ( E ) = l i m v v-+ + 00 v Em
fL E
d V
c
00
=
f O
= C(
i i i ) Let of
V.
fC ( {XEU ;
= l im v-+ + 00
L
0
E
( x » s } ) ds = V
00
l im c ( { xE U ; L > 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 ) )
C ( E )
i
( E ) ) + (/2 . \I
(
( E ) ) +£/2 Th e n
and
U.
- 90 ) + E < C ( E,,v ) + E
C ( E )
and i i i )
i s p roved .
S i nc e
a ( E u E 2 ) =a ( E ) u a ( E ) ; 1 2 1 i i i
iEI ,
we have ,
for
c
subadd i t i ve
C
wh i ch p r ov e s t h a t
i s s ub a dd i t i ve .
A s s ume n ow t h a t
c
K v ' v Eill ,
i s a c a p a c i t y . Let
U.
d e c r ea s i ng sequence o f comp a c t s u b s e t s o f
be a
We have t o p r ove
that 00
i nf C ( Kv ) = C ( n Kv ) . vEN v= 1 Choo s e
so that
C(K ) < c(a (K ) ) n n n S i nc e a �a ' n O
(a ) i iEI
+ 1n
i s norma l , a nd c ompl e t e , we c a n a s s ume t h a t
u n i forml y on c omp a c t s u b s e t s o f
ao (
00
00 n
00
K n ) :::> 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 )
n
K
(a
-1
of
such that
a{E)
(K»
+E
a n d s i nc e
a
-1
(K)
i s a comp a c t s u b -
t h e proo f i s c omp l e t e .
E,
F a v o r ov ' s c a p a c i ty i s b y d e f i n i t i o n
r
whe r e
c
F n
{ E ) = s up { c
n
{a{E»
i
a c omp l ex u n i t a r y t r a n f ormat i o n } ,
i s t h e l oga r i thm i c capa c i ty o n
�.
- 94 -
Coro l l ary IX : 4 .
The a n a l ogy o f C o r o l l a r y I X : 5 h o l d s t ru e f o r
F r ' n Fo l l ows f rom Theorem I X : 6 a n d C o r o l l a r y I X : 4 .
Proof ,
Propos i t i on I X : 2 .
If
E
is
u n i tary t r a n s f ormat i on s . y ( E ) =O , n
�
r ( E ) =O .
n � - p o l a r i s i nv a r i a n t u n d e r c omp l ex
The property b e i ng
Proo f .
n � -po l a r t h e n
I t i s t he r e f o r e e n ou g h to p rove t h a t
A s s ume the propos i t i on t o b e t r u e f o r
n- 1 .
T h e n by
Theorem I X : 6 we have
whe r e
� t - oo
EC { �=-oo } .
i s a p l u r i s u b h a rmo n i c f u n c t i on s uch t h a t
B u t a s u b h a rmo n i c f u n c t i o n
out s i de a s et o f
caP = O 2
( t -oo )
on
�
is
> -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
f l c [ rz , � ] f ( � ) da ( � ) I P da ( z ) < +00
whe r e 1 fEH ( a , aB )
extends t o
The s p a c e
LH
1
IS
t h e C a u chy k e r ne l . Hence , each
H(B) .
c o n s i s t s o f the f u n c t i on s
f
in
H(U)
- 1 00 -
IfI
s u ch that 1
LH ( U )
is
Il f ll whe r e
h a s a p l u r i s u bharmo n i c ma j o r a n t . A n orm o n
PH
LH
1 = inf{h( 0 ) ;
h E PH ( U ) ,
I f I �h }
d e n o t e s the p l u r i s u bha rmon i c f u n ct i o n s . Th i s n o rm i s
c a l l ed the Lume r norm . For e a c h po i n t
z EU
we d e f i n e t h e c a pa c i t i e s
Q ( E ) =s u p u t E ) z u EM z R ( E ) = s up u t E ) . z EN \J z
O b s e rve that
R
a nd that
R
Q - c a p a c i t y . S e e examp l e b e l ow . We now w i sh to s tudy f u n c t i on a l . Put in
L
P=Re A ( U )
Q
c a n v a n i s h a t s e t s o f pos i t i ve R=R
( We wr i t e
and
o
Q=Q o ' )
a n d t h e r e f o r e e x t e nd i t t o a a nd l e t
P
be t h e c l o s u r e o f
P
1 l oc ( U ) . Let
�
b e a r e a l -va l u e d f u n c t i on o n
� ( z ) = s up u EM Theorem X : l .
If
and put f o r
f* �dU . z
�
i s uppe r s em i - c o n t i n uous o n
� ( z ) = i n f { h ( z ) ; h E P , h�qJ o n a u } , Proo f .
au
The p r o o f o f Lemma 1 1 1 : 3 a p p l i e s .
zEU .
au,
then
z EU
- 1 01 Lemma X : 1 .
If
O�cp
i s upp e r s em i - c on t i nuou s , t h e n
c on t i n u o u s a n d p l u r i s ub h a rmo n i c o n
- cp
is
u.
By Theorem X : 1
Proo f .
- 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 z '�z r ema i n s t o prove t h a t - cp i s c o n t i nuou s , a n d the n e x t p r opos i A
t i on s h ows t h i s . (h ) i iEI
be a f am i l y of ha rmon i c
Propos i t ion X : l .
Let
f u n c t i on s o n
u n i f o r m l y bounded a bove . Then
U,
c o n t i n uo u s .
1
iEI ,
a r e n e g a t i ve .
Put that
H
H=sup h . . 1 iEI
S i nce each
1
h.
i s con t i n u o u s i t i s c l ea r
i s l owe r s em i - c on t i n u o u s s o w e h a v e t o p r ove t h a t
a l s o i s upper s em i - c on t i n u o u s . L e t Zo
upp e r s em i - c o n t i nuous at
be g i ve n . I f
f or s ome
c a n choose
h
Take
so that
r >O O
Zo
v
H
H
i s not
there i s a sequence
w i th l i m i t
put
is
W i t h o u t l o s s o f g e n e r a l i ty , we c a n a s s ume that a l l
Proo f .
h. ,
s up h . iEI 1
E>O .
We
w i th
B(z
O
,r
O
)
i s r e l a t i ve l y c ompa c t i n
U
a nd
- 1 02 We then have compact i n
l i m r = r O and B ( z v , r \) ) , v E TIiJ, v-++oo \) U f o r v l arge enough . S i nc e
a re r e l a t i ve l y
w e have l im h) z O ) v-+ +oo =
>
=
�
J
h) z ) dz = l im m ( B ( Z , r ) ) o +oo v-+ B ( zO , rO )
m(B( z ,r ) ) 1 h ( z ) dz > im m ( B ( zv , rv ) ) m ( B ( z O O v-++oo v , rv ) ) B ( z , r ) v O O
J
1
m( B ( z , r ) ) hv ( z ) d z = l im m ( b ( z v r v O , O ) ) m ( B ( z v , rv ) ) v-++oo B ( zv , rv )
J
m ( B ( z v , rv ) ) = l im h ( z » H ( Z O ) + E: . v-++ oo m ( B ( z O , r O ) ) \) v H e n c e , there i s a
v
s o that
wh i ch i s a contrad i ct i on and the theorem i s p r oved . Exampl e .
ba l l lO n
Let E= { ( z , 0 ) E cr: 2 ; I z 1 = 1 } C S= Cl B '"� 2 . The n R ( Z , w ) ( E ) = 0 , � O tw
On the other hand , by Theor em
X: l
wher e ( s i nc e
and Lemma
X: l ,
B
i s the u n i t i s pol a r ) .
E
Q(
O, Z , w) ( E »
� ( z , w ) EB . Lemma X : 2 .
f unct i on s o n
Let U.
( In
) be a sequence o f negat i ve s u bha rmon i c 'f' i i E I I f there i s a po i n t z O E U such that
- 1 03 i n f - 00 t h e n t h e r e i s a s u bh a rmon i c f un c t i o n iEI l 00 ( (Il . ) . 1 s u b s equ e n c e so that F u r t he r mo r e , J= f u n c t i on s
converge n c e
Proof . inf h . ( iEI 1
z
i
'
't' l .
J iEI ,
a r e h a r mo n i c t h e n
i f the
i s ha rmon i c a n d t h e
i s u n i f o rm .
S i nce
U
0 ) > - 00
i mp l i e s t h a t
inf iEI
h
and a
f
K
i s a s s umed t o b e c o n n e c t e d ,
l
h. (
z
) d z > - 00
K
f o r e v e r y comp a c t s ub s e t
of
we a k l y c o n ve r g e n t s u b s eque n c e
Hence
U. (h
1. .
)
J
l im j -+ +oo
t h e c o nd i t i o n
s o t h e weak l i m i t
Th i s c omp l e t e s t h e p r oo f ,
because
00
(h
If
J. = l '
i
)
iEI
conta i n s a then
i s a s u b h a rmo n i c f u n c t i o n .
it
i s a we l l - k n own
fact that
a w e a k l y c o n v e r g e n t s e q u e n c e o f h a rmo n i c f u n c t i o n s c o n v e r g e s u n i f o r m l y o n c ompac t s . Lemma X : 3 .
If
au
s em i c o n t i n u o u s o n
By Lemma X : 1 ,
t i n uo u s
for every
l < v <m
5 0 s i nc e
-
v
(
-00 ,
z
)
v
a r e n o n - n eg a t i v e a nd upp e r then
-
i 5 c o n t i n u ous
U.
i s p l u r i s u bh a r mon i c and con-
m.
-�
and i f
a n d p l u r i s u b h a r mo n i c o n
Proo f .
wh e r e
Moreover
=
-
- 5 up WEM
f z
5 up
�
-
i s p l u r i s u b h a rmon i c on
m-+ +OO
U.
- 1 04 -
I t r ema i n s to prove c on t i n u i t y . We p u t and c l a i m t ha t
A= { hEP ;
h�-� }
I S non -empty a nd
A
h = -�
s up hEA
wh i ch w i l l prove t h e cont i nu i ty , by P r op o s i t i on X : l . By de f i n i t i on , a sequence
h EP m
s up h < - � hEA
so t ha t
h
a nd Lemma X : 1 prov i de s u s w i t h ----
-
< - s up � m v l < v <m
1 -- � ( o ) < hm ( 0 ) . m �
and
Lemma X : 2 shows t h a t we c a n s e l ec t a s u b s eq u e n c e
(h
m.
