Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann Series: California Institute of Technology, Pasadena Adviser: C. R. DePrima
593 IIIIII IIIIIIIIIIIII
Klaus Barbey Heinz KSnig
Abstract Analytic Function Theory and Hardy Algebras III II
I IIIII III Inllll
I
Springer-Verlag Berlin-Heidelberq • New York 1977
II
II III I
Authors Klaus Barbey Fachbereich Mathematik UniversitAt Regensburg U niversit~tsstraBe 31 8400 Regensburg/BRD Heinz K6nig Fachbereich Mathematik Universit~t des Saarlandes 6 6 0 0 SaarbdJcken/BRD
AMS Subject Classifications (1970): 46J 10, 46J 15
ISBN 3-540-08252-2 Springer-Verlag Berlin • Heidelberg • New York ISBN 0-38?-08252-2 Springer-Verlag New York • Heidelberg • Berlin This work is subiect to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin • Heidelberg 1977 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
Preface
The p r e s e n t work w a n t s tional-analytic
theory.
classical analytic
It is an a b s t r a c t v e r s i o n of those parts of
f u n c t i o n t h e o r y w h i c h can be c i r c u m s c r i b e d by b o u n d a r y
v a l u e t h e o r y and Hardy the
to be the s y s t e m a t i c p r e s e n t a t i o n of a func-
spaces H p. The f a s c i n a t i o n of the field comes from
fact that famous c l a s s i c a l
theorems
of t y p i c a l
vor a p p e a r as i n s t a n t o u t f l o w s of an a b s t r a c t are s t a n d a r d r e a l - a n a l y t i c m e t h o d s and m e a s u r e
theory.
such as e l e m e n t a r y
in papers of Arens
and Singer,
and went t h r o u g h
c h l e t algebras, and i l l u m i n a t e
Gleason,
functional analysis
H e l s o n and L o w d e n s l a g e r ,
several steps of a b s t r a c t i o n
logmodular algebras,...). the c o n c r e t e c l a s s i c a l
concept
theory.
We p r e s e n t
the u l t i m a t e
is the a b s t r a c t Hardy a l g e b r a situation.
IV-IX.
It is
to b u i l d up a cohe-
rent t h e o r y of r e m a r k a b l e and p l e a s a n t w i d t h and depth. in C h a p t e r s
(Diri-
for about ten years.
c o m p r e h e n s i v e as w e l l as pure a n d s i m p l e and p e r m i t s
present a systematic account
Bochner,
It never c e a s e d to r a d i a t e back
step of a b s t r a c t i o n w h i c h has b e e n u n d e r w o r k
The c e n t r a l
fla-
The a b s t r a c t t h e o r y s t a r t e d a b o u t t w e n t y y e a r s ago
Bishop,
Wermer,...
complex-analytic
theory the tools of w h i c h
We a t t e m p t
to
The a b s t r a c t Hardy alge-
bra s i t u a t i o n can be looked upon as a local s e c t i o n of the a b s t r a c t function a l g e b r a situation.
To a c h i e v e the l o c a l i z a t i o n is the m a i n b u s i n e s s
of the a b s t r a c t F . a n d M . R i e s z d e c o m p o s i t i o n procedure. voted
t h e o r e m and of the r e s u l t a n t G l e a s o n p a r t
These
are c e n t r a l
themes in C h a p t e r s
to the a b s t r a c t f u n c t i o n a l g e b r a situation.
c o n c r e t e unit disk s i t u a t i o n concepts.
Chapter
in such a spirit as to p r e p a r e
C h a p t e r X is d e v o t e d to s t a n d a r d a p p l i c a t i o n s
theory to p o l y n o m i a l and r a t i o n a l a p p r o x i m a t i o n
II-III de-
I presents
the
the a b s t r a c t
of the a b s t r a c t
in the c o m p l e x plane
and
is the m o s t c o n v e n t i o n a l part of the book.
In c o m p a r i s o n w i t h the r e s p e c t i v e parts of the e a r l i e r t r e a t i s e s on u n i f o r m algebras, present work
the m o s t c o m p r e h e n s i v e of w h i c h
contains n u m e r o u s new results.
tion it is shaped after
the w o r k of K6nig.
s u b s t a n t i a l new m a t e r i a l .
Riesz
t h e o r e m VI.4.1.
estimation
Pichorides
for the a b s t r a c t
and Notes
M o s t of the c h a p t e r s
the
contain
i n d i v i d u a l n e w result is p e r h a p s
Let us also quote
the
S e c t i o n VI.5 on the M a r c e l
conjugation after f u n d a m e n t a l results of
in the unit d i s k situation.
Introductions
[1969],
and s y s t e m a t i z a -
A p r i m e p o i n t is the s y s t e m a t i c use of the asso-
c i a t e d a l g e b r a H #. The m o s t i m p o r t a n t approximation
is G A M E L I N
In c o n c e p t s
For m o r e d e t a i l s we refer to the
to the i n d i v i d u a l
chapters.
IV
In its o v e r a l l les ~ne l e c t u r e s California part
pation
and
likewise
in S e a t ~ t l e / W a s h i n g t o n Galen
Seever
and
tion,
a n d he w a n t s
to p a r t i c i p a t e step
seminar, with
active
to i n c l u d e
we want
care
Above
the p r o o f s
with
their
assistance.
his
all h e
of
sends
student
1970/7]
our
care,
and
of W a s h i n g t o n thanks
Barbey notes
to
cooperawho
started
which
formed
text.
thanks
in c o n n e c t i o n
Schirmbeck
thoughtfulness,
his h o s t s
deepest
Klaus
in his
the p a r t i c i -
and pleasant
lecture
sincere
work
and which
who were
his
resembat the
to e x p r e s s
to w h o m he o w e s
valuable
work
1967/68
He w a n t s
DePrima
of the p r e s e n t
and Gisela
in
at the U n i v e r s i t y
former
to e x p r e s s
distinctive
form.
Lumer
for m o s t
and valuable
and
the p r e s e n t held
and Charles
the e l a b o r a t i o n
interest
parts K~nig
in P a s a d e n a / C a l i f o r n i a ,
Seminar
in 1970.
to U l l a F a u s t
impressive
kind
Algebra
in the e v o l u t i o n
In c o n c l u s i o n for h i s
which
to G u n t e r
to K S z 6 Y a b u t a
with
in c e r t a i n
in a p r o v i s i o n a l
to W i m L u x e m b u r g
days,
in the F u n c t i o n
the n e x t
and
algebras
of T e c h n o l o g y
distributed
thanks
in t h o s e
function
Institute
had been
warmest
structure
on
who
typed
to H o r s t
to K a r l a
May
to M i c h a e l with the
Loch who and Gerd
Neumann
a common
final
text
read most Rod~
for
of
Contents
Chapter
I.
Functions
I.
Boundary in
the
Harmonic Pointwise
3.
Holomorphic The
Unit
Theory
Disk
Functions
2.
4.
Value
Harmonic
and
Holomorphic
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . .
Convergence:
The
Functions
Function
for
Fatou
Theorem
and
its
Converse.
I .
. . . . . . . . . . . . . . . . . . . .
Classes
HoI#(D)
and
H#(D)
. . . . . . . . . . .
16
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter
II.
Function
I.
Szeg6
2.
Measure
3.
The
Algebras:
Functional
abstract
4.
Gleason
5.
The
and
Theory:
The
Fundamental
Prebands
F.and
Bounded-Measurable
M.Riesz
21
Situation.
22
. . . . . . . . . . .
22
Bands
. . . . . . . . . . . . .
26
Theorem
. . . . . . . . . . . . .
31
and
Lemma
Parts . . . . . . . . . . . . . . . . . . . . . . . .
abstract
Szeg~-Kolmogorov-Krein
34
Theorem . . . . . . . . .
36
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter
III.
Function
I.
Representative
2.
Return
3.
The
4.
Comparison
to
the
Gleason
Algebras:
Measures
and of
two
Compact-Continuous
Jensen
F.and
Harnack
the
The
and
abstract
Measures
M.Riesz
Situation
Part
I.
IV.
The
Abstract
Hardy
Algebra
44
44 47
Decompositions
Situation
.
. . . . . . . .
Metrics . . . . . . . . . . . . . . . Gleason
42
. . . . . . . . .
Theorem
. . . . . .
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter
6 12
. . . . . . . . .
48 54 58
59
Basic Notions and Connections with the Function Algebra Situation . . . . . . . . . . . . . . . . . . . . . .
60
2.
The
Functional
66
3.
The
Function
4.
The
Szeg~
~
. . . . . . . . . . . . . . . . . . . . . .
Classes
Situation
H # and
L#
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter
V.
Elements
I.
The
Moduli
2.
Substitution
of
of
the into
Abstract
Hardy
invertible entire
Algebra
Elements
Functions
Theory
of
H #.
69 76 79
. . . . . . .
81
. . . . . . . .
81
. . . . . . . . . . . . .
84
VI
3.
Substitution
into
4.
The
Class
5.
Weak-L
]
6.
Value
Carrier
Function
Functions of Class Hol#(D) . . . . . . . . + H . . . . . . . . . . . . . . . . . . . .
Properties
of
and
the
Functions
Lumer
Spectrum
in
H+ .
.
.
.
.
.
.
.
.
.
1. A
VI.
The
Abstract
Representation
2.
Definition
of
Theorem
the
3.
Characterization
4.
The
basic
5.
The
Marcel
6.
Special
7.
Conjugation
102
of
Return
Riesz
to
the
108
. . . . . . . . . . . . . . . . . .
110
E with
and
Situations
106
. . . . . . . . . . . . . . .
abstract
Approximation
Conjugation the
means
. . . . . . . . . . .
111
M . . . . . . . . . .
115
of
Theorem . . . . . . . . . . . . . . .
Kolmogorov
Estimations
119
. . . . . . . . .
. . . . . . . . . . . . . . . . . . . . .