)
00 J. = l '
converg i ng
J un i f ormly on compact s u b s e t s . I t i s c le a r t h a t t h i s
l im i t
hO
be l o n g s t o
shown
that
A
a n d that
A
h ( O ) = -� ( O ) . O - � > - oo
i s non -empty a n d t h a t
eve r ywh e r e .
can repeat the a bove a r g um e n t for any po i nt i n Let
Theorem X : 2 .
� au .
a b l e f un c t i o n o n
be If
c
and
l im
v-+ +00 and con t i nuous on U . S i n ce
Proo f .
�
cp ( z ) = s u p \.l E M where
g
S o we
U.
n o n - n eg a t i ve a nd u n i ve r s a l l y me a s u r -
t h e n t h e r e i s a n i nc r ea s i n g
� ( O ) < + 00
sequence o f upper sem i - c o n t i n u ous f u n c t i o n s � <� v
T h u s we have
Fu r th e rmor e ,
-�
� ' v E lli , v
such that
I S p l u r i s u bh a rmo n i c
i s u n i ve r s a l l y mea s u r a b l e we have
z
s up f CP ( � ) d \.l ( E� ) = g��
va r i e s a�ong the upper
= inf - g( z ) g��
\.l E M
z
semi -cont i nuous f u n c t i o n s .
Hence , - cp ( z ) = - sup g ( z ) g� cp
sup
-
so
-�
1 05 -
i s upper sem i -con t i nuous by Lemma X : l .
By Choquets Lemma there i s a denume r a b l e subset of
{ g�� }
s o that i f
l< i n. f - g . J
then
J
1_ < -� .
1
Now
- sup g . ( z ) = sup
J
j
-
I1 E M
< i n f - sup I1 EM z j
f
00
( g . ) .=1 J J i s l ower semi -conti nuous and i f
z
f sup j
g . d 11 � s up sup g . d 11 < J I1 E M j z
fJ
-
g j dl1 = i n f- g
j
J
.
.
By Lemma X : 3 the l e ft -hand s i de i s a c on t i nuous funct i on so i t f o l l ows that -------
- s up g . ( z ) < -� ( z ) . J j �
But on the other hand ,
s up g . < �
j
- � = - s�. up g . .
J
so
s up g . < �
-
J -
and there fore
J
I f we take
� m= sup g . l < v<m
we get a s equence o f funct i ons w i th
J
the r equ i red propert i e s . Corol l ary X : l .
a b l e f u n c t i on on Proo f .
If
�
au
j "
then
a non-nega t i ve and u n i ve r s a l l y mea sur� ( z ) = i n f { h ( z ) i h El' , � � h } .
U s e Theorem X : 2 t o f i nd an i nc r ea s i ng s equence
of upper s em i - cont i nuous funct i o n s such that l im � v = � . v-+ +oo Lemma X : 3 .
� v < � and App l y now the a r g ume nt i n the end o f the proo f o f
-
1 06
-
F a - s e t i n a u o f van i sh i ng Q-capac i ty . Then t h e re . i s a s eq u e n c e ( f v ) oov= l of funct i on s in A ( U ) w i th I f v l � 2 s uch that Let
Theorem X : 3 .
be a
E
f) z ) = O ,
1)
l im
2)
l im v-++ oo
out s i de a s e t o f van i sh i ng
3)
l im fv ( � ) = 1 V-+ +OO
on
zEU,
We know that
Q-capa c i ty ,
E.
E=
u K whe r e i s an i nc r ea s i ng v= l v u n i on o f compac t s et s . By Lemma X : 1 , we can choose g v E A ( U ) ,
Proof .
1m g v ( 0 ) = 0 ,
P v =e
- vg
v
h v =Re g v
Then l im
hv > X ; h ( 0 ) < 1 Iv3 . v K
such t h a t
I P v 1 -< 1
P v ( z ) = l im
v-+ +oo
s i nce e
v
h v -> 0
-vh ( z )
v
and i f
zEE
Put
we g e t
=O.
Furthermore ,
j
O -< ( l -Re P v ) d ].l = l -Re P v ( O ) = l - e f o r eve r y
].l EM O
l i m Re P v ( z ) = l V-+ +OO
S i nce l i m P v= 1 V-++OO
].l
and ever y a.e.
( ].l )
vE N .
so
< 1 /v2
I t f o l l ows that
l im P ( z ) = l v-++oo v
wa s any mea s u r e i n
out s i de a set o f
- vh v ( 0 )
MO
a.e.
( ].l )
s i nce
i t f o l l ows that
Q-capa c i ty z e ro . I f we put
f v= l -P v
we get a s equence o f funct i on s w i th prope r t y 2 ) a nd 3 ) a nd s i n c e l -Re P v > 0 ; l - l im R e P v ( 0 ) =0 , v-++oo
Lemma X : 2 p r ove s that
- 107 -
l - l im Re P v = 0
on
Aga i n , s i nc e
U.
l i m f v ( z ) = l im 1 - P V ( z ) v-+ + ao v-++co o f Theor em X : 3 . Let
Theorem X : 4 .
�
=
0
on
U,
I p v 1 -< 1 , wh i ch completes the proo f
be a non-neg at i ve s u bharmo n i c f u nct i on
on the u n i t ba l l . Then
�
h a s a p l ur i harmo n i c ma j orant i f and
only i f s up sup O < r < l \.l EM O Proo f .
If
� � hE PH
f � ( r� ) d� ( � ) then
�
+ co .
� ( rU � h ( rU
s up � ( rg ) ( 0 ) � h ( a ) O
<
so
•
� ( r� )
i s subha rmon i c then
upper semi con t i nuou s . H ence , by Lemma X : 1 ,
l as
is
�
� ( r � ) ( z ) = i n f { h ( z ) i h E Re A ( B ) , � ( r U � h ( U } so choo s e
w i th
sup h ( 0 ) < +co . O
( hr . ) j = l
App l y Lemma X : 2 a nd choose a s ub sequence verg i ng
un i f o r m l y on compact s u b s e t s o f
B
to
J
h.
S i nce
i s s ubha rmon i c , � ( r . z ) - hr . ( z ) < O J J on
B
so Zo Zo � ( z O ) - h ( z O ) = � i m Ir � ( r . ) -h r j ( ) r r J j J -++ co j
] < 0, -
con-
'tj z 0 E B .
�
-
For
1 08
-
the rest o f th i s sect i on we r e s t r i c t o u r s e lves t o t h e un i t
ba l l al though some o f the r e s u l t s a r e t r ue on mor e g e n e r a l doma i n s . LH 1 ( B )
Reca l l that
i s the Banach space o f a na l yt i c
funct i ons w i t h modul u s hav i ng a p l ur i subha rmon i c ma j or a n t . LH 1
The norm on
Il f l l
LH
is
1 :;;; i n f { h ( 0 ) ; h E P H ( B ) ; I f l.s. h } .
Observe that Theorem X : 4 s hows that on l y i f
s up I f ( r U I ( 0 ) < O
Theorem X : 5 .
Il f ll
f E LH
If
fEH ( B )
LH 1
is in
i f and
+00 .
1
then
�
1 :;;; l im I f ( r U I ( O ) . r-+ 1 LH with
I f h E PH . ----. h ( O )� l f ( r� ) I ( O ) . Proo f .
h� l f l
then
so
h ( r� ) � l f ( r� ) 1
H ence
wh i c h g i ves that Il f l l
------
� l i� I f ( r U I ( 0 ) . LH 1 r-+ 1
On the other hand ,
-------
l im I f ( r f,; ) I ( z ) > r-+ 1
fQ ( z
,
U 1 i m I f ( r U I d lJ ( U
r-+ 1
>
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
0
L
wE � ;
I f(z)
be g i ve n , choose K n r 2 K > 2"1 s uc h t h a t with Iwl=l
in
s o that
co
and put
VK
1 � 1 +€ so that
s i nce
nK r 1 < 1 /4
we n ow have
if
z EV K
€ =
1
16
for
and
- 1 10 -
The r e f or e
f l f ( r 1 � ) - f ( r2 � ) I d� � �
sup � EM O
s i nce we c a n take
�
{ e i 8 � K ' 8 t= :ffi } .
Th i s
to be the norma l i z ed Lebe s g u e mea s u r e on prove s that
f
has prope r t y 2 ) .
To prove 1 } take
�ES
f i x , then
w i th un i f orm convergence , s i n c e
� E VK
f o r a t most one
Howeve r , we have the f o l l ow i ng c h a r a c t e r i z at i on o f the funct i on s i n the c l os u r e o f
A(B)
K. ALH 1 ,
w i t h re spect t o the
Lumer norm . A s s ume that
Theorem X : 6 .
f EH ( B } .
Then
f E ALH 1 ( B )
i f a nd
only i f \)
l: \)
Ref =
\)
L
and 1m f
=
\jJ V v 1 L
-
L
l./J \)2
whe re cp v� , \jJ \)� ,
v E JN ,
�= 1 , 2 ,
are non-po s i t i ve and p l u r i h a rmo n i c on
B,
c o n t i nuous up t o the
bounda r y . Proo f .
�)
choose
t.
J
Choose
f nE A ( B )
s o that
s o that
II fm- f n i l
LH
2 l < 1/j ,
m , n) t . , -
J
Il f n - f ll
LH 1
-+ 0 ,
n-++oo ,
and
- 111 -
and put
Then
II f t
s
- ft
II
s - l LH
1
<
1
( s- l )
2
hs
s o we c a n choose p l u r i harmon i c ma j or a n t s hs ( O ) <
-
1
( s- l )
w i th
2 ·
I f we de f i ne v
= Re ( f t - f t ) -h " , v v- ' v
2 = -hv
v
,
and
we get f unct i on s w i th the requ i red p r opert i e s . c )
As s ume now f ( z ) = l:
v
G i ve n
(>0
00
v=N
(
fEH ( B )
ha s the r ep r e s entat i on
choos e
- ( l:
that
N
(
v
v
s o t hat
v v v v
< (
•
- 1 12 Then N
NE
I v=E l ( � � ( r � ) _ � � ( S � ) ) _ v=E l ( � � ( r � )
I f ( r� ) - f ( s� ) I <
E
NE
-
NE
- � � ( s � ) ) + i ( E \(J � ( r t,; ) - 1jJ � ( s � ) ) - i ( E \(J � ( r U v= l v= l
- 1jJ V2 ( s � ) )
I
ex>
-
v
� ( � ( rt'; V N l E
v v v ) + � l ( s t,; ) +� 2 ( r � ) + � 1 ( s U
+
The f i r st term a t the r i ght-hand s i de conve rg e s u n i f o rm l y t o z e r o when
r,s�l .
wh i ch i s l e s s than
T h e second s u m i s a p l u r i harmon i c f u n c t i on 2E
at zero . Hence
( f ( r� ) ) a < r < is a l a n d we have p roved that f ( r � ) tends
Cauchy s equence i n LH 1 t o f in L H 1 a nd the proo f i s c omp l et e .
f E ALH ( B ) d e f i ne i t s v boundar y va l ue s : The t e rm i n the s e r i e s E � 1 a r e non-po s i v= l t i ve a n d cont i nuous up to t h e boundar y . I f ).l E M a then
Remark .
so
E
By Theorem
v � 1 ( � ) > -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
fEH
<Xl
( B ) then ( f ( r U ) O
L
2
( I.l , S ) ,
r """ 1 .
- 115
-
Notes a n d re f e r ences Theorem X : 3 i s due t o F . Forel l i , A n a l yt i c mea sur e s . Pac i f i c J . Mat h . 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 p l u s i eurs va r i ab l e s comp l exe s . C . R . A . S . Pa r i s 2 7 3 ( 1 9 7 1 ) . The examp l e i s a s i mp l i f i ca t i on o f an examp l e o f W . Rud i n , Func t i on theory i n the u n i t ba l l o f � n . Spr i nger Ve r l ag 1 9 8 0 , pg . 1 5 0 . I t has been proved by W . Rud i n i n " New construct i on s o f funct i on s hol omorph i c i n the u n i t b a l l o f � n " , AMS reg i on a l 1 con f e r e n c e s e r i e s i n mathema t i c s No . 6 3 on pag e 6 4 t h a t ALH ( B ) cont a i n s no i nn e r f unct i on s . Theorem X : 8 i s due t o G . M . Henk i n , Banach spaces o f a n a l yt i c f unct i on s on the ba l l and on t h e b i c y l i nder a r e not i somo rph i c . Funct i on a l Ana l . Appl . 2 ( 1 9 6 8 ) . Theorem X : 9 i s proved by U . Cegr e l l i n On the s t r ong con ver g e n c e of d i l a ta t i on s of bounded a n a l yt i c f unct i on s i n the un i t ba l l of � n . Manus c r i p t . Tou l ou s e . 1 9 8 6 . For mo re r e f erences , see Ann . Pol . Math . ( 1 9 8 5 ) .
U.
Cegre l l ,
Sma l l sets i n
�n ,
XI
Szego Kernels
We keep the notation from Section 1.
x.
Approximation of the identity
Let U be a positive measure on a � and let ( P i ( z ' � ) ) i = l ' ( z , U E a � x a � be a fami ly of functions in L 1 ( U®U , a � x a � ) such that b ) sup fP i ( Z , � ) dU ( Z ) =M< +oo i,� 2 ) P . > O , i E JN Lemma XI : l . 00
1-
Then fP i ( Z , � ) � ( � )dU ( � ) tends to � in every � E LP ( U , a n ) .
LP ( U , a � ) ,
i �+oo
for
Assume first that � i s continuous . Then
Proo f .
l im i-++oo and since
SUI? l fpi Z,l
by 2 ) if Then
� sup l � ( � ) I � the lemma follows by dominated convergence . I f � E LP and i s given , choose � E E C ( a � ) with f l �-� E I PdU « / ( M+ 1 ) .
( >0
( z , � ) � ( U d u ( f; )
I
-
+ ( f I fp l. ( z , U ( cp £ ( � )
1 17
-
cp
£
-
( z » d � ( � ) I pd � ( z ) ) .
Fubini 's theorem and property 1 ) show that the f irst two integrals are smaller than and the last i ntegral tends to z ero when i�+oo by the f irst part of the proof . Since £ > 0 was arbitrary , the lemma follows . £
00
Let ( P i ( z , � ) ) i be a fami ly of functions as i n Lemma Assume that ( ,I, . ) l. == 1 l S a sequence of L ( � , a � ) functions such that l\J . dw I\Jd� where I\J E L 1 ( � , a � ) . I f Lemma X I : 2 .
=
XI : 1 .
l
then
I\J . l
tends to
I\J
't' l
1
00
1
�
in L1 ( w , a� ) .
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 ( fp i ( z , � ) I\J ( � ) d� ( � ) - l\J i ( Z » d� ( Z ) . Proo f .
By Lemma X I : l , the first integral tends to zero when i�+oo. By weak convergence , we have the same conclus ion for the second integral .
- 118 2.
s z e g o a nd Poi s s on k e r n e l s �
Let
b e a pos i t i ve m e a s u r e on
K
compact subset s up I f (
(*)
zEK
of
n
<
CK
z) I
A(n)
( and s o i n
there i s a c o n s t a n t
(f
an
2
I f I d� )
1 /2 ,
ex>
g(z) = CD
+
For
ex>
ex>
1:
v=o
�
v=O
such that
CK
f EA ( n ) .
( e ) v= l u ( d v ) v= l i n L 2 ( � , a n ) . V then has a u n i que repr e sent a t i on
O N-ba s i s
s u c h t h a t t o every
( e v ) V= l ' e v E A ( n ) o f f u n c t i on s d e n s e 2 C omp l et e the s ystem to a n H ( � i an» .
Choo s e an O N - s y stem In
an
fg ( � ) ev ( � ) d � ( � ) e v ( z )
f g ( � ) dv ( � ) d � ( � ) d v
(z) ,
Every
gE L2 ( � , a n )
+
zE an.
mE 1N ,
so by ( * ) s up zEK
Thus of
CD
m 1 , m 2 -+ +CD
f
•
1: ge v e v converges u n i f o rm l y on every c ompa c t subset v=O n a nd there fore repr e s e n t s a n a n a l yt i c funct i on there .
If
f EA ( n ) ,
we can app l y the s ame a rgument on
- 1 19 -
f -
m L
v=O
f fev d u e v
t o conclude that CX)
f(z)
ffev d
L
=
1
ue
v( z ) ,
zEQ ,
w i t h u n i f orm convergence on c ompact s u b s e t s o f
Q.
I f we con s i de r the mapp i ng s L
2
p
CX)
( u ) 3 g 1-+ l:
V==O
n nd
zE Q , i t i s c l ea r that they a r e cont i nuous so the i r compo s i t i on T ==V o P 1 S g i ve n by a n e l ement i n L 2 ( u ) : z z gEL2 ( u )
.
I t 1 S e a s y to s e e that T
CX)
L
z(�)
v= l
ev ( z ) e v ( � ) '
a n d we have seen that T
CX)
z(w)
=
T
z
extends to a n a n a l yt i c f u nct i on on
L e (z)e (w) v v= l v
Now , a c r uc i a l p r operty o f
�EaQ
,
wEQ .
on
i s : . Are the
l i near l y i ndependent as a n a l y t i c f un c t i on s on
Q?
e v ·. S
Q,
-
I n other word s : �
a =0 ,
must
1)
If
f EA ( � ) , v
f vd
l im
�
f l f) 2 d� � 1 ,
v E JN ,
and
if
l im f ( z ) = O , v v-+ +oo
z,. O .
't1 v E � ,
There ex i s t s an ON-ba s i s f lln c t i on s i n 1)
Proof .
2)
be a pos i t i ve meas u r e sat i s fy i ng p r oper-
The f o l l ow i ng s tateme n t s a r e equ i va l e nt .
then
2)
0 -
If
�
Let
Theorem XI : l . ( *) .
2
't1. v f JN ?
v
ty
1
=>
V-++OO
v
2).
If
2)
A s s ume that
1 ) . f
=>
{ z ) =O ,
l i ne a r l y i ndepende n t on
A{ � ) ,
zE� .
is
false ,
1
f . EA ( � ) ,
put
f 1. ( z ) =
I l f J. 1 2d� -< 1
i
E a e v v v=O
�.
(z) .
a nd that
Select a weak l y convergent subs equence ( wh i c h we aga i n denote by f . { Z )= l
f
( f 1. )
f . ( U1 I Z
wi th l im i t ( t;; ) d� ( t;; )
By a s s umpt i on , of
00
( E . ) . 1 1= 1
Now
so
fe d � = O , v
't1 z E JN ,
wh i ch shows that the weak
l im i t
i s zero a nd the theorem i s p r oved .
Corol lary XI : l .
(*)
f
fEH2 { � , d� ) .
Every pos i t i ve mea s u r e on
and pos s e s s e s a bas i s
for H 2
dB
wh i ch s at i s f i e s
s o that
- 121 h a s t h e p r ope r t i es i n the t heorem . The S z eg6 ker ne l ( re l at i ve l y
Def i n it ion .
00
S(z,�)
L:
=
v= O
w
a nd
n)
is
e ( z ) e ( � ) , zEn , sE a n , v
v
a nd the Po i s son kernel i s
P( z,�)
Is( z , � ) 12 S( z,z)
=
I t i s c l ea r that
zEn , sE a n .
P( z , s ) � O.
If
f EA ( n ) , then
so
f
=
f(z)
Note t h a t s i nc e
S(z,s ) f( s) S( z, � ) dw ( � ) 8( z,z) P
i s real ,
f
Re f ( z ) = p ( Z , � ) R e f ( � ) d J..t{ s ) , f E A ( Q ) .
A s sume that t h e Choquet bo�ndary o f
Theorem XI : 2 .
t i ve l y
an
A(n)
equal s
h av i ng p r operty 00
( F � ) �. = 1 de f i ned n e a r TI f am i ly
.
If s up i EJN ll E a Q
f
an (*) .
a nd l e t
w
rela-
be a pos i t i ve mea sure on
F u r thermor e , a s s ume that there i s a
n,
o f a na l yt i c mapp i ng s i nt o such that
n
l. im F �. ( z ) = z , � .... + oo
I S ( F �. ( z ) , l1 ) 1
� z ETI .