Marcel
Riesz
and
Kolmogorov
Estimations.
Disk
VII.
Analytic
Situation
. .
and
Isomorphisms
with
the
I.
The
Invariant
The
Maximality
Subspace
3.
The
Analytic
4.
The
Isomorphism
5.
Complements
6.
A
of
Theorem
Theorem Disk
on
I.
VIII.
The
149 151
. . . . . . . . . . . . . .
155
Theorem . . . . . . . . . . . . . . . . . . .
Theorem.
160
the
simple
Compactness
Decomposition
2.
Strict
3.
Characterization
. ..
Invariance
of
H
. . . . . . . . .
Theorem
Convergence
of M . . . . . . . . . . . . . . . .
of
Hewitt-Yosida
. . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . Theorem
and
Main
Result
. . . . . . . . . .
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter
IX.
Logmodular
Densities
I.
Logmodular
Densities
2.
The
Subgroup
3.
Closed
Small
Extensions
149
. . . . . . . . . . . . . . .
Examples . . . . . . . . . . . . . . . . . . . . .
Weak
146
. . . . . . . . . . . . . . . . . . .
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter
144
Unit
. . . . . . . . . . . . . . . . . . . . . . . . . .
2.
Class
Disks
126 138
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter
91 97
. . . . . . . . . . . . . .
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter
85
and
Small
Extensions
. . . . . . .
165 167 170
172
172 175 177 180
181
. . . . . . . . . . . . . . . . . . . .
181
Lemma . . . . . . . . . . . . . . . . . .
186
. . . . . . . . . . . . . . . . . . . . . .
190
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . .
197
VII
Chapter
X.
Function
I.
Consequences
2.
The
Cauchy
Algebras
of
the
on
Compact
abstract
Transformation
214
5.
On
. . . . . . . . . . .
218
6.
The Logarithmic Transformation Logarithmic Capacity of Planar
of Measures and the Sets . . . . . . . . . . . . .
221
~7. T h e
Walsh
for
R(K)
. . . . . . . . . . . . . . .
199 204
Basic
Parts
cA(K)
Theory . . . . . .
. . . . . . . . . . . .
On the annihilating and the representing Measures f o r R(K) a n d A ( K ) . . . . . . . . . . . . . . . . . . . . . . Gleason
P(K) o R ( K )
Algebra
Measures
198
3.
the
on
Hardy
Sets . . . . . . . .
4.
8.
Facts
of
Planar
and
A(K)
Theorem . . . . . . . . . . . . . . . . . . . . . .
Application
to
the
Problem
of
Rational
Approximation
....
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
I.
Linear
Functionals
2.
Measure
3.
The
Theory
Cauchy
and
the
Hahn-Banach
Theorem
. . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
Formula
via
the
Divergence
Theorem . . . . . . . .
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References Notation Subject
. . . . . . . . . . . . . . . . . . . . . . . . . . . . Index
209
227 231 234
236
236 239 241 244
245
. . . . . . . . . . . . . . . . . . . . . . . . . .
255
Index . . . . . . . . . . . . . . . . . . . . . . . . . . .
257
Chapter
Boundary
for H a r m o n i c
The basic
present model
where
the
the a b s t r a c t
will
then
rems
a r e valid.
in H a r m ( G ) ,
Riemann
HoI(G),
The
sphere
situation
into
the
which
It leads
action.
The
Disk
forms
G~,
the
plane
Harm~(G)
class G~.
abstract
individual
theory
classical
q let Harm(G)
the c l a s s
of t h o s e
denote
kernel
value
theory
and unit P:D×S~.
if G is u n b o u n d e d .
= Re
for the
circle
above
classes
the
which
of G r e l a t i v e
Furthermore
s -
--
s-z
function
S = {s6~: IsI=1}
It is d e f i n e d
s+z P(z,s)
let
of h o l o m o r p h i c
for u n i t by the
to be
sz
1-1zl 2
I -s~
[s-z 12
+
s-z
classes
is d o m i n a t e d
V zED a n d
s6S,
I-R 2 P ( R e Z U , e Iv)
=
V 0~RO.
of 2.5:i)
V h6A = R e C ( [ O , ~ ] )
is obvious.
There
exists
and O<s O such that O ~ ( x ) < _ M x
for all
In fact we have
F(s) which
~ _ _s
2
d~(x)
~ ~ o s2+x 2
> 72
=
7 2-~
~(S, r,s
d~(x)
o
for
s>O,
in v i e w of the a s s u m p t i o n s i m p l i e s the result, ii) L e t us f o r m I ~t:~t(x)=t--~(tx) for x~o. T h e n ~ t : [ O , ~ [ ÷ ~ is l i k e w i s e
for t > O the f u n c t i o n monotone
increasing
and b o u n d e d w i t h ~ t ( O ) = O ,
x~O. N o w a f t e r p a r t i a l
F(s) It f o l l o w s
integration
2 [ = ~ o
2 = ~ £
for s>O.
2sx (s2+x2)2 ~ t ( x ) d x
for s>O and t>O.
to p r o v e ~ ( X ) ÷ a for x%0 w e c o n s i d e r a s e q u e n c e t ( n ) % O for I the v a l u e s t(n--~(t(n))=~t(n) ( 1 ) c o n v e r g e to some l i m i t c. In v i e w
of i) c m u s t be finite. the f u n c t i o n s
~t(n)
val of
Therefore
[O,~[.
to W I D D E R
[1946]
and a s u b s e q u e n t pointwise
on
increasing
T h e n we have to s h o w that c=a.
(n=I,2 .... ) are e q u i b o u n d e d the H e l l y
Chapter
1.16)
diagonal
[O,~[
selection
applied
selection
to a f u n c t i o n
and s a t i s f i e s
theorem
to
[O,N]
lead to a s u b s e q u e n c e
@:[0,~[+ R which
2sx 2 (s2+x2)2
The operation
in q u e s t i o n .
F(s)÷a
is l i k e w i s e m o n o t o n e
Now
from
for s>O
for s+O w e d e d u c e
~(x) dx = 2 i x ~
I t ~ /...ds o
(N=I,2...)
which converges
via d o m i n a t e d
that
2 i a = ~
subinter-
(for w h i c h we r e f e r
successively
2 ~ 2s x2 = ~ / , I O S2+X 2 S2+X 2 ~ ~t(n) (x)dx
and f r o m the a s s u m p t i o n
In v i e w of ii)
on each bounded
O~8(x)=O. We k e e p the n o t a t i o n
for the s u b s e q u e n c e
F(t(n)sl
gence
2sx ~(x)dx (s2+x2)2
In o r d e r
which
~t(n)
for all
that
F(ts) iii)
and a l s o OO leads
oo 2 [ 1 1 1 _ x_x__] a = ~ o ttx t2+x 2 ~ e ( x ) d x
e(x)dx
for s>O.
to
oo 2 [ t ~(X)dx = ~ o t2+x 2 x
for t>O
conver-
12
2 ~ t i~)_ald t2+x 2 - Thus
f r o m 2.6 we o b t a i n
@(x~)=ax
for all x>o,
for all x>O.
We introduce
classes
for I ~ o
and the f u n c t i o n
HP(D)
:= {F6LP(I) : 6 H o I ( D ) }
A(D)
:= {F6C(S)
H~(D)
the b i j e c t i v e
correspondences
c { :F6H I (D)} c Hol I (D)
c
c H 1 (D) ~ H I ( D ) I c
HP(D)
theorem
to the h a r m o n i c
is an algebra.
function
Therefore
and the H ~ I d e r
inequality
l e a d s to o b v i o u s
for
In p a r t i c u l a r
1 and f=. f(z)h(Rz)
Then F86an(D)
For O~R1 =
the boundary
Therefore
fn÷1 on D implies
that there exists
f:D+~ such that fnf=gn
for all n>1. This
implies
now ~6H #(D) be the radial boundary FnF for all n>1.
In view of Fn÷l
f~F under consideration 4.2 COROLLARY:
is surjective.
Let f6Hol#(D).
that f6Hol#(D).
function produced
this implies
function
a function Let
by f. Then Fn~=Gn =
that ~=F.
Thus the map
QED.
Then for l-almost
all s6S the angular
limit F(s):=lim f(z) on ~ ( s , e ) : = { z 6 D : - R e ( z - s ) ~ Z÷S z6~(s,~) exists
for all O<e
iii)
(EAF) ^ =
iv)
(E+F) ^ = E ^ n F ^.
e 6
=
bE
I)
It
is
e
Ve6ca(X,Z).
we
0 =
where
<E>0 @ =
= E^ +
F ^ with that
we
(E+F) ^ =
2.6
+
<E>
REMARK: i)
Proof:
2.7 {0}
It =
all
I
<E>
+
For
bands =
= 0
Let s~t
]I811
is
<ENF>.
Now
implies
+
-
E,F
+
observe that
from
6 ca(X,Z)
to
prove
i)~ii).
and
hence
e =
I.
be
Put
< ~
I)
a
F ^.
that
<E>e
6
F.
For
the
obtain
For
e6E
6 F ^.
bands Then
=
8
will
be
written
@BS=B
trivial
B^
=
that
( U B s) ^ = s6I
n Bs s6I
•
and
and is =
ii)
and
this
(EQF) ^ =
iii).
2)
a band,
It
and
= iv).
QED.
properties
are
^.
of
e
i) E+F
subsequent
Compare
<E>
<E+F>
proves
FcE
@
(ENF) ^. =
proves
( U Bs)^^" s6I
I
and
<E^>@,
Therefore
we
iii)
family
B:=
<EAF>
This
. This
ii)
+
6 F^ + E^ c that
= E+F.
<E>
EC
<E>@
<E">.
F ^^
. Also
(Bs)s61 in
+ <E^>8 follows
<E> E ^^
+
s6I this
It
-
a band.