2
d w ( z ) < + 00
each o f them
- 1 22 whe r e
S
i s the S z eg6 ker n e l r e l at i ve l y
f o l l owi ng prope r ty . For eve ry s equence lim f ( z } =O , s-+ +oo s
�zE�
Mo reover , i f L1
it fEH
f o l l ows that 00
f
then
a�
f sEA ( � ) S
z:,.
d�
0,
00
then
( f ( Fi ( � ) ) ) i= 1
�
h a s the
I f s I -< 1
,
w i th
s-++oo . conve r g e s i n
( w , a� ) .
Proof .
Put
P 1. ( z , � ) =P ( F 1. ( z ) , � )
kernel r e l a t i ve
a� .
whe r e
P
We f i r s t prove that
i s the Po i s son 00
( P 1. ) 1. = 1
i s an
approx i mat i on o f t h e i dent i ty i n the s e n s e o f Lemma 1 . F i r s t , each
P 1.
S i nce
i s i nt e g r a b l e on
f
f ( F . ( Z ) ) = p 1. ( Z , U f ( U d � ( U , � f E A ( � ) , � z ETI , 1 i t i s c l ear that
P 1. ( z , � ) d w ( � )
�
8 z , i-+ +oo
( Th i s i s s o because every po i nt i n
a�
f o r every
zE a� .
i s i n the Choquet
bounda ry . ) Thus 3 ) of Lemma X I : 1 i s va l i d . I t is t r i v i a l that 2 ) and the f i r s t part o f
1 =
fp
l'
1 )
( Z , U d� ( � ) =
h o l d . We a l so have
f
2 I S ( F 1. ( z ) , � ) 1 S ( F 1. ( z ) , F 1. ( z ) ) d w ( � )
H ence S ( F 1. ( z ) , F 1. ( z ) ) =
f I S ( F . ( Z ) , n I 2d� ( n ) 1
•
- 1 23 -
so
I S ( F . ( z ) , � ) 1 2dW ( z ) sup fp 1· ( z , � ) dW ( Z ) =SU P f S ( F 1. ( z ) , F 1. ( z ) ) i E lN i E lN 1
�Ea�
=
=
�Ea�
Up i E lN
S
I S ( F 1. ( z ) , � ) 1 2
f
d W ( z ) < + 00
�E a�
by a s s umpt i o n s o the l a s t part o f 1 ) h o l d s true . Now , l et
f EA ( � ) , S
I f S I � 1 be a f s dw does not
g i v e n sequence w i th
l im f s ( z ) =O , �zE� . I f converge weak l y to z e r o , s-+ +oo then we c a n s e l ec t a s ubsequence ( wh i ch we aga i n denote by
f s dIJ )
such
that f s d W � fdlJ
wher e
o t f EL
00
.
W e now w i sh to prove that
f (Z
l im P i s-++oo
,
U f s U dW
(
( � ) = fPi ( z , � ) f ( � ) dW ( � ) '
�zE� ,
because t h e n l im f s ( F i ( z ) ) =0 , f P i ( z , � ) f ( � ) dIJ ( � ) = s-++oo
�zE� ,
by a s s umpt i on . On the other hand , we have shown that Lemma app l i e s 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
S J. ( F 1. ( z ) , � ) =
zE�
j
L
v=O
eV
XI : 1
a n d con s i de r
( F 1. ( z ) ) e V ( U .
- 1 24 Given
(>0
choose
so that
j
I I s . ( F 1. ( z ) , � ) -S ( F 1. ( z ) , � ) 1 2 d\.J( U < ( S ( F 1. ( z ) , F . ( z ) ) 1
J
and then s so that I I l s . ( F 1. ( Z ) , � ) 1 2 ( f ( U -f S ( � ) ) d\l ( U I < E: S ( F 1. ( z ) , F 1. ( z ) ) . J
Then
I S(F. (z) , � ) 1 2 < I I ( S ( F . �Z ) , F . ( Z ) ) 1
I S . ( F 1. ( z ) , � ) 1 2 S ( F . ( z ) , F 1. ( z ) ) ) ( f ( � ) J
1
1
s . ( F 1. ( Z ) , U I 2 l - f s ( U ) d \l ( � ) I + I I S ( F 1. ( z ) , F 1. ( z ) ) ( f P'; ) -f s ( � ) ) d\l ( � ) I < J
I S . ( F . ( z ) , U -S ( F . ( z ) , � ) I 2 d \l ( � ) + € < 3 E: < 2 I J l S ( F . ( Z ) , F . �Z ) ) 1
1
and the proof of the first part of the theorem i s complete . It remains to prove the last statement . I f fEHm then f 1. =f ( F 1. ( z ) ) is a uni formly bounded sequence in A ( Q ) . We can find a function f E L ( \l , a Q ) and a sequence 1 . ) m. 1 so that f 1. . d \l gd\l and by the proof above J m
( f .
�
]
J=
l im f p J. ( z , � ( F 1. ( � ) ) d ( U = I ( Z , � ) ( � ) d \l ( U . i-++m ) f
\l
p . J
9
- 1 25 -
Hence
f
f ( F . ( Z ) ) = p . ( z , � ) g ( � ) dU ( � ) J
s o a nother appl i c at i on o f
J
Lemma X I : 1 comp l e t e s the proo f o f the theorem .
3. We n ow r e t u r n to the un i t ba l l i n i z ed Lebesgue mea sure o n I t i s c l ea r that
a (�) ca aE:lt'P ca
B
a nd
a , the norma l -
dB.
a
s at i s f i e s prope rty 2 H ( a , dB )
i s a n O N-bas i s f or
)
(* .
The set
whe r e
1 ) !a! = f I � a l 2 da ( � ) = ( n( n- 1 +l a l ) ! dB
The Cauchy k e r n e l i s then S ( z , U = C [ z , E;: ] =
a -a E;: c a a E
z
=
' zEB, �EdB, ( 1 -
and the c o r re spond i ng Po i s s o n Y G r n e l = IS( z , U s(z,z)
12
=
( 1-lzI2 )n 2n . 1 1 _ 1
O b s e rve t h a t
f
sup p ( r z , � ) da ( Z ) = s u p E;: E d B � E dB O
P r z ( z , E;: ) = P ( r z , � )
2 n
f 1 1 -( <1 _z r, r E;:) > 1 2 n
da ( z ) = 1
s at i s f i e s t h e cond i t i on s i n Lemma X I : 1
- 1 26 and the Theorem X I : 2 . Corol l a ry X I : l shows that
a
h a s the
equ i va l e nt proper t i e s i n Theorem X I : 1 . Th i s proves the f i r s t p a r t i n the f o l l ow i ng propos i t i on . Propos i t ion XI : l .
a) If
l i m f s ( z ) = O , \:1 z E B s�+ oo
then
b ) Let
v
f sEA ( B ) , f s da
4 0,
f l f s l 2 da � 1 , s� +oo .
be the Lebesgue mea sure on L 1 ( " ) � A ( B ) �"J f v
r f --+
has a c l o s ed exten s i on
r
l
a BE H
and i f
B.
The r e s t r i c t i on map
1 ( �B) 0 , 0
with
Dom r = { f E H ( B ) ; sup O
f l f ( r U l da ( U < +oo }
and r = H1 ( a , aB ) .
Range Proo f .
and L
We f i r st reove ( t he we l l -known f a c t ) that i f
s up O
f I f ( r � ) I do < +00
the n
( f ( d� ) ) 0
fEH( B )
con v erg e s i n
1 ( a , aB ) .
B E ] O , l [ and c on s i de r the u n i f o r m l y i ntegrabl e B f am i l y ( I f ( r U I ) o < r < l ' � E d B . Choose r v 1 , v +oo a nd B T E L 1 ( y ) s o that I f { r v � ) I da � Tda . Let Q be the c l a s s i ca l Poi s son kernel f o r the u n i t ba l l i n �n . Then Choose
A
I f(z) I
B
f
� Q { z ' � ) T ( t; ) da ( u , z E B ,
�
- 1 27 -
SQ T d O
wher e
There fo r e
T
6 If1 .
i s the sma l l e s t h a rmon i c ma j o r a n t o f CXl ( r v ) v= l
i s i ndependent o f
and Lemma XI : 2 p r oves
that
By the R i e s z - F i s c h e r theo rem , we c a n s e l ect a seque nce s v A l , V-+ +CXl
s o that
I im
V-++CXl
I f ( sv� ) I 6 = T ( � )
a.e.
(0)
•
Fatou ' s l emma g i v e s
so
and
I f(z) I
=
( I f ( z ) I 6 ) 1 /8 <
S
Q ( z , � ) T ( � ) 1 / 6 do ( U
by J e n s e n ' s i nequa l i ty . Anot h e r app l i cat i on o f Lemma X I : 2 g i ve s the de s i red con c l u s i on . Now , i f a s the
rtf) a nd i f
f s-+ g sup O
so
rt f)
fEH ( B )
with
s up O
L 1 ( O , a B ) -l imit of in
H 1 ( o , aB )
S I f ( r � ) I d o ( � ) < +CXl f ( r� )
.
If
f s-+ f
we de f i ne in
L1 ( v)
then i t i s c le a r that
S l f ( r � ) I do ( � ) < +CXl
i s we l l d e f i ned . By Lemma X I : 1 ,
S p ( r Z , U g ( U dO ( U = l i m S p ( r Z , U f s ( 9 ) dO ( � ) = f ( r U S-++CXl
-
so
� ( f ) =l im f ( r � ) =g ( � )
in
' 28
-
' L ( da )
by Lemma X I : ' . Thus
r
i s a c l o s ed ope rator a nd the p r o o f i s c ompl ete .
4.
A-me a s u r e s I n t h i s sect i on , w e s tudy exte n s i on p r ob l ems r e l a ted t o
Theorem X I : 2 . W e a l so s tudy t he i r c o n n e c t i on w i t h t h e behav i ou r of t h e S z eg6 ke r n e l r e l a t i ve l y a c e r ta i n mea sure . Con s i de r the r e s t r i ct i on operator
r,
r L oo ( dw ) :;) H oo ( n ) :;)A ( n ) �A ( n ) a n = A ( a n )
l
and l et
u
be a g i ven mea s u r e on
Lebesgue mea sure on ' spa c e s o f ( L ( dw ) )
n. ) I
an .
We t h i nk of ' ( L ( U , an ) ) I
a nd
( Here A( n)
dw and
i s the A( an)
a s s ub
r e spect i ve l y a nd make
the f o l l ow i ng de f i n i t i on s . Def i n i t i on .