S6I Proof:
<E>.
e6F
<E>@
It
s6I Symbolically
is
{0}.
suffices
REMARK:
=
<E>0
=
EAF
8
for
and
E^ Q F^ =
ENF
Therefore
@.
(E ^ N F ^ ) ^ =
equivalent,
<E>9
that
<E^>9
follows have
+
6 ENF +
.
have
<E>0
<EDF>8
<E^>
obvious
thus
=
Then =
+
F ^.
+ i~1
that n
n
11oli = I1Oli + l=~flIelI
IIell ~ llel]
Therefore
~ s6 1 and hence
summable := s61
< B >8 s
~
which
3. T h e
We
since
return
chapter. M(~) v c
I, w e
F.
Let
see
algebra
intersection
And
BNM(~)
[ 8 is a b s o l u t e l y s6 I ca(X,Z) is n o r m - c o m p l e t e . Of c o u r s e
band that
= 0
projections 8-e 6 B E
Vt£I
are c o n t i n u o u s
Vt6I
and hence
linear
e-~ 6 B ^.
QED.
us fix ~ 6 Z ( A ) .
ca(X,Z).
that
= S6~I0
and M.Riesz
to the
so
since
nontrivial
t h a t e = @.
abstract
~,
V86ca(X,Z) .
from
= < B t > s 6[I e
of n o r m =
It f o l l o w s
= < M ( ~ ) v >
: ca(X,Z)
that
eM(~0 ) 6 A ±.
÷ M(~0) v m a p s
32
It is n a t u r a l an(~)
the class with
to i n t r o d u c e
= an(A,Z)
of a n a l y t i c
~d0 = O. T h e n
3.3 C O R O L L A R Y : m6M(~)
w i t h m O. For each s>O then e - S K 6 A and hence
l(1-e-SK(u)) s
For s%O it follows
>
1-Ct(U'V) l+ct(u,v)e-SK(v)
that
I I e -sK(v) ~( -
).
52
1-ct(u,v) K(U)
~
K(V)
or
1+ct(u,v)
K(u) -K(v)
1-ct(u,v)
1-t 1+St(u,v)
1+ct(u,v)
1+t 1-St(u,v)
A
5) N e x t we p r o v e ~ in 3.3. Let f6A w i t h sume that f(v)
=
l+f
If(u) l~t, and as-
I-If I 2
k :=
6 A
with
K := Re k
> 0 .
I-f It f o l l o w s
llflll
the
lemma
f(O)=O
Vz6DUS
algebra
to A(D)
of the
and
h£A with
I:f6A w i t h =
the
=
Ilfl O. T h u s F(v) F(u)
of DUS
to c o m p u t e
= S u p { l h ( z ) I :h6A w i t h
different for
consider
f6A w i t h
f(z)=h()
= So(O'O
T6M(~)
for A in the G e l f a n d
it is the w e a k , t o p o l o g y
the e s t i m a -
o°6M(~)
T > O,
is d e f i n e d
Gelfand
Let For
situation.
parts
on Z(A)
f 6 A the
- ~ = B~O so t h a t ~ iii)
I is > ~ H(@,@)
(T-T O)
for A is a c o u n t a b l e
for e a c h
ii)
as r e q u i r e d
compact-continuous
for e a c h
Hence
unit ball
compact
measures
a n d is as r e q u i r e d ,
tion with
tinuous.
and H(~,~)T
are s a t i s f i e d ,
I := o O + - H(~,~)
a
The
- T = ~O
in ii)
6 M(~) .
H2(~,~)-I
follows
from
58
Notes
The Notes
present
Notes
to C h a p t e r
by s e v e r a l present
II.
authors,
proof
of
1.5
to be r e a d of J e n s e n
the
1.1
existence first
of w h i c h
is f r o m K O N I G
as to its d e d u c t i o n Lemma
are of c o u r s e The
from
is a v e r s i o n
the
in c o n n e c t i o n measures
appears
[197Ob]
standard
was
with
to be B I S H O P
[1963].
a n d is of u t m o s t
theorems
of the m o d i f i c a t i o n
due
The
shortness
of f u n c t i o n a l
technique
the
discovered
analysis.
to L U M E R
[1968]. The due
first
part
to R A I N W A T E R
well) 2.3
and
are
the
of 2.1 [1969].
theorem.
applied
in S e c t i o n
[1969]
GLEASON
definition the
The
4.4
Chapter
that
are due
results
Notes
BEAR
(for o n e - p o i n t
topological
characterization
is such
M.Riesz
variant of the that
[197Ob].
in G L I C K S B E R G
relation
after
Z(A) him.
to B I S H O P [1967].
[1970].
2.2
are
T as theorem
of the F o r e l l i abstract
it can
Hahn-Banach
versions
It r e p l a c e s [1967],
F.and
a l s o be
the
RAINWATER
II.
the
are
consequence
F.and
u s e of the
as
spectrum
named
the L e c t u r e
2.1
to K O N I G
theorem
and B E A R - W E I S S
3.3 of the p r i n c i p a l
of 2.1
abstract
systematic
on the
of the p a r t s
fundamental [1965]
noticed
relation
and its
for the p r o o f
f o r m of
is due
of the m i n i m a x
[1957]
part
of the
basis
The
[1969]
T)
It is an a m e l i o r a t e d
present
X.5.
and G A M E L I N
equivalence
BEAR
[197Ob].
in the A p p e n d i x
application
second
proof
is a t r a d i t i o n a l
M.Riesz
described
The
subsequent
from KONIG
lemma which
(for o n e - p o i n t
The
[1964].
The
due
G(~,~)
essentials Explicitely
quantitative
to K O N I G
In c o n n e c t i o n
of G l e a s o n
= II~-~II < 2 is an
of A a n d took
parts
this
as t h e
of 3.5
3.5 is in
versions
3.2
[1966a][1969c]. with due
4.5 we
a n d of
See
refer
to G A R N E T T
and also
to the
[1967].
Chapter
The A b s t r a c t
The a b s t r a c t Chapters
IV-VI
ters V I I - I X essential cept
Hardy
to p a r t i c u l a r
We fix a finite The a b s t r a c t subalgebra
Hardy
Hc-L~(m)
weak • topology functional sented
topics
algebra which
measure
o(L~(m),L1(m)),
~:H÷~ w h i c h
IV-IX,
III.1.1
assumptions.
in the proof
to
is w e a k • continuous,
consist
of a c o m p l e x in the
multiplicative
that
is w h i c h
for some F6LI(m).
linear
can be repre-
It will
by n o n n e g a t i v e
(ex-
w i t h m(X)>O.
and is closed
and of a nonzero
that ~ then can be r e p r e s e n t e d
It m a k e s to
of V.6.1).
(X,Z,m), of course
is d e f i n e d
with
and Chap-
iII is not referred
the c o n s t a n t s
Vu6H
Chapters
of the theory
additional
space
situation
contains
aspects
Chapter
of J e n s e n m e a s u r e s
in the form ~ ( u ) = f u F d m
come clear
under
II, w h i l e
positive
Situation
theory will o c c u p y
to the u n i v e r s a l
use of C h a p t e r
the e x i s t e n c e
Hardy Algebra
algebra
devoted
IV
soon be-
functions
O~F6LI(m).
There
are o b v i o u s
situation
in both directions.
the H a r d y a l g e b r a function
connections
algebra
situation
with
the a b s t r a c t
The direct
image
as a l o c a l i z a t i o n
situation:
it
arises
and w h e n
a fixed ~6I(A)
m and ~ in a d e q u a t e
relation
to each other.
results
can be a p p l i e d
The abstract
F. and M ° R i e s z
bra p r o b l e m s
can always
problems
in this w a y
A ± which however
The complex We use
be localized,
Under
this c o n n e c t i o n
implies
algebra
Hardy
theory.
that f u n c t i o n
is r e d u c e d
to H a r d y
singular
The v e h i c l e
II.4.5.
modulo
of course w i t h
of f u n c t i o n
completely
consequence
is r e d u c e d
is chosen,
can o f t e n be shown to be=O).
image
subalgebra
construction
of those
this c o n n e c t i o n algebra
rem thus o b t a i n e d
will
not be used
then
alge-
algebra
measures
in
of r e d u c t i o n
We shall be m o r e
transforms
in B(X,Z)
the m a i n
The a b s t r a c t form the source
In contrast,
after
simply
functions
to transfer
situation.
sults of the theory. will
that
exhibits
specific
X.I°
inverse
the H a r d y
II.3.2
(at least m o d u l o
is the m a i n F. and M . R i e s z in S e c t i o n
to p r o b l e m s
theorem
construction
algebra
of the b o u n d e d - m e a s u r a b l e
w h e n AcB(X,I)
a fixed m£Pos(X,Z)
algebra
function
theorems
into the
of C h a p t e r
in H. II into
Szeg~-Kolmogorov-Krein of m o s t of the d e e p e r
the a b s t r a c t
the transfer,
an HcL~(m)
w h i c h m o d m are
F. and M . R i e s z
as it can be expected,
theore-
theorem
but for an
60
isolated
particular
localization). be to introduce algebra
L~(m)
subalgebra abstract
(to wipe out the remainders
the Gelfand
and thus
algebra
of C h a p t e r
for the inverse
structure
to t r a n s f o r m
of C(K)~L~(m).
Hardy
situation
purpose
An alternative
This
theory
III-
image
space K of the c o m m u t a t i v e
an HcL~(m)
connection
into a closed
would
permit
from the w e l l - e s t e e m e d
but the price w o u l d
space K and thus to loose quite
adopt
approach.
After classes
the f u n d a m e n t a l s
H#cL#cL(m). The
of H into
the d o m a i n
be r e s p o n s i b l e n o r m closures
The chapter Szeg6:
ends w i t h
main
true
in the same evaluation
specialization
Szeg~
In C h a p t e r situation
(H,~)
where
VII
special
can be apart
be a H a r d y a l g e b r a
with
will
M
:={O~F6L1(m):~(u)
:= r e a l - l i n e a r
theorems H~(D) a6D).
re-
(with ~= The imme-
h o w far the
H~(D).
Algebra
Situation
We i n t r o d u c e
Vu6H},
= /uFdm Vu6S}
= {F6K:F~O
and of r e p r e s e n t a t i v e
span(M-M)
named
O~F6LI(m).
and /Fdm=1},
functions
(=densities),
and N
theoe.
will be always
be treated
the F u n c t i o n
situation.
:={f6L1(m) :/ufdm = ~ ( u ) / f d m
of a n a l y t i c
situation
from the classical
and C o n n e c t i o n s
use to
The m a i n
situation
or ~a in any p o i n t
to the Szeg~
extension
The LP(m) -
function
classical
So we
the function
3.9 for the functional
prominent
then the q u e s t i o n
in a c o m p l i -
of the theory.
form as for the unit disk algebra ~o in the origin
K
the classes
introduces
representative
the most
the
some directness.
less important.
look at the
a unique
of the theory
I. Basic N o t i o n s
Let
a first
to deduce
and its systematic
formulation
1O]) Suplexp( f (log~)Vdm):V6M with [F>O] If dP(h)>O then there exist F6M with c;[F>O] V V 6 M w i t h [V>O]c[h>O]
[V>O]c[F>O]
[F>O]c[h>O]
and 3}.
such that [V>O] c
as required.
1.7 UNIVERSAL JENSEN INEQUALITY: Let h6K. Assume that F6M is such that [V>O] c [F>O] VVEM with [V>O]c[h~O].Then
[F>O] If fhdm+O then there exist F 6 M w i t h [F>O]c[h#O] such that [V>O]c[F>O] VV6M with [V>O]c[h~o] as required.
64
1.8 COROLLARY:
For u6H and F£M we have
logic(U) I ~ S u p
{S(ioglul)Vdm:V6M
A set E£Z is defined [V>O]cE. exist
Application
functions
F6M with
independent
[F>O]cE
[h>O]
such that
[V>O]c[F>O] V V 6 M
V6M with There
with
over E. The subset Y(E) := [F>O]cE dominant
If O~h6Ll(m)
CONSEQUENCE:
Assume
[V>O.]cE. is of
F6M.
with dP(h)>O
is full. And if h6K with /hdm~O then
1.9 F. a n d M. RIESZ
with
functions
to xEm and Mm then shows:
from the particular
From the above we know: then
[V>O]c[F>O]}.
to be full iff there exist
of II.2.2. ii)
These F6M are called dominant course
with
for some
[h~O]
I~< ~
is full.
that E6Z is full. For hEK
[h+O]cE then hXy (E)6K and /hXy (E)dm=/hdm.
Proof: dominant
The band B:={XEm}v over E. Then after
@BDMm = 0Fm
satisfies
V 06ca(X,E),
(fm) BAMm = fX[F>O]m 4-
4-
NOW hm 6 BNan(H,q0) and M.Riesz assertion.
BNMm=BNM(H,~)~.+ ÷ -
Choose
an F6M
II.2.2.ii)
in particular
= fXy(E)m
V f6L I (m).
and hence hm-(/hdm)Fm 6 BN ([)i. From the abstract F.
theorem
II.3.1
thus hXy(E)m-(Shdm)Fm6 (H) 4- ± . But this is the
QED.
1.10 COROLLARY:
Assume
that E£Z is full. Then
HXy(E ) c HxEweak*. In particular Proof: SuxEhdm=O
HXy(x)CH , which
Follows
from 1.9 by duality.
Vu£H or hXE6K with /hXEdm=O.
/hXy(E)dm=O.
But this means HXy(E)±h.
polar theorem.
that Xy(x)6H.
For h6(HXE)±CLI(m)
we have
From 1.9 thus hXy(E)£K with The result follows
from the bi-
QED.
The Hardy algebra Y(X)=X,
simply means
situation
that is iff there exist
(H,~)
is defined
functions
to be reduced
F6M which are
iff
>0 on the
65
whole
of X. A n e q u i v a l e n t
in H be c o n s t a n t . /(u-~(u))2Fdm
implies F6M
that
condition
In fact, =
(/(u-~(u))Fdm]
u=~(u)=const
is c h o s e n
is t h a t
for r e a l - v a l u e d
on
2 =
real-valued
functions
(/uFdm-~(u)] 2 = 0
[F>O],
to be d o m i n a n t
all
u 6 H a n d F 6 M the e q u a t i o n
hence
on X. A n d
on X if
the
(H,~)
converse
is r e d u c e d
follows
and
from
Xy(x) £HImportant formulate tion
algebra
c a n be m a d e 16M.
parts
in the
situation to l e a d
In C h a p t e r s
tuation. assumed
of the H a r d y
reduced
But
via
in c e r t a i n
algebra
(H,~)
can be t r a n s f o r m e d
1.11
THEOREM:
H.:={uIY(X)
The
tinuous
space
The
1.12
situation
u=O
cannot
extension
from
of
1.10 t h a t
simply
si-
be
each
by the w i p e -
on X-Y(X) }
of L ~ ( m l Y ( X ) )
Vu£H
defines
functional
is a r e d u c e d
nontrivial
on H~.
Hardy
which
contains
the
a nonzero
weak • con-
We h a v e
M~={F]Y(x) :
algebra
situation
on the
For
point
be in
f6K w i t h Thus
is the w e a k ~ c l o s e d n e s s
the w e a k ~ /fdm=O
closure then
of H~.
fUXY(x)fdm=O
O=/v(f] Y ( X ) ) d m = / V f d m .
of H~.
Extend Yu6H
It f o l l o w s
To p r o v e
v to V6L~(m) from
1.10
that V6H
and
QED.
RETURN
TO T H E
to the u n i t
tion
~a in some
disk point
situatZon
In p a r t i c u l a r
the
reduced
(Y(X),Z I Y(X) , m I Y(X) ] .
lied
algebra
with
even with
to the
We d e d u c e
func-
II.4.5
of X.
subalgebra
linear
(H~,~)
f] Y(X)6H~.
v6H~.
situation,
reducedness
to
of the
consequence
ourselves
IX.
easier
restriction
complex
by V] (X-Y(X))=O.
hence
algebra
theory
a reduced
X-Y(X)
are m u c h
localization
in c o n n e c t i o n
~:~(uIY(X))=~(u)
let v6L~(m] Y(X))
and h e n c e
of the
into part
multiplicative
Proof: this
And
Therefore
measure
Hardy
restrict
:u6H} = {ulY(X) :u6H w i t h
is a w e a k • c l o s e d
F6M}.
parts
the
F.and M.Riesz
s i t u a t i o n S as in C h a p t e r
inessential
constants.
shall
in p a r t i c u l a r
Hardy
o u t of the
we
theory
Also
the m a i n
to the r e d u c e d
V-VIII
a priori,
algebra
situation.
the
UNIT
DISK:
algebra a6D,
(H~(D),~a) situation
The
direct
image
A(D)cC(S)cB(S,Baire),
and
to L e b e s g u e
because
measure
of 1.3.3.iii).
is r e d u c e d .
construction
1.3 app-
to the p o i n t i leads
evalua-
to the H a r d y
We h a v e M a = { P ( a , - ) } .
66
2. T h e
Functional
The
functional
~(f)
~ : R e L ( m ) ÷ [-~,~]
= Inf{-iogI~(u)
In fact,
from both
u6H with
f+loglu 1 bounded
definitions
two d e f i n i t i o n s
c a n at
functional
~ is to r e f l e c t under
liminaries.
The main of the
We
list
some
immediate
tone,
that
iv) ments
claim
is fO.
iii)
e is iso-
e(f)~a(g).
for all
u6H
In particular
×
:= the
set of i n v e r t i b l e
ele-
~ ( R e u ) = R e ~ ( u ) V u6H b y e x p o n e n -
tiation.
2.1
REMARK:
ii)
Let
V6M with
i) L e t V 6 R e L I ( m ) .
f6Re L(m) w i t h
[V>O]c[F>O]
Inf{ffVdm
: V6M with
Proof:
In o r d e r
implies
i)
Let
that
to p r o v e
us r e m a r k with
inequality
that
the m a i n
a n d F6M.
and
~ note
details 1.8.
the a b o v e
inequality
ffVdm~(f) Then
¥ f6Re~
there
(m)~V6MJ.
exist
functions
/ f + V d m < ~. A n d
[V>O]c[F>O]
t h a t V>__O. T h e o t h e r
from the Jensen
pared
~(f)0
therefore
sin(tu) 6 H # D L ~ ( m ) = H
u is r e a l - v a l u e d .
Now ~sin(tu)÷u
and
for t+0
to note
the
subsequent
variant
which
of c o u r s e
im-
assertion.
PROPOSITION:
Proof:
But
QED.
the a b o v e
2.4
since
F ( t ) ~ e t Vt>0.= 2.1.
Assume
f:f(z)=
that
u6H # is>0
(-1)
~ z~ ~
For
t>0
and e / U 6 L #. T h e n
V z 6 ~ we h a v e
F(t)<e
/~
u=const.
Yt>0
and of
Z=0 course that
f(z2)=cos
=const=c(t)
In the functions that
z V z6~.
f(tu)6H # from
Szeg~
u>0.
real-valued find
respective
always
3. S u b s t i t u t i o n
We w a n t
(except
into
n o t be p o s s i b l e
If f(M(u))
to o b t a i n
the
This
us s e e k
2.1 w h e r e
to e x p l o r e
lu!~ some for all
in the
Functions
to s u b s t i t u t e
theorem
t ~ 6 L # so
This
trivial that
2.3
says
that
the r e a l - v a l u e d
assertion
is s h a r p
case
exist
H=~)
IulT6LI(vm)
It f o l l o w s
for all
that
2.4
in the
sense
nonconstant 0
if f is of w e a k - L P ( m )
Let F:[O,~[+~
type
be c o n t i n u o u s
+o0
/F(f)dm
And
for t+~.