The mea sure
c l o s ed exten s i on Def i n i t i on .
r
U
lS
s a i d t o be c l osed i f
r
has a
I n the weak * -topol og y .
We d e f i ne
00
H ( U , an )
to be the weak * - c l os u r e o f
A( an ) . Def i n i t i on .
f E H oo ( n )
We say that
n
has pr oper t y
i s the weak * - l i m i t ( re l at i ve l y
funct i on s i n Lemma XI : 3 .
( ** )
i f every
dw ) o f a s equence o f
A(n) . Con s i der the f o l l ow i ng cond i t i o n s .
- 1 29 i)
�
ii )
If
i s a c losed mea sure ; conver g e s i n the weak * -topo l ogy on
f s E A ( &1 ) , s E lN , L
1
( g s ) s=
1
conve r gent i n s equence
00
s u c h that
A ( &1 )
in
and equal to
f
and
a &1 .
on
(�)
a.e.
( � , a &1 )
&1
ever ywhe r e o n
( f s ) : = 1 1 S weak * and t h e r e i s a u n i f orml y bounded
then
f,
dw ) to
( re l at i ve l y
( i i i ) For every sequence l im f s ( z ) = O , 'tt z E &1 , s-++oo
&1
f s E A ( &1 )
l im 9 ( z ) s-++oo s l im 9 ( z ) s-++oo s
,
s ElN , s u p i f s ( z ) i .s. 1 s ElN z E &1 we have
ex i s t ex i st
w i th
I
l im qJ ( Z ) f s ( Z ) d� ( Z ) = 0 , 'tt qJ E C ( a &1 ) ; s-++oo
i)
Then iii)
�
Proo f .
�
ii)
i). i)
�
�
iii)
and i f
i i ) . As s ume that
&1
h a s p r operty
( ** )
then
i s c l o s ed and that 00
s EJN , 1 S a weak * - convergent s equence i n L ( dw , &1 ) . L 1 ( dw , &1 ) i s comp l e t e , ( f S ) : = 1 i s u n i f o r m l y bounded on But s i nc e
L
1
con-
i s s epa r a b l e and c omp l et e ,
( � , a &1 )
t a i n s a s u b s equence that conve r g e s to a f u nct i on 00
TI .
f
in the
is But the a s sumpt i on that c l o s ed n ow g i v e s that the sequence ( f ) oo s s = 1 i t s e l f conve r g e s to f i n the weak * -topol ogy . By the Banach- Saks theorem , we m can s e l ec t a s u b s equence s o that 9 m = m1 r f S j=l j tends t o f i n L 2 ( � , a &1 ) a n d by the R i e s z - F i scher theorem , we
weak * -t op o l og y i n
L
( � , a &1 ) .
-
- 1 30 00 ( g m ) j 1 so that . J == To complete the proo f we o b s e rve that
can s e l ect another subsequence a.e.
( �) .
( f s ) := 1 i s .
a u n i formly bounded sequence s i nce That
�
ii ) iii)
i i i ) i s c l ear . i ) . A s s ume that
�
f E H oo ( Q )
�
h a s prope r t y
l. im g m = f . J -++oo J 00 is ( gm . ) j = J
( ** ) .
there i s a sequence f s E A ( Q ) converg i ng t o wea k * -topo l ogy . S i nc e L 1 ( dw , Q ) i s comp l e t e ,
1
Thus , i f f
i n the
s up I f s ( z ) I < +00 s E1N zEQ s o , by
ii ) ,
l im s-++oo ex i s t s f o r every
�EL
1 ( � , aQ )
and the l im i t i s i ndependent o f 00 We c a n n ow the par t i cu l ar c ho i ce o f the s equence ( [ s ) s= l 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 de f i ned i s a c l os ed ope rator . Def i n i t i on
( c f . Sect i on X ) .
aQ .
v
Then
Let
v
be a comp l ex mea sure on
i s c a l l ed a n A-mea s u r e i f for every u n i f o r m l y oo bounded s equence ( f s ) 8 = 1 of f u n ct i on s i n A ( Q ) w i th l i m f ( z ) = O , � z E Q , i t f o l l ows that l i m f s dV= O . s-++oo s s-++oo
f
Examp l e s o f
A-me a s u r e s a r e mea s u r e s i n
MO
a nd c l osed
mea s u re s . Theorem XI : 3 .
M
Let
�
be a r e g u l a r ( compl ex ) Bor e l mea s u r e and
a weak * - c ompac t a nd convex s e t of r eg u l a r Borel probab i l i ty
- 1 31 U
mea s u r e s on a compact H a u sdor f f space s . Then decompo s i t i on
whe r e in
M
u1
i s a b s ol utely c o n ti nuous w i th r espect to a measure
and
i s car r i ed b y an
s up { U ( E ) i
Jl
vEM
van i sh i ng
=
O
'
F -set a
such that
E
uEM} = O .
U
Let
Theorem XI : 4 .
whe r e
h a s a u n i que
be a n
A-measu r e . Then
gdv+n gEL
1
n
and
(v)
i s c a r r i ed by a n
Q-c apac i ty . Furthermo r e ,
n .l
A(�)
F a - set o f
( i .e.
Ifdn:;; o ,
'tI f E A ( � ) ) . Proo f .
The s et
decompose
M
i s weak * -c ompac t . U s e Theorem XI : 3 t o
O
a n d u s e Theorem X : 3 to prove
Jl = f d v + n
Cor o l l ar y XI : 2 .
M
O
fdv+ n ,
vEM
O
'
fEL
1
(v),
n .l
wher e the decompo s i t i o n i s u n i qu e . I c l os ed s o
�d n
A(�) .
.
We k n ow f rom Theorem X I : 4 that i f u=
n .l
Ever y c lo s e d mea s u r e i s a b s o l ut e l y c on t i nuous
w i t h r espect t o a mea s u r e in Proo f .
t hat
A(�)
c o n t i n u ous f u n c t i on
by u n i qu e ne s s . �
so
n � O
u
i s c losed then
A(�)
f
�EC ( a � )
then
Hence I�dn= o
�dJl
is
f o r ev�ry
wh i ch proves t he corol l a r y .
-
Co rol l ary X I : 3 .
If
�
is
absolutely
co n t i n u o u s w i th
Proo f .
The o r em X I : 3 ,
By
pos i t i v e
so
The
me a s u r e need
a
A-mea s u r e
�
then
is
MO '
in
]J = fd v + n ,
n .L A ( S1 )
0= J l o d n
so
n = O.
the
n o r ma l i z e d L e b e s g u e
and
�
s i nce
is
examp l e
not t o be in
01
( wh e r e
I z2 1 =1 )
on
a n y mea s u r e
s hows
c l o s ed o r
MO '
To
at
is the
s ame
absolutely see
thi s ,
t i me
that
cont i nuous
wEC� (
t ak e
A - me a s u r e s
w i th r e s p e c t t o
Iz
2
w ( O ) �O ,
1 <1 ) ,
c on s i d e r
If
Theorem XI: 5 . A - me a s u r e
is
Thus depe nds
a
the
of
S1
the
s h ape
that
a s s oc i a t e d
funct i on s
on
s t r i c t l y p s e ud o c o n v e x
equ i v a l e n c e
p r ope r t y o f
its
is
a�
2
.
every
that
and
t heorem
on
H
2
s hows
that
a
a
� EM O
s uc h
( � , a� )
maps
smooth
ex i s t e n c e
.
c l o s ed m e a s u r e s
of
f u nc t i on s .
i s a mea s u r e
such
the
nex t
-prO ] ec t l on
and
1 1 8)
The
is
A s s ume
there
A - me a s u r e s
a�.
of
L
smooth
of
Theorem XI: 6 . that
then
c l o s ed m e a s u r e .
r e l e va n t
pg
p o s i t i ve
r e s p e c t t o s ome me a s u r e s
Thus
� = o ® o O - o ®0 1 l 1
and
and
n.
is
1 32 -
�
�
that
has
has
� E MO
property w i th
n o p o i n t ma s s .
( ** )
p r op e r t y
( cf (*)
F u r t h e rmor e ,
.
pg
1 28)
( cf . a s s u me
t hat
- 1 33 1)
�
1S
2)
S[z,�]
3)
( ( � ( z ) -� ( � ) S ( z ' � ) ) zEQ for e v e r y �EC oo ( � n )
Then every Proo f .
Let
for every
c l o s ed ; i s cont i nuous o n
TIx 8 Q \ { z= � } ;
is a
� - u n i formly i nteg r a b l e f am i l y
A-me a s u r e i s a c l o s ed meas u r e . 00
� E C ( � ) . Property 2 ) and 3 ) show that
z ETI .
Furthermo r e , by 3 )
and
i s cont i nuous for every
00
fEL ( � , 8Q ) .
As s ume that
a u n i f o r m l y bounded sequence o f f u nc t i o n s i n l. i m f 1. ( z ) = O , � z E Q , 1-++00
prove that
1 im . 1 -+ +00
a nd l e t
I f . �dv= O 1
i s dense i n Put
and
C ( 8Q )
�E C ( 8 Q )
f o r every
A( Q)
00
( f 1. ) 1. = 1
is
with
b e g i ve n . We want t o A-mea sure
we can a s s ume that
v
a nd s i nce 00
�EC ( � n ) .
-
I . (z 1
We have a l r eady seen that
TI
t i on o n
a nd s i nce
f 1. ( z ) ep ( z ) = I 1. ( z )
J 1. ( z ) ,
so does
1 34 -
)
extends t o a cont i n uous f u n c -
+ J 1. ( z )
and mor e ove r ,
s up I 1 . ( z ) I + I J 1· ( z ) I < +00 . 1 i E lN zEn
�
We h ave a s s umed that
i s a c l osed mea s ur e , s o s i nc e , f o r l i m 1 1. ( z ) = O i -++00 I . dv=O f o r every 1
i t f o l l ows that
z ETI ,
J
l im i -++00 and
and dom i nated convergence g i v e s that
an .
mea s u re on ag a i n s i nc e A-measure
v
l im i-+ +oo
]..I
Furthermor e , i s c l osed . T h u s
lim i -++00
J
l. i m J 1. ( z ) = O , � z E n , 1-++00 J . dv=O for every 1
wh i ch mea n s that
Jepf . dv 1
=
l. im
1 -+ +00
Jl 1. dv
+
l. i m 1 -+ +00
JJ . dv = 0 1
wh i c h proves the theorem . Coro l l ary XI : 4 .
]..I
A s s ume that there i s a mea s u r e
on
an
w i th the propert i e s i n Theorem X I : 6 a nd that every po i n t h a s van i sh i ng
Q-capa c i ty . Then a l gebra o f L00 ( ]..I , a n ) . Proof .
As s ume f i rs t that
i s a c l o s ed sub-
00
f E H ( ]..I ,
an)
and
00
ep E e ( 0:
n) .
Then
-
1 35 -
and by the proof o f Theorem X I : 6 i t f o l l ows that the f i r s t term TI
exter.ds c on t i n uous l y to
c lo s ed and th i s f o l l ows f rom
every po i n t i n
A s s ume tha t an
ii)
H ( jJ , a n ) +C ( jJ n )
is
i n the next theorem .
has prope rty
n
H ( jJ , a n ) .
co
By appr ox i ma t i on , i t rema i n s t o prove that
Theorem XI : 7 .
co
wh i l e the s econd i s i n
has van i s h i ng
( ** )
a nd that
Q-capac i ty a nd that every
A-me a s u r e i s a c l os ed meas ur e . Then i)
if
JJ E M
O
w i th prope r t y
(*).
Then
j
�
i s a n i s omet ry f o r every ii)
if
w i t h prope r t y
c l osed subset o f
is a
then
(*)
co
\! ,•
L ( jJ , a n ) .
i ) I t i s c l ear tha t
Proo f .
A-mea sure
j
i s a con t i nuous map f rom a
Banach a l gebra i nto a B a n a c h a l gebra s o i t i s e nough to prove that
j
i s b i j ect i ve . F i r s t
the Choquet boundary o f st r i ct l y sma l l e r than mea s u r e �OE an
e
A( n) )
wou l d be
a nd there wou ld then be a pos i t i ve s o that
on
and s i nc e every
for otherw i se , by ( * ) ,
( wi th r e spect t o
an an
s upp jJ = a n
f or a poi nt
A-me a s u r e i s a s sumed t o be a c l osed
mea s u r e t h i s wou l d i mp l y that
o�
Coro l l a r y X I : 2 s hows that t h e
Q-capac i t y o f
pos i t i ve , a contrad i ct i on .
D
i s a c l osed measure . { �b}
i s s t r i ct l y
-
Hence , the sup norm a n d t i nuous funct i on s on funct i on s i n
A(n)
-
1 36
e s s s up ( � )
a r e equal f o r con-
So i f
an.
( f s ) : = l i s a sequence o f conve rg i ng wea k * i n L ( � , a n ) then the OD
fam i l y is u n i formly bounded on Lemma X I : 3 and property
(*)
and it f o l l ows now f rom
n
OD
that
( f s ) s= 1
converg e s weak * i n i s c l osed by
a s s umpt i on . I t fol l ows that
j
i s a b i j ect i o n wh i ch comp l et e s
t h e proo f o f i ) . i i \ By the above part o f the proo f we c a n appl y Theorem 2 and P r opos i t i on 1 i n Aytuna and Cho l l et
[ 1 ] ,
wh i ch c omp l et e s
the proof o f t h e theo r em . We f i n i sh th i s s e c t i o n w i th two examp l e s whe r e Theorem XI : 6
and Coro l l a r y X I : 4 c a n be app l i ed .
Example 1 .
We choose
n =B ,
the u n i t ba l l i n
the norma l i z ed Lebe sgue mea s ur e on
aB o
Then
pseudoconvex a nd Theor em X I : 5 shows that every
�n n
a nd
�=o ,
i s str ictly A-mea sure i s a
c l osed mea sure . Howeve r , we want t o s how that the cond i t i on s i n Theorem X I : 6 a r e s at i s f i ed . ( He n c e , Coro l l a r y X I : 4 appl i e s i n th i s c a s e . ) We know that
! l - < z , f,; > !
n
Thus prope r t y 2 ) i n Theorem X I : 6 i s sat i s f i ed and by P ropo s i t i o n X I : 1 we know that t rue .
a
i s a c l o s ed mea s u r e , s o 1 ) h o l d s
- 1 37 I t r ema i n s to understand that 3 ) h o l d s t r up . Choose �E ] l ,
If
n-
7/2 [
and cons i de r
s up l ( z ) < +oo zEB
then i t i s c l ear that 3 ) h o l d s . But
=
< 2 1 /2
2 1 1 - < Z , c:: > l n - ( 1 / 2 ) sup l ( z ) < oo . zEB
By Rud i n [ 7 , P r opos i t i o n 1 . 4 . 1 0 ] Example 2 .
]0, 1 [
Let
be
n
number s i n the i nterva l
a nd put
Then
i s a ps eudoconvex s e t w i th
a s a t l e a s t one
a J.
C 2 -bounda ry , but a s soon
1 , 0
i s l e s s than
a
i s not s t r i ct l y
p s eudoconvex . We sha l l n ow s e e that t h e cond i t i on s i n Theorem X l : 6 are s at i s f i ed f o r a c e r t a i n mea s u r e For
n= l
we d e f i ne
u=o l =
u
that we de f i ne i nduct i ve l y .
t he norma l i z ed Lebe s gue mea sure on
I z 1=1 . A s sume now that
i s de f i ned for a l l
a ' = ( a 1 , · · · , a n- 1 ) ·
- 1 38 -
f
a Da
f ( z ) dWa
. ( 1 - 1 zn 1 whe r e
dZn
a= ( a ' , a n )
for
We then de f i n e
=
2 /an
as
f
f ( ( l - l zn 1 a Da . x { l z n 1 < 1 }
2/a
n ) 1 /2 z ' z ) , n .
) 1 a 1 - 1 dwa ' dz n ' •
denotes the Lebe s g u e mea s u r e o n
c l ear that
w aE MO G1
the fact that
a n d that
{ l zn i < l } . It i s i s c l o s ed i s e a s i l y seen f rom
wa
i s c losed .
Bonami and Lohoue h ave c a l cu l ated the c o r r e -
For
a nd they p r ove i n P ropo s i t i on 2 : 1 Wa that i t extends to a n i n f i n i te l y d i f f er e n t i ab l e f u n c t i on on spond i n g S z ego k e r n e l
D
a
xD
a
S
out s ide the d i a go n a l o f
a D a x a D a . ( Bonami and Lohoue [ 2 ] . )
Fur thermor e , they prove i n P r opos i t i on 5 . 3 that 1 S u n i f ormly i nt e g r a b l e in p<
1 - 21 n 1. n f a k
is a
LP ( W a )
for ever y
wh i ch s hows that
W a -un i f ormly i nt e g r a b l e f am i l y f or every L i p s ch i t z
f u nc t i on
�.
I t f o l l ows that the cond i t i on s of Theorem X I : 6
are sat i s f i ed and we c a n a l s o app l y Coro l l a ry
XI : 4 .
-
1 39 -
The Cauchy t r a n s form o f mea s u r e s
5.
W e c on s i de r aga i n t h e u n i t ba l l norma l i z ed Lebesgne mea sure on
B
�n
in
We wr i te
aB.
a nd C
and
a ,
P
the for
the c o r r e spon d i ng Cauchy a n d P o i s son k e r ne l , and we say that a funct ion
f EH ( B ) g(z) =
HP ,
belongs t o
if
r ( f ) E HP ( a , a B ) .
If
f P ( z , � ) d lJ ( � ) E H ( B )
aB
lJ
f o r a comp l ex mea s u r e s upp lJ = a B
then
lJ = f d lJ
whe r e
fEH ( o , oB )
and
( or empty ) . Th i s i s so bec a u se
a nd
so
( g ( r U ) O < r < l conve r g e s i n ( Pr opos i t i on X I : l ) . O n t he other h a nd ,
conver g e s weak l y to
lJ
Reca l l now that the
so
L1 ( a , aB )
to
g*
lJ=g * do .
Cauchy
k e r n e l r e l a t i ve l y
B
and
a
is
C[z,�] zEB,
t ions
If
�E a B . (�
o.
)
=
lJ
(
1 -
=
L: 0. > 0
i s a ny mea s u r e o n
aB
a r e pa i rw i se o r thogo n a l the n
f C [ z , � ] d ]..t{ � ) = lJ ( l ) .
f o r wh i c h the f u n c
-
For i nstance , let by
1 40 -
be defi ned on the unit sphere in � n
w
2n
I � ( � ) d w ( � ) = 2TI I � ( e iG , 0 ) dG dB o ( i . e . w =o ® o O ) . Then W i s s i ngular relatively but the in Cauchy transform of W � l . But there are restrictions on order to have 0
�
�
Let be a real measure so that g ( Z ) =IC [ z , � ] dW ( � ) EH 1 . Then W i s a closed measure .
Theorem XI : 8 .
Let f 1. EA { B ) , iEill , be a uni formly bounded sequence such that l. im f 1. ( z ) = O , �zEB. Then
Proo f .
1-++00
� im If i { U dW { � ) = � im l im I ( Jc [ r z , � ] f i { Z ) dO ( Z ) ) dW ( � ) = 1-+ +00 1-++00 r-+ 1 =
1
z
im If 1. ( z ) g* ( ) do ( z ) = 0
. 1 -++00
by Propos ition X I : l where g* i s the L 1 ( o ) -l imit of g { rz ) , which exists since gEH 1 . �
i s an A-measure and we know that We have proved that every A-measure is a closed measure . Corol l ary XI : 4 . If JC [ z , � ] d � { � ) EH l for a real measure w { t O ) then the Q - capacity of the support of w is pos itive .
- 1 41 Proo f .
vEM
a
By Theorem X I : 8 and Corollary X I : 2 , w = f d v where and fEL 1 ( v , d� ) . Thus supp w = ( supp v ) n ( f � O ) .
There i s no converse of Theorem X I : 8 . Choose n= l and a non-negative function TEL 1 ( a , d B ) \L log L . Then Tda I S a closed measure but g ( z ) = fC [ z , � l Tda�H l because i f gEH 1 , Re g�O then Re g*EL log L . Now, s i nce 2 Re C - l = P we get that Remark .