For O
< IPI
(qP)
2
and ~(fo)+~(h)
to prove
2 =
0 s u f f i c i e n t l y small.
QED.
4.2 REMARK:
We cannot
lhE-h I ~ e M a x ( 1 , 1 h ply that jugate
expect
(l-e)lh I ~ lh I+~
function
in 4.1 an e s t i m a t i o n
I) for ~>O i n s t e a d
of ii).
V £ > O and h e n c e
P~, a fact w h i c h
In fact,
of the f o r m this w o u l d
the b o u n d e d n e s s
is k n o w n
im-
of the c o n -
to be n o t a l w a y s
true in the
unit d i s k s i t u a t i o n . Let us turn to the m o s t mation
theorem.
E~ c where
We h a v e
important
consequences
of the a b o v e a p p r o x i -
the c h a i n of i n c l u s i o n s
{P6ReL~(m) : 3 P n 6 R e H
IPnl ~ IPI and Pn + p} c ~ w e a k * c N
with
the f i r s t o n e is 4.1 and the last one is 3.4. Of c o u r s e N
denotes
the a n n i h i l a t o r
of N c R e L I ( m ) .
Thus we obtain
~ = E ~, cReL
the s u b s e q u e n t
(m) theo-
rem. 4.3 T H E O R E M :
We have ±
E~ = {P6ReL~(m):
3Pn6ReH
4.4 C O N S E Q U E N C E :
We h a v e
with
K A ReL1(m)
whenever
F 6 M is d o m i n a n t
Proof:
The i n c l u s i o n
tion P £ R e L ~ ( m ) S P V d m = O VV£M. IRehnl ~IPI Thus
which From
m is o b v i o u s .
annihilates
and Reh n÷P,
/Pfdm=O.
s p a c e of ReLl(m)
4.3.
.
In o r d e r
to p r o v e = take a func-
the s e c o n d m e m b e r , a sequence
which means
of f u n c t i o n s
that
hn£H with
a~d a l s o w i t h R e ~ ( h n ) = / ( R e h n) F d m ÷ / P F d m = O .
we have
/(Re h n) fdm =
(Re~(hn))/fdm ,
S i n c e the s e c o n d m e m b e r
the a s s e r t i o n
The n e x t t h e o r e m in t h e o r e m
=N
o v e r X.
/hnfdm = ~(hn)/fdm , and h e n c e
= ~weak*
= ~F + N ReL1(m) ,
4.3 w e o b t a i n
for f £ K A R e L I ( m )
IPnl ~ IPl and Pn÷P}
follows.
is the c o m p l e x i f i e d
is a c l o s e d
linear
sub-
QED.
version
of the last e q u a l i t y
122 4.5 T H E O R E M :
Assume ± H = N
where
that F 6 M is d o m i n a n t ± N (Hq0F)
with H :={u6H:M(u)=O},
N :=E~+iE ~ is the a n n i h i l a t o r
The p r o o f we s h a l l
uses
the s u b s e q u e n t
come b a c k
the p r o o f
in the n e x t
is the a p p r o x i m a t i o n
4.6 REMARK:
Let h = P + i Q 6 H
/Q2Vdm : /p2Vdm -
Proof
of 4.6:
o v e r X. T h e n
of NcLI (m) in the c o m p l e x
s i m p l e but f u n d a m e n t a l
section. theorem
with
Apart
to w h i c h
the h e a r t
of
Then
(~(h)) 2 < / p 2 V d m
/h2Vdm=~(h2)=(~(h))
remark
f r o m this,
4.1 as before.
Im~(h)=O.
VV6M.
2 is real and ~O.
/h2Vdm=/(p2-o2+2iPO)Vdm=/p2Vdm-/Q2Vdm.
L~(m).
It f o l l o w s
that
QED.
P r o o f of 4.5 : The i n c l u s i o n c is obvious. In o r d e r to p r o v e m con± h 6 N N ( H ~ F ) ~ c L ~ ( m ) and put h - / h F d m = : P + i Q . We h a v e to
sider a function prove
that h6H.
1) We have P , Q ± N and h e n c e
P , Q £ E ~. F r o m 2.8 and
3.4 we
know that u:=P+iP~6H #
w i t h ~(u)
= e(P)
= / P F d m = O,
v:=Q+iQ~6H #
w i t h ~(v)
: e(Q)
: / Q F d m : O.
A n d f r o m 4.1 we o b t a i n
functions
Un=Pn+iFn6H
and V n = Q n + i G n 6 H
such that
IunI ~
luI,
IPn I ~
IPI,
u n ÷ u,
~(Un)
real and ÷~(u)
= O,
Ivnl ~
ivl,
IQnl ~
QI,
v n ÷ v,
M(Vn)
real and ÷~(v)
= O.
2) F r o m 4.6 w e see that / l U n ] 2 F d m ~ 2 / P n 2 F d m ~ 2 / p2Fdm,
/ I v n l 2 F d m ~ 2 / Q n 2 F d m ~ 2 / Q2Fdm,
so t h a t
/Iui2Fdm ( a ) ) I / f d m = /uf°dl so that
+
(a)f(a)
Thus
for u6H~(D)
we
= f (u,(a))fdm
= /ulf°+f(a)P(a,-)!dl,
f°+f(a)P(a,-)6K(H~(D),q~o)=H~(D)
L I (I)cL°(I)=L#(H~(D)).
In fact,
4) implies
LI
(m)cH#(D)
that
in v i e w of 4.8 and
126
G o
Also
4) i m p l i e s
G°
that
1-a
G=
so that we c o n c l u d e =f(a)/G(a).
Thus
IV. 3.4 i m p l i e s
6
(D) w i t h q0a
that
(f/G)° = f~/G°6H#(D)
3) shows
w i t h ~a((f/G) *) =~-j~ 1+a f (a) =
that f / G f H #. N o w f l f / G [ G d m = / I f l d m < ~
so t h a t
that ~ ( f / G ) = / ( f / G ) G d m = / f d m .
6) We have K = ~ LI (m) . In o r d e r 5). T h u s
l-a"
t h e r e are f u n c t i o n s
to see c let f6K so that f / G 6 H # a f t e r
fn6H w i t h
Ifn1_1,
furnishes
examples
namely
for 1O 2k+T~n
Thus with A and B the LnP(vm)-norms of P and Q we obtain 1 Bn
= (flQ[nPVdm)P~2R(p,V)
1
[ [2#+1) [ l i p I (2k+l)p IQt (n-l-2k)Pvdm) p k>O 2k+~n
2k+I n-1-2k 2R(p,V) ~ [2#+1) (/iplnPvdm) np (flQinPVdm) np k>O 2k+~n = 2R(p,V)
(2#+I)A2k+1B n-1-2k = R(p,V) {(B+A)n-(B-A) n) • k>O 2k+T__=O and =~ n-1 I n>2. Let us put O<x: O and ~B = ~ =1< 3nR(p,V)
dt>
I ~ + S2,
S ~2S_n-I ~
3
3n
1 R(p,-V~'
as claimed, ii) Let h = P + i Q £ H
with real
a:=tp(h) = fPVdm. Then lal < /IPlVdm < IIPllLnP(vm) , so that from i) applied to h-a= (P-a)+iQ we obtain IIQI~ np < 3nR(p,V)llP-allLn p < 6nR(p,V)II L (Vm) (Vm) P llLnP(Vm)
130
It follows
that R(np,V) ~ 6 n R ( p , V ) .
5.5 COROLLARY:
QED.
Let P6E ~ and h:=P+iP*6H #. For each
I ~ < ~ then
i) @(lhl p) = Sup{/lhlPVdm:V6M}< ~. ii)
If the he6H Ve>O are as in the approximation
theorem
4.1 then
e(lhe-hl p) = O ( £ p) for ~+O. Proof:
i) follows
then clear for all
from 5.4 for the exponents
10 in ]O,~p[
in
in
[~,~[.
with
f:f(t)
f strictly
1-Bpf(2~)=O.
increases
Thus
so that F strictly The a s s e r t i o n
In the case 2 < p < ~ w e c o n s i d e r =
V~-~
O
in ]2
strictly increases in ]~
p'2 ~ ~ 2-p[ ~ and < O in ] 2 - ~p,~[, ~ ~ so that F ~ ~ 7T 1T 71 p'2 ~p] and strictly decreases in [ ~ - ~ , ~ [ .
The assertion follows. QED. Proof of 5.12 ~ 5 . 1 1 : z =Izleit6 ~ as follows:
i) In the case IO and -4: Re z O and hence O ~< t ~ O}.
The
functions
in 6.1
are
an e>O such
F 6 M which
called
internal
d i m N O. F r o m ii) we have an e>O such that W: =
=F-e(V-F)=(I+s)F-eV6M.
It f o l l o w s
= c(U + W - ( I + E ) F ) =
f = c(U-V) iii) = i )
Let V6M.
It f o l l o w s
c 1+a ( ~ - F ) 7
T h e n F - V 6 N and h e n c e F - V = c ( U - F )
that F - V ~ - c F
6.2 REMARK:
that
or V ~
(1+c)F.
For F 6 M c o n s i d e r
"
for some U 6 M and c>O.
QED.
the s u b s e q u e n t
properties.
o) L°(Fm) c L #. I) If O~f n £ ReLI(Fm) ~) If O~f n 6 ReLY(m)
and ~ f n F d m ÷ O and / f n F d m + O
I~) If O~f n 6 ReLI(Fm) ~+) Then
If O~f n 6 ReLY(m)
The f u n c t i o n s
implies
that ~ ( f n ) + ~ ( f ) .
~) i m p l i e s exp(-XB)
the e q u i v a l e n c e
but we do not
implies
I) ~ ) .
and ~ ( u ) = e x p ( - ~ ( X B ) ) = l .
properties enveloped
I) and functions.