2 Re g * - f ( U da ( � ) T
= T ,
a contradiction , since the left-hand side i s in L log L . For the rest o f thi s section , let w denote a pos itive measure so that the functions ( � ) 0:> 0 are pairwi se orthogonal and remember that C denotes the Cauchy kernel relative to B and a . By construction , i f
. f 0 o: Furthermore , i f L: d = +00 for a sequence .
Theorem X I : 9 .
0
a
w .
�
=
0: .
J
- 1 42 ca � ( d j ) 1 / 2 < + 00 then there is a continuous ( a . ) . = 1 wi th J J j = 1 a J. function on d B such that 00
T
( Remember that we have assumed that fC [ z , � ] dU ( � ) =U ( 1 } > o . ) Proo f .
If
cpEL 2 ( u , d B ) then
cp ( z )
=
2:
a>O
where
Hence fC [ z , U CP ( S ) dU ( S ) = f 2: ;}- cp ( � ) d ]J ( S ) a- a a
=
za dU ( � ) 172 ca sup :a < +00 then fC [ z ' � ] CP ( S ) dU ( � ) EH 2 . Assume now that a a d sup c a = +00 and choose a sequence ( J ) 00J. = 1 such that a a c a_._ 002: � ( d J ) 1 / 2 < +00 and cons ider J. = 1 a j k so
if
-
a
.
- 1 43 c a1 /. 4
co
L
j=l
d
3;4 (,;
a .
a .
c 1 /. 4 a
co
L
J =
j=l
(d
J
a J.
)
�
J 1 /4
d
a J. 1 /2 a . J
Then
fI
I
2
00
dw = . L ( J=l
c . a
1 /2
r)
> + 00
a . J
but
fe [ z
, � ]
a . a . 1 /4 J z J ) ) d w ( t,; ) = L ( _ c . 172 a . ca . J= 1 J 00
(,;
d
J
d
a . _ J = c a . j �+oo J
s i nce
wh i ch i s not i n
l im
+ co
•
F i n al l y , a s s ume that 00 .
L
J=1
ca .
(�)
d
1 /2
a .
< +co
00
but
J
L d
j=1
a
= +00 . j
Then 1.jJ ( z ) =
00 E
j=l
i s con t i nuous on
c
a.
(�)
1 /2
a .
Z
a .
1
J
aB
but c
d
1 /2
a J.
1 /2
a J.
z
c
a .
J
a .
J
.
co E
J= 1.
d
a.
( --.J. )
c
a .
J
1 /2
z
a .
J
- 1 44 -
so . [ , s ]\(J ( s ) d w ( s ) I d ( U
I Ie z I
2a
=
00 L:
j= l
d
aj
= +00
wh i ch comp l etes the proof o f Theorem X I : 9 .
v=a , x o O
If
Example .
d ( m , O ) = l , �mElli ,
then
vEM O
I�a� S dV= O , a � S .
and
s o Theorem X I : 9 s hows that t he cont i nuous
f unct i o n s do not operate on t he c l a s s
of mea s u r e s
T
P r opos i t i o n XI : 1 s a y s t h a t i f ( f ) 00s= l A ( B ) , bounded i n H 2 ( a , a B ) and such that then
f
da
S
:4
0,
But
s-++oo .
is a s equence i n l im s-+ +oo
f
s
( z )=O , �z EB
We do not know i f th i s c a n be extended to a r b i t r a r y mea sures i n
MO
but w e h a v e the f o l l ow i ng theorem .
Theorem XI : l 0 .
S uppo s e that
t i ons i n
such that
A( B) s up . J
II
f .
J
00 ( f . ) .=, J J
i s a s equence o f f u n c -
1 2 n + € da < + 00
zEB.
a n d that
�EM O
l im f . ( z ) ex i st s for every If then j-++oo J i s convergent i n the s e n s e o f d i st r i bu t i on theory . I f
f .dw J fur thermo r e Proo f .
sup j
I I f J. I d w < +00
S uppos e that
00
( f . ) .=, J J
then
f . dw J
i s weak l y c onverg e n t .
s at i s f i es t h e a s s umpt i o n s l n the
-
1 45 -
theorem and that l -++co im f . ( z ) = O , �zEB . We then have to prove j J that l -+im+ ff j epdlJ = O , 'ti
where we have used that lJEM O ' I t follows from Proposition X I : l that l im ff j ( s ) ep ( r j s ) dO ( S ) = O for every sequence r J 1 , j -++co .
.A
•
It remains to prove to take care of the first term . Note f i rst that it follows from the calculation i n Example 1 , that
where
a=2n+E
and
a
+
a'
=
1 .
Therefore , for zEB f ixed , 1
im f f J ( s ) ( ep ( r J. z ) -ep ( r J U S ( r J. z , U d o ( U = 0
j-+ + ro
.
.
- 1 46 1 , j-++oo . Thu s , by dom i n ated conJ vergence , the f i r s t term a t the r i ght hand s i de tends to z er o .A
for every sequence
r
for every sequence
r .->1 1 , 1 -++00 . J
.
.
To f i n i sh the proof o f Theor em X I : l 0 , we need o n l y o b s er ve that every con t i nuous funct i on on mated by funct i on s i n C oo ( � n ) .
aB
c a n be u n i f ormly approx i
Not es a nd re ferences Gen e r a l r e f e rences f o r t h i s sect i on a r e Bungart [ 3 ] and Rud i n [ 7 ] . Theorem XI : 3 was proved by a comb i nat i on o f argume n t s o f Gl i cksberg , Kon i g , Seever and Ra i nwa t e r . See Rud i n [ 7 , pg . 1 9 4 ] for deta i l s . Theorem X I : 5 i s due to H e nk i n , s e e H enk i n and C i rka [ 4 ] . We have no exact d e s c r i p t i on o f the s e t s that s at 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 ea r that eve ry s t a r shaped doma i n sat i s f i es
( ** ) .
I t i s k nown that t h i s i s true a l s o f o r
smooth s t r i c t l y pseudoconvex doma i n s , c f . : N . Kerzman ; H o l d e r a n d L P e s t i mates f o r s o l ut i on s o f
aU= f
i n s t rong l y ps eudo-
convex doma i n s , Comm . Pure App l . Mat h . 2 4 ( 1 9 7 1 ) , 3 0 1 - 3 7 9 , and B r i a n Cole
and R . Mi chael Range ;
f o l d s a nd some appl i c at i on s , That
H oo +C
J.
A-mea s u r e s on c omp l ex man i Func . An . 1 1 ( 1 9 7 2 ) , 3 9 3 - 4 0 0 .
i s a c l osed s u b a l g e b r a o f
Loo
wa s f i r s t
proved by S a r a son [ 8 ] on the u n i t c i r c l e , by Rud i n [ 6 ] o n t h e
- 1 47 -
u n i t spher e i n
�n ,
by Ayt u n a and Cho l l et [ 1 ] on t h e boundary
of s t r i c t l y p seudoconvex doma i n s . Jewe l l a nd K r a n t z [ 5 ] proved the r e s u l t s f o r convex sets in
2
w i t h r ea l ana l yt i c boundary and they a l so remarked that 1 n w h e re a= ( a , . . . , a ) , t h ey h a d t h e r es u It f o r "'0 0 a cfT" \L, n a k ElN , 1 1
U n e Exten s i on d ' un r e s u l t a t de W . Rud i n . Bul l . S oc . Mat h . f ra n c e 1 0 4 ( 1 9 7 6 ) , 3 8 3 - 3 8 8 .
[1 ]
Aytuna , A and Chol l et , A . -M . ,
[2]
P r o j e c t e u r s de Be rgman et S z ego pour une c l a s se de doma i ns f a i b l ement pseudo-convexes et e s t i ma t i ons LP . Compo s . Ma th . 4 6 ( 1 9 8 2 ) , 1 5 9 - 2 2 6 .
[3]
Bunga r t , L . ,
[4]
Boundary proper t i es o f h o l o morph i c f u n c t i o n s o f s e ve r a l c omp l ex v a r i ab le s . J . Sov i e t
Bonami , A . and Lohoue , N . ,
Boundary k e r n e l funct i ons f o r doma i ns on c ompl ex man i f o l d s . Pac i f i c J . Math 1 4 ( 1 9 6 4 ) , 1 1 5 1 - 1 1 6 4 . Henk i n , G . M . and C i rka , E . M . ,
Math . 5 ( 1 9 7 6 ) , 6 1 2 - 6 8 7 . (5]
Toep l i t z operators and r e l a ted f un c t i o n a l g e b r a s o n c e r t a i n p seudoconvex doma i n s . Tran s . Ame r . Math . Soc . 2 5 2 ( 1 9 7 9 ) , 2 9 7 - 3 1 2 .
(6]
Rud i n , W . ,
Jewe l l , N . P .
( 1 975 ) ,
[7]
[8J
and Krantz , S . G . ,
Space s o f type
00
H +C .
Ann . I n s t . Four i er 2 5
99-1 25.
Rud i n , W . ,
Funct i o n t heory i n the u n i t ba l l o f [ n o Spr i nger-ve r l ag . New Yor k , H e i de l berg , Ber l i n 1 9 8 0 .
Genera l i z ed i nt e r po l a t i on i n Ame r . Mat h . Soc . 1 2 7 ( 1 9 6 7 ) , 1 7 9 - 2 0 3 . Sarason , D . ,
.00
H
.
Trans .
XII
Complex Homomorphisms
I n the l a s t sect i on , we s aw that one could a s s i gn " bounda ry va l ue s " to c e r t a i n a n a l yt i c f u n c t i o n s by c on s i de r i ng c l o s ed exten s i o n s o f the r e st r i c t i on ope r at o r . Here we u s e the a lg e br a i c s tructure o f H oo ( � } a nd con s i de r e l emen t s i n H oo ( � } a s c o n t i nuous funct i o n s on a compact
�.
set " conta i n i ng "
We start with a very b r i e f rev i ew of the
e l emen t s of Ge l f a nd r ep r e s e n t a t i on . Let
A
be a u n i form a n d c ommuta t i ve Banach a lgebra ( wi th
i dent i ty ) . A comp l ex homomo rp h i sm ( o r l i n e a r mU l t i p l i ca t i ve funct i o na l ) on
A
i s a n e l emen t
O �mE A '
such that
m ( f g } =m ( f } m ( g ) , � f , g E A . We wr i t e MA
MA f o r the c ompl ex homomo rph i sms o n is cont a i ned in the u n i t sphere in A ' .
A.
Note that
Ther e f o r e , by the Banach-Al aog l u theorem ,
MA
i s c ompact
i f we g i ve it the weak * - topo l ogy , wh i ch i s g e ne rated by n e i ghborhoods of the f orm
O ne can show that every max i ma l i de a l i n o f a n e l eme nt i n
MA ,
max i ma l i de a l space o f
the r e f or e A.
MA
A
i s the k e r n e l
i s s omet i me s c a l l ed t h e
-
1 49 -
With every element fEA , we associate f EC ( MA ) by f ( m) =m ( f ) , mE MA and i t follows from the definition of the weak*-topology that f is always continuous on the compact space MA o I f we denote by A the set of f , f E A we have an algebra homomorphism f4f , sup I f ( m ) I =sup I m( f ) I � " f l l and f i s m E MA m E MA called the Gel fand transform of f . A
A
closed subset
K
of
MA
if
is called a boundary for
A
f = sup I m ( f ) l , 'tffE A . mEK One can prove that there is smallest boundary; this boundary i s called the Shilov boundary of A . Let now n be an open and bounded subset of a: n , n�l . Let A be a uni form Banach a lgebra of conti nuous functions on n ( with supnorm) . Then every z O E n gives r i se to an element II l !
a
TI
We denote by ( m ) = ( m ( z 1 ) ' " weak*-closure of
. , m ( z n ) ) ' mEMA ,
and TI* = the
- 1 50 Propos i t ion XI I : 1 .
in
The S h i l ov boundary o f
H oo ( n )
i s c on t a i ned
IT* .
Proo f .
s UE I f ( m ) I < I f ( rn a ) I mE n *
If
l / f - f ( mo )
and
then
are i n
00
H ( B) .
Ther e f ore
wh i ch i s a contrad i c t i on . We w i l l need the f o l l ow i ng r e f i nemen t s o f Lemma
a nd
III : 2
Theorem X : 3 . Theorem XI I : 1 .
K
Let
G
be a convex s u bset o f s ome vectorspa c e ,
a convex subset o f some t opo l og i ca l vector space a nd
F : G x K�m
a f u nct i on such that �
for every
yEK .
K3y
for every
xEG .
Then
F(x,y)
G3x
�
F(x,y)
i s convex o n
G
i s c oncave and con t i nuous on
K
sup i n f F ( x , y ) = i n f sup F ( x , y ) . xEG yEK yEK xEG
T o formu l a t e t h e next t heorem , w e need s ome mo re notat i on . Let
X
be a compac t H a u s do r f f -space ,
r e g u l a r Bore l meas u r e s o n a c lo sed s ub a l g e b r a o f A
X.
C(X)
M(X)
i s t he s e t o f a l l
A funct i on a l gebra ( wi th sup- norm ) .
A
on
h
i s a mU l t i p l i ca t i ve l i n ear f u nct i on a l on
A,
X. then
it
fol l ows f r om the Hahn-Banach theorem a n d R i e s z repr es e n t a t i on theorem that there
that
ex i st s a probab i l i ty mea s u r e
is
We a s s um� that
conta i n s a l l the constants a n d sepa rates po i nt s o n
If
X
p EM ( X )
such
- 1 51 -
f
h ( f ) = f d p , tf f E A a n d we then s a y that
represents
p
h.
We de f i ne
a l l the probab i l i ty mea s u r e s that r epr e s e n t s
F a - s e. t i n { f m } oo m= 1
such that
X
h.
W i t h notat i on a s a bove , suppos e
Theorem X I I : 2 .
o f f u nc t i on s i n
i s an
E
Then there i s a sequence
sup p ( E ) =O . p E Mh ( X ) A
to be
Mh
such that
II f m II -< 1 , tfmElN l i m f ( x ) =O , m-++ oo m l im f ( x ) = 1 m--)o+ oo m Let n ow H oo ( B )
B
tfxE E a.e. (p) ,
be the u n i t ba l l i n
i s a c losed s u b a l g e b r a of
We i e r s t r a s s theorem , fore , i f
W EM O
[n ,
C ( MA )
( fg ) f , g E oo H ( B)
Then
and by the Stone
generates
C ( MA ) .
There
then
d e f i ne s a c on t i nuous l i ne a r ope r ator o n d e n o t e s t h e weak * - l i mi t o f
( f ( r� ) ) O
Theorem
XI : 3 .
X: 8,
00
n � 1 , A= H ( B ) .
XI : 5
and Lemma
e ( MA ) .
He r e
r e l a t i ve l y
f* w .
See
S o by the R i e s z represent a t i on theorem , there i s a regular Borel mea s u r e
0
on
MA
s o that
- 1 52 In particular , L( f ) = f f*d�= f ( O ) =O ( f ) so OEMO . On the other hand , i f DEMO ' consider for f , gEA( B )
fgdD . Then , this determines a continuous l i near functional on = 3 B . Again , since f -+ f f dD=f ( O ) = O ( f ) , Ries z represen tation theorem gives a measure �EM O such that
C(S) ,
S
f gd = f f�dD , b'f , gEA ( B ) . (
�
and M�o contai ns only one element . When When n= 1 , n> 1 , thi s is not so and the following two questions are natural to ask . 1
•
Given DEMO ' determine �EM O as above . Is it then true that f f *g*d � == f f�dD , b'f , gEH ( B ) ? 00
2
i t then true that l im f r ( � ) ==f ( � ) , r-+ 1 DEMO ? this would be an analogue of Fatou ' s MA ) . In view of Henkin 's theorem ( Theorem X I : 5 ) 2 ) would imply 1 ) . But 1 ) i s false - assume for a moment that 1 ) is true .
•
Is a . e . ( 0 ) for all ( Here f r ( � ) =f ( r� ) ; theorem extended to
- 1 53 Then ,
by T heor em X I : 5
a� inf g EA ( B )
f I g - f * 1 2 d ].l �
inf g EA ( B )
f I ;-f 1 2 d O
s o Theorem X I I : 1 wou l d g i ve
a ::: s up i nf O EMo g EA ( B )
=
i nf gEA ( B )
sup ].l EM
f I gA - fA I 2 d O
=
sup i nf g E A ( B ) O EM O
f I gA - fA I 2 d O
=
f l g - f * 1 2 d ].l . a
Not e t h a t
IfI
i s c on t i n u o u s s i n ce <X> fEH ( B ) .
lS
c on t i n u ou s on
M
A
f or a l l
Th i s g i v e a c o n t r ad i c t i o n , s i n c e t h e r e ex i s t
<X> fEH ( B )
wi th inf sup g EA ( B ) ].l E M
a
f l g - f * l d ].l > a
( c f . T h e o r em X : 6 a nd the examp l e p r e c eed i ng i t ) .
N o t e s a nd r e f e r e n c e s For r e s u l t s a n d r e f e r e n c e s c o n c e r n i ng B a n a c h a l ge b r a s 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 . Academ i c Pres s ,
1 98 1 .
Aspects of MathEmatics
E ng l i sh-lang uage subseries ( E )
VoI. E 1 :
G . H e cto r / U . H i rsch , I ntrod uct i on to t h e G eo metry of F o l iations, P a rt A
Vo l . E 2 :
M . K n ebusch / M . K o l ste r , W i tt ri n gs
Vol. E3:
G . H e ctor I U . H i rsch , I nt rod uct i o n t o t h e G eo metry of
Vo l . E 4 :
M . Laska, E l l i pt i c C u rves o ver N u m b e r F i e l d s w i t h P resc r i bed
F o l i a t i o n s, Part B Reduction T y pe
Vol. E5 :
P . St i l l e r , Automorph i c F o r ms a n d t h e P i ca rd N u m b e r of a n E l l i pt i c Su rface
Vol . E 6 :
G . F a l t i n gs / G . Wiist h o l z et a I . , R at i o na l Poi nts
(A
P u b l i ca t i o n o f t h e M a x - P l a n c k - I n s t i t u t f u r M a t h e ma t i k . B o n n )
Vo l . E 7 :
W. Sto l l , Va l ue D i st ri b ut i o n T heory for M e ro m o rph ic M a ps
Vol. E8:
W . von Wa h l , The E q u at i o n s o f N av i e r-Stokes a n d A bst ract Pa rabo l i c Eq uations
Vo l . E 9 :
A . H oward / P .- M . Wo n g ( Ed s . ) , Co n t r i b u t i o n s t o Severa l C o m p l e x V a r ia b les
Vol. E 1 0 :
A . J . Trom b a , Se m i na r o n N ew R esu l ts i n N o n l i n ea r P a rt i a l D i fferentia l E q u a t i o n s ( A P u b l i ca t i o n 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 i d a , F uc h s i a n D iffe rent i a l E q uations ( A P u b l i ca t i o n of t h e M a x- P l a n c k - I n st i t u t f u r Mathemati k , B o n n )
Vo l . E 1 2 :
Vo l . E 1 3 :
Vol. E 1 4 :
R . K u l ka r n i , U . P i n ka l l ( Ed s. ) , Confo r m a l Geometry
(A
P u b l i ca t i o n of the M a x - P la n c k - I n s t i t u t f u r Mathemat i k . B o n n )
Y . And re, G - F u n ct i o n s a n d Geo metry
(A
P u b l i ca t i o n of the M a x - P l a n c k - I n s t i t u t f u r M a t h e mat i k , B o n n )
U . Cegre l l , Capac i t i es i n C o m p l e x A n a l ysis
Aspekte der Ma1hanatik Oeutsc hsprach i ge U nterre i h e
B a nd
01 :
(0)
H . K raft, G eo m et r i sche M et h oden i n d e r I n va ria nte ntheorie
1
B a nd 0 2 :
J . B i ngener, Lok a l e M od u l ra u me i n d e r a n a l ytischen G eo metr i e
B a nd 0 3 :
J . B i ngener, Lok a l e M od u l raume i n d e r a n a l yt i schen G e o metrie 2
B a nd 0 4 :
G . B a rt h e l / F . H i rzebruch / T . H ofer, G e rade n k o n f i g u rationen u nd A ige bra i sc h e F la c h e n
( E i n e Veroffen t l i c h u n g d e s M a x - P l a n c k - I nstituts f u r M a t h e m a t i k , B o n n )
B a n d 05 :
H . Stieber, E xi stenz sem i u n i ve rse l l e r O efo rmati o n e n i n
der k o m p l exen A n a l ysis