The i m p l i c a t i o n
1)~
from IV.3.12
t h a t OO on X. If B £ I w i t h
F r o m IV.3.6 we o b t a i n
f e x p ( - x B) V d m = 1 - ( 1 - ~ ) ~ X B V d m (H,~)
conclude
It is o b v i o u s
that ~)
that ~(XB)=O.
the e q u i v a l e n t
plus F > O in 6.2 are c a l l e d
F o r the c o n v e r s e
ii) We n e x t p r o v e
to o) plus F > O on X
~) ~ + ) ,
(note that ~+) does not d e p e n d on F!).
F 6 M which possess
i) We s t a r t w i t h
is trivial.
and are e q u i v a l e n t
e v e n if F > O
plus F>O and I+)
Proof: ~)
and fn+O then e(fn ) ÷ O .
to I+) plus F > O on X. Of c o u r s e
c l a i m the c o n v e r s e
~), o)
then e(fn ) ÷ O .
and fn+O then e ( f n ) ÷ O .
I) and ~) are e q u i v a l e n t ,
and e q u i v a l e n t
then e ( f n ) ÷ O .
a function
/XBFdm=O
For each V 6 M it f o l l o w s
and h e n c e
/XBVdm=O.
then
u6H w i t h that
Hence m(B)=O
lul~
l=/uVdm
since
is reduced.
iii) We d e d u c e If e ( f n ) ÷ O
1) from o) and F>O.
is false
Let O ~ f n 6 ReLI(Fm)
then a f t e r t r a n s i t i o n
I > O. A n d w e can a s s u m e
that / f n F d m ~
to a
with ffnFd~O.
subsequence
1 It f o l l o w s n2 n"
we h a v e e ( f n )÷
that G:= [ nf 6L 1 (Fm) n=1 n
140
(and in view of F>O on X is well-defined after 0). Hence
for ~>0 we obtain
hence e(fn ) ~ s+~((G-sn)+). hence a contradiction, that
14) ~ o ) .
=(f-fn)+40
even modulo m) so that e G 6 L # I ~ + ~ (G-an)+ ~ s+(G-en)+ and
f n ~
From IV.3.13
iv) Since
Let O ~ f E ReLI(Fm)
it follows
that I ~
I) ~ 14) is trivial and put fn:=Min(f,n)
so that e((f-fn )+) + 0 .
Vs>O and
it remains Vn~1.
From IV.3.13 we conclude
to prove
Then f-fn = that e f 6 L #.
QED. In order to illustrate
condition
We shall come back to this context 6.3 REMARK:
Consider
i) M is compact ii)
the subsequent
conditions.
in o(ReL1(m),ReL~(m)).
If O ~ fn 6 ReLY(m)
iii) =o+)
~4) above we insert the next result. in Chapter viii (see also IV.4.5).
and fn40 then 8(f n) ÷O.
If O ~ fn 6 ReLY(m)
Then i) ~ i i ) ~ i i i )
and fn40 then e(fn ) +0.
(let us announce
that also ii) ~ i )
as it will be
seen in VIII.3.1). Proof:
i) ~ i i )
continuous
The functions
real-valued
functions
fn + 0 the Dini theorem implies IV.3.9.
f n : V ~ /fnVdm are 0(ReL1(m),ReL~(m)) on M with supnorm llfnll= @(fn ) . Since
that 8(fn)÷O.
ii) ~ i i i )
is obvious
from
QED.
For O ~ F
6 ReL1(m)
and I ~ p < ~ let us now define
R p(Fm):=R-~-~eLp(Fm) :={f6ReL(m) :Bfn6Re H w i t h
/If-fnlPFdm÷O},
so that likewise --ReL p (Fm) R p (Fro) = E °° The final result of the first part of the present
section
then reads
as follows. 6.4 PROPOSITION:
For F6M consider
i) F is internal. ii) F is enveloped.
the subsequent
properties.
141
iii)
RI (Fm) c E
iv)
and e(f) = S f F d m Vf 6 RI(Fm).
RI(Fm) c E.
v) RI(Fm) A ReLY(m) c E ~ and h e n c e vi)
NoN
N F(ReL~(m))
T h e n i) ~ i i ) ~ i i i ) = i v ) ~ v ) ~ v i ) . c l o s e d t h e n i) - vi) Proof: ~c/fFdm
i) ~ i i )
fn÷f a n d
If V ~ c F
Thus c o n d i t i o n
and vi)
e(f) ~ 8 ( f ) ~
f
n
ii)
£ Re H such that
and h e n c e e G 6 L°(Fm) c L #. F r o m 2.4.i) iii) ~ i v )
the e q u i v a l e n c e
and iv) ~ v )
v) ~ v i ) .
i m p l y that F > O on X. For vi)
from v) n o t e
is L 1 ( m ) - n o r m
I) in 6.2 is o b v i o u s ,
S i n c e F > O o n X we h a v e a s e q u e n c e
we see that f £ E and a ( f ) = / f F d m , to p r o v e
if N N F ( R e L ~ ( m ) )
V V 6 M then f r o m I V . 3 . 9 w e o b t a i n
IfnI~G w i t h G 6 ReLI(Fm)
it r e m a i n s
Hence
are e q u i v a l e n t .
for all O~f 6 ReL(m).
iii) L e t f £ RI(Fm).
=E ~.
ReL1(m).
that the c h a r a c t e r i s t i c
this
is obvious,
func£ion
are trivial.
Now observe
Thus
that b o t h v)
and to d e d u c e
XB of B : = [ F = O ]
it
is in
RI(Fm) N ReLY(m) c E ~ so that S X B V d m = / X B F d m = 0 V V 6 M a n d h e n c e m(B) = O since
(H,~)
is reduced.
u n d e r the a s s u m p t i o n
Therefore
w e c a n p r o v e the e q u i v a l e n c e
v) ~ vi)
that F > O on X. N o w we have
N N F(ReL~(m))
= {f 6 K N F ( R e L ~ ( m ) ) : /fdm = O}
= {f 6 F ( R e L ~ ( m ) ) : f I H }
= {f6F(Re~(m)) : f L R e h}
= {f 6 F ( R e L ~ ( m ) ) : f ± R 1 (Fro) }, (I~ N ) N R e L ~ (m) = {f 6 ReL~ (m) : flFR1(Fm)~.--~ S i n c e FRI(Fm) c R e L 1 ( m ) bipolar
theorem
linear
subspace
it f o l l o w s
F R 1(Fm)
= {f 6 ReL l(m) : f ± (1N) N R e L Y ( m ) } ,
RI(Fm)
= {f6ReL(m):
R 1 (Fro) N ReLY(m)
We c o m p a r e
is a c l o s e d
from the
that
fF6ReL1(m)
a n d fiN A F ( R e L ~ ( m ) ) } ,
= {f 6 ReLY(m) : f i N N F ( R e L ~ ( m ) ) }.
the last e q u a t i o n w i t h
142
E ~° = {f 6 ReL°~(m) : f i N } . It f o l l o w s
f r o m the b i p o l a r
N A F(ReL~(m)) valence
theorem
v) ~ v i )
becomes
obvious.
situation.
6.5 LEMMA: t h a t T c FL~(m)
Proof:
Let
red r e s u l t
ii)
and T c L P ( m )
i) We can a s s u m e
then r e a d s for some
c l a i m that for each
be a c l o s e d
as follows:
If a l i n e a r
1 ~ p < ~ then d i m T < ~
This
the c o n d i t i o n
d i m N < ~.
to the a b s t r a c t
linear
subspace
IFI+I
and h e n c e a s s u m e
Har-
case.
such
subspace
Thus
of F.
The d e s i -
TcL~(m)
is LP(m) -
We s h a l l p r o v e
closed.
instead
that F=I.
this v e r s i o n .
f r o m the c l o s e d g r a p h
c>O such that Ilfll~ ~ clIflILp v f £ T. We L(m) (m)
1~sO s i n c e we can take
to ~ T c L P ( F P m )
a constant
the e q u i -
T h e n T is f i n i t e - d i m e n s i o n a l .
It is c l e a r t h a t T is L ~ ( m ) - n o r m
t h e o r e m we o b t a i n
Thus
in b o t h the real and the c o m p l e x
for some F 6 LP(m).
T h e n w e can pass o v e r
norm closed
centers
lemma which
It is true
I ~pO
such that IiflI ~ < L (m)= for p<s O
is t h e r e
In v i e w of
are no
4) we
determined
Ibnl=1.
after
2) and
Buton
X - D we h a v e
that
XD6H.
6) We tion ties.
this
since
ii)
f~O.
that
assume
the
such
t h a t m(D)
is m i n i m u m .
in the
of
assertion
since
6A(f). nimal
It f o l l o w s
of
of
then
Of c o u r s e
in H.
shall
Define
I) T c o n t a i n s
then
fn÷XD
sense
of
We h a v e F ~ =
IfnI=F n w h i c h
Thus
Fn=fnbnwhere
Then bnlD=const
and h e n c e
But
TcL~(m) B.
f =F =I n n
for n ÷ ~
on D.
and h e n c e
iv)
We
proper-
each
D6A(f)
we h a v e
If Dn6A(f)
for n~1
to be
show
to 5).
false
obtain
of
the
f and
that
D must
be m i -
if U 6 A
is
fXu#f
or
fXD_u=fXD-fXu=f Thus
fXx_u+O.
and h e n c e
closed
lots
functions
subsequent
mi-
QED.
subalgebra
us w i t h
D-U6
D is i n d e e d
contradiction.
be a w e a k *
for
a set D6A(f)
In fact,
and h e n c e
2.3 p r o v i d e s
the
func-
immediate
v)
so t h a t m ( U ) = O .
to c o n s i s t
a nonzero
and v) we
is the d e s i r e d
that
Fix some
For
assertion
then
T h e n we h a v e
functions
iii)
m(D)>O.
L e t BcL~(m) see
list
UNV6A(f).
t h e n U~A(f)
2) w h i c h
2.1.iv) ~ i ) :
fF a n n i h i l a t e s
XD~H.
one.
lemma.
We
in c o n t r a d i c t i o n
assumption.
of
We
t h a t O<m(U)
O
can
so t h a t
are m i n i m a l
we o b t a i n
factor
Now assume
I f n l = F n = e -n.
can n o w p r o v e
f6L(m)
From V.I.6
3) so t h a t we
But
exists
D6& w h i c h
that m(D)<m(X)
up to a c o n s t a n t
b n 6 B x of m o d u l u s
t6~,
assumption.
there
sets
can a s s u m e
e x p ( - n ( 1 - X D ) ) E B × for n~1. are
for all
to o u r b a s i c
of
such
z e r o di-
f6L~(m)
such
facts.
40.
2) BTcT. 3) T c H cH. To
see
1) t a k e
a nonzero
c L ° ( F m ) c L #. H e n c e 6L~(m}." "
It f o l l o w s
is an a l g e b r a .
there that
To p r o v e
h6L1(m) are all
which
functions these
3) we
annihilates u~6H with
h u £ ~ are
invoke
the
in T.
B.
h I T h e n ~--6L (Fm)~
lu~l~1, 2)
u~÷1
is o b v i o u s
fundamental
theorem
and u~Fh-£ since VI.4.8
B in
155
the
form VI.6.9.iii).
fF£K and
/fFdm=O.
proves
3).
Now
the
such
time
that
fF a n n i h i l a t e s fEH#NL~(m)=H
2.3 has
fXx_D
functions
t h a t we can
iterate
in H i n d e e d .
2.4
then that
for
fXD a n d
at n o n z e r o
If f6T
It f o l l o w s
in H the p r o d u c t
to the U N I T
from
isomorphic
to H o I ~ ( D )
of the b a s i c
1.3.8.
facts
less
which H
analytic
linear
subspace
variant
from
theorem
it is c l e a r
of
zero d i v i s o r s
II.3.6
H
that
The
analytic
unit
since
zero
I
(~)
disk
iii)
as a c o n s e q u e n c e
The properties
properties
answer
i) and
of H~(D)
to c e r t a i n
situation.
nontrivial
satisfies
it is a l g e b r a i c a l l y
divisors
theory.
is i m m e d i a t e
a6D.
H always
functienals
and
disk
as well.
questions
One question
invariant
time
the
weak~
from
is on
closed
the d e p t h s
coordinate
nonzero
weak~
than ~ and how
Szeg~
continuous these
the a n s w e r
in q u e s t i o n
is a r e d u c e d
of
function.
to d e s c r i b e
(H,~)=(H~(D),~o)
functionals
(H,~ a)
comes
Z the
admits
other
situation
IV.I.12:
Each
but
it is H = Z H w i t h
is w h e t h e r
are
is
the ~a
situation,
and
= z,F9 H. ~a 1-aZ disk
questions.
l i n k s (H,~)
1.1
In the u n i t
see
the
simplest
(D)cL
a l w a y s s i m p l y i n v a r i a n t ? In the u n i t disk m t h i s is true. T h e r e p r e s e n t a t i o n a f t e r the in-
theory:
linear
for the p o i n t s
the a b o v e
a set D6&
T h u s we arrive
And
lots
which
of L~(m) : is H
question
functionals ?
we
the
no
function
is a b e a u t i f u l
in v i e w of to be
H
iv)
are w e l l - k n o w n
theorem
function
The other
is =0.
obtain
algebra
contains
but
(H,~)=(H~(D),~o)
multiplicative
clear
which
disk
subspace
analytic
The
of a n a l y t i c
Theorem
appears
situation
f6T we o b t a i n
of T as well.
of w h i c h
and thus
it s a t i s f i e s
Disk
are n a t u r a l
which
DISK:
And
immediate
3. T h e A n a l y t i c
The
a nonzero
members
H so t h a t
QED.
RETURN
are
For
the p r o c e d u r e
as we k n o w
ii)
come.
are n o n z e r o
B and hence
and ~ ( f ) = / f F d m = O
theorem
discloses
It d e p e n d s
to the u n i t
disk
upon
the
intimate
a fundamental
situation
connection
between
construction
which
(H~(D),~o) . L e t
us s t a r t w i t h this
construction.
3.1 tion
THEOREM:
I£H
such
Assume that
that
H =IH. m
H
is s i m p l y m
invariant
and
fix an i n n e r
fun~
156
i) For each u6H # t h e r e e x i s t s an(n=O,1,2,...)
a unique
n u - [ a£I ~ 6 In+IH # ~=0 called
sequence
the T a y l o r
numbers
for all n~O,
coefficients
of the
function
ii) For e a c h u6H # the p o w e r
series
expansion
an=/u[nFdm
of c o m p l e x
such that
u. If u 6 H # N L I ( F m )
then
Vn~O.
AA u:u(z) and d e f i n e s
=
a function
~ anZ n=o
n
converges
~6HoI#(D).
: Su
for all zED,
If u 6 H # N L I ( F m )
Fdm : Su
Fdm
then
Vz D
II-zl iii)
If u 6 H # n L P ( F m )
for some
i
Np~
6 H o I ~ ( D ) .
u(z) = f P ( z , . ) U d t S
function
after
IV.3.15
For z6D then
= f((P(z,-)U)oI)
Fdm
= f(P(z,-)oI) ( U o I ) F d m = /(UoI)
I-]z12 F d m =
(UoI)^(z).
Iz-zl 2 ii) For f6H#(D)
write
U f=~ w i t h U , V 6 H
(D) and V invertible in H#(D). Then U=Vf
and UoI=(VoI) (fol). F r o m ~z (U) =~z (V)O with
B~(f) ~ c f f F d m V O ~ f 6
8~(fn)÷O
from i) and hence
iii) Now 1.6 implies
The assertion
we list several properties
follows.
of loglHXl
that
QED. and of NL which
in the sequel. Assume
that
IHXI is an additive
(H,~)
subgroup,
is reduced, ii)
i) E~clog[HX[cReL~(m),
and
loglHXl={f6Ren~(m):e(f)+~(-f)=O}
=
186
={f6ReL~(m) :e(f)+~(-f)~O}. and
Ifnl ~ some G w i t h Proof:
we see that
+~(-f)~O
and hence
If d i m N
that
Assume
fn61OglH×I
with
is a direct
consequence
of V.I.3.
~(fn ) and e ( - f ) ~ l i m i n f
that
fn÷f£ReL~(m)
f61ogIHXl.
f61oglHX I from ii).
~Renl (m)
ii)
ii)
~(f)~liminf
1.9 PROPOSITION: NLcNL
Assume
eG6L #. Then
i) is obvious,
IV.3.12
iii)
(H,~)
iii)
From
~(-fn ) . Thus
~(f)+
QED. is reduced,
i) We have
(NL) ±± c (ioglH×l) i c N II = N ReL1(m)
-
O from V I . 6 . 1 . i i i ) .
that U6ML.
to prove
of ML,
Hence
f6NL as claimed.
that r e a l - l i n e a r
In
span(loglHXl)
c real-linear
to be finite. closed.
Subgroup
In the proof
cReL~(m).
dual we have
sum of N±=E ~ with
is weak~
2. The C l o s e d
span(loglHXl)
= d i m ( ( R e L I (m)) '/N I] = dim (ReL~(m) /N ±) ,
is a s s u m e d
is the d i r e c t and hence
f=c(U-F)
':= the L 1 ( m ) - n o r m
dimN = dimN' and this
point of M as well.
point
is weak, closed. We have
N ± : E ~ c loglHXl But with
IoglHXlc(NL) I and N A = E ~ c l o g l H X I . ii)
let F£ML be an i n t e r n a l
i). Thus
the last a s s e r t i o n cReL~(m)
from
s p a n ( l o g [ H x] )weak~= real-linspan (logl~I).
It follows
some
that
real-linear
finite-dimensional
linear
span(logI~ subspace
QED.
Lemma
of the m a x i m a l i t y
theorem
3.5 we shall
need a lemma
from
I)
187
topological
algebra which deserves
be e s t a b l i s h e d
in the p r e s e n t
Let V be a fixed a closed
additive
D(S) :=
N tS t>O
REMARK:
Proof:
span (S)
and D(S)
the c l o s e d
is c l o s e d
that d i m E ( S ) < ~ .
consists
of i s o l a t e d
a sequence
of n o n z e r o
or 1-I1uzIit])< ~ Vt>O. Put M = {(x,t):O f l U
is ho-
lomorphic. iv)
If Uc~
is o p e n such that h l U is h o l o m o r p h i c
then fh+fIu
is
holomorphic. Proof:
In v i e w of 2.5 the f u n c t i o n s
exist everywhere 2.6 w e see that iii)
f r o m 2.11 ifl
=
and are c o n t i n u o u s 8 C = h and h e n c e
and iv)
f r o m 2.4.
9 C and
(fS) C and h e n c e
f
on ~ and tend to 0 at infinity.
fh + f = In o r d e r
(fS) C. T h u s we h a v e
to p r o v e
From
i) , and
ii) we e s t i m a t e
If (u)-f(z) ~ ~-~ I u-z dg(u) [ < ~ ( f , V ( a , e ) ) I ~h
I
dL (u)
7V~7 V (a,e)
< 2e ~ But s i n c e
~(f,?(a,e))
f is h o l o m o r p h i c
to 0 at i n f i n i t y
3. B a s i c F a c t s
outside
the same e s t i m a t i o n
L e t K be a f i x e d c o m p a c t
in v i e w of iv)
and tends
subset
%@ of ~. For the r e m a i n d e r
the b o u n d a r y
~K of K,
K°
the i n t e r i o r
of K,
the c o m p l e m e n t the u n i q u e
components
K. In the p r e s e n t
of the
the n o t a t i o n s
X
~
of V(a,e)
is true all o v e r ~. QED.
on P ( K ) c R ( K ) c A ( K )
chapter we introduce
The bounded
Vz6V(a,e).
of ~
~-K of K,
unbounded
component
(if t h e r e are any)
s e c t i o n w e s t a r t to e x p l o r e
of ~.
are c a l l e d
the h o l e s of
the a l g e b r a s
P(K)cR(K)c
210 CA(K)
defined
tions
we p r e s e n t
in the
Introduction.
several
basic
After
some
applications
simple
direct
observa-
of the C a u c h y
transforma-
K = {z6¢:Izl~1}
= DUs we
tion.
3.1
EXAMPLE:
In the u n i t
have
A(K)
= CHoI(D)
that
P(K)
= R(K)
disk
situation
per definitionem.
= A(K)
3.2 P R O P O S I T I O N :
(see
We h a v e
The Taylor
series
expansion
shows
1.3.3.i)).
P(K)=R(K)
iff ~ = ~ ,
that
is iff K has
no
holes.
I 6 R(K) We p u t F u = Z-u
Proof: a i m is sult
to s h o w
from
obvious +C(K). then
the
that M=Q ~ which subsequent
estimation
shows
2) M is open.
1 = z-u
Fu(Z)
with
the
series
3) M N ~ # ~
four
remarks.
that
the m a p
T o see
~/~u-a< @ / d i s t ( a , K ) < 1 1 z-a
this
the
But
I) M is c l o s e d
this w i l l
in ~.
u ~-~ F u is s u p n o r m
re-
In fact,
continuous
l e t a 6 M and O < 6 < d i s t ( a , K ) .
1
I
and hence
M:={u£~:F u6P(K)} " Our
assertion.
For
an ~ ÷
u6V(A,6)
and h e n c e
u-a z-a
uniformly
lu l > M a x { Izl :z6K}
for u60~. D e f i n e proves
Fa(Z)
k=O
convergent
~cM
after
(u-a) k (Fa ( z ) ) k
on K.
I) and
Thus
2).
VzEK,
V(a,6)cM.
In fact,
if u E ~ ~ w i t h
then k 1- _z u
with
the
each
hole
Pn÷Fu
series
uniformly
on G s i n c e
We n e x t define
uniformly
G of K.
~GcX.
But
determine
the h u l l
Thus
K = ¢-~
that
the
then this
the
so t h a t
(Z-U) Pn+1
spectra
phism
P(K)+P(K)IK
tains
the p o i n t
= P(K).
since
= @ for
(Z-u)P n v a n i s h e s
of K a n d of
Therefore
4) M N G
on K a n d h e n c e
of P(K)cR(K)c-A(K).
with
C(K)÷C(K):f
evaluations
u6M.
in M a n d Pn p o l y n o m i a l s
uniformly
is n o n s e n s e
K is c o m p a c t map
on K. T h u s
if u 6 G w e r e
K of K to c o n s i s t
restriction
u { ~ u for u6K.
convergent
In fact,
on K,
- k=O u
the
~ u in the p o i n t s
u6K.
It f o l l o w s
a supnorm
Z(P(K))
P(K)
of its h o l e s .
~K = 9~ ~ c ~Q = X.
spectrum
at u. QED.
To s t a r t w i t h
the u n i o n
~--~ flK p r o d u c e s
with uniformly
isomor-
= ~(P(K))
As u s u a l
we
con-
identify
211
3.3 P R O P O S I T I O N :
We have
Z(P(K))
= {~u:U6K}
= K. ^
Proof:
We have
to p r o v e
that
each ~6~(P(K))
~£Z(P(K))
and put
u:=~(Z).
Then
u6K
6P(K)
after
= ~(I)
the p r o o f
= ~((Z-U)Fu)
polynomial
= ~(Z-u)~(Fu)
f and hence
3.4 P R O P O S I T I O N :
Proof: u6K.
As a b o v e
= f(u)
The course proof
P(u)
= ~u(f)
that
deeper
each ~6Z(A(K))
3.5 A P ~ N S
this
LEMMA:
Proof:
Fix
choose
for
off
V(a,¢)
such ~h
and
form
spectrum
e>0
is = ~ u
a6K
the
of a}CA(K)
some
3.6 P R O P O S I T I O N :
function = f(u).
is m u c h
more
requires
set
{flK:f6C,(~)
is s u p n o r m
he6C~(~)
some
constant
with
poles
Hence
~(f)=
difficult. u6K.
But
Of the
the s u b s e q u e n t
holomorphic
dense
in A(K)
it to some with
hc=1
in K °
(observe
function
in V(a,~)
QED.
We have
E(A(K))
3.5 p r o v i d e s +
independent
that
f6C,(~). and h£=O
of e,
f := f + < @ e > f
and satisfies
as r e q u i r e d
and
c>O
It f o l l o w s
6C,(~).
Fix ~6Z(A(K))
f(u)
for some
or ~(f)
u6K
and extend
as in 2.12.
function
lemma
is = ~ u
Now ~(P)=P(u)
2.12.
f6A(K)
functions
that
as above.
~ u in the p o i n t s
function
Thus
such
for e a c h
for a£K°!).
with
@e6ca,(~)
the A r e n s
Z(A(K))
from
in K O a n d in v(a,~)
Proof:
= ~u(f)
that
c ~ ~
we obtain
I =
= f(u)
= K.
u6K
for s o m e
is h o l o m o r p h i c
a suitable
Fu6
is a r a t i o n a l
evaluations
For each
a function
Now
Fix
contradiction
each ~6Z(R(K))
Then
= ~(f)Q(u)
be d e d u c e d
neighborhood
is t r i v i a l
u6K.
QED.
of the
will
that
if f = P / Q
= ~(f)~(Q)
the p o i n t
fact w h i c h
a n d in s o m e that
u:= ~(Z).
determination
some
u6~ ~ and h e n c e
to the
= {~u:U6K}
to p r o v e
Vf6R(K).
for
QED.
Z(R(K))
P. T h u s
otherwise lead
= O. N o w ~(f)
and put
= ~(P)
it c o n t a i n s
since would
we have
polynomial
off K then
Vf6P(K).
We have
Fix ~6Z(R(K))
for e a c h
of 3.2 w h i c h
is = ~ u
upon multiplication
= {~u:U6K}
of t h e
u:= ~(Z)6K
as above.
us w i t h
a sequence
of
on K.
fe w i t h
= K.
conclude
(Z-u)f n ÷ f u n i f o r m l y
6 CB(~)
~fe-fll ~ 2 c ~ ( f , V ( a , £ ) ) .
For
functions
It f o l l o w s
that
f6A(K) fn6A(K) f(u)
=
212
= qg(f(u)) = m ( f ( u ) + ( Z - u ) f n )
÷ re(f). T h u s ~(f)
= f(u)
= ~u(f)
Vf6A(K).
QED.
The n e x t t h e o r e m e x p r e s s e s Cauehy
the d e c i s i v e
relation
between
3.7 T H E O R E M :
i)
F o r 06ca(K)
ffd@=O Vf6R(K),
the s u b s e q u e n t
that
iii)
0C(x)=O
Proof: as above.
ii)~i)
properties
Consider
We can a s s u m e
a function
that F6C~(U)
the a s s u m p t i o n s
and i)~iii)
an e x t e n s i o n
implies
are obvious,
f6C(K)
with
3.9 C O R O L L A R Y =
to some
and h e n c e
F6C~(~).
Then
F6CI(u)
2.6 com-
that
QED.
(Hartogs-Rosenthal) : Assume
F6CI(u)
that L ( K ) = O .
to some
T h e n R(K)
=
C(K).
Proof:
For each
3.7 and h e n c e
06R(K) ± we have
0=0 from 2.8. T h u s
The n e x t r e s u l t t h a t the i d e n t i c a l 3.10 T H E O R E M :
Proof: functions
result
Assume
closed neighborhood
that
U(x)
with
hl,...,hr6C~(~) of K. N o w
We h a v e
= R(K) ±± = C(K).
everywhere
from
QED.
Bishop
localization
theorem
application
in the p r o o f
of 8.4. O b s e r v e
is t r i v i a l
We c h o o s e p o i n t s
neighborhood
0C=o L e b e s g u e - a l m o s t
R(K)
is the i m p o r t a n t
It w i l l h a v e a b e a u t i f u l
0CI~=0.
F6CI(u)
an e x t e n s i o n
3,8 C O R O L L A R Y : A s s u m e that f6C(K) has an e x t e n s i o n open set UDK such that ~~FI K = O. T h e n f6R(K)
A(K)
are e q u i v a l e n t .
for all x6~.
iii)~ii)
bined with
R(K).
and the
is 06R(K) ±.
ii) I f d 0 = O for e a c h fEC(K) w h i c h has 9F o p e n set UDK such t h a t ~-~IK = O.
=
R(K)
transformation.
f6C(K)
for A(K). is such that e a c h p o i n t x6K has a
flKNU(x)
Xl,...,Xr6K
6 R(KDU(x)).
Then
f6R(K).
w i t h K c U ( X l ) ° U . . . U U ( X r )O and
w i t h h k = O off U ( x k) a n d h 1 + . . . + h r = 1
fix 66R(K) ± so t h a t
to s h o w that ffdS=O.
for
06ca(~)
Let us put
in some
lives on K and
213 1 3hk
e k := hke - ~ ~ - ~ eCL 6 ca(~) , so that from 2.10 i/3hk )C 8 C = (hkS)C - ~k3-~Z-- @CL = hk@C
It follows
that e~l (~-KnU(Xk))=O
and 0k6R(KNU(Xk)) ± after
so that 0 k lives on KDU(x k) after 2.8
3.7. Thus ffdek=O.
and e C live on K. It follows For the measures
L-almost'everywhere.
that ffdS=O.
Now %1+...+8 r = @ since
@
QED.
@6A(K)±cR(K) 1 we have the subsequent
addendum to
3.7. 3.11 PROPOSITION: points
for Lebesgue-almost
all
x6X.
Proof: Then
If 86A(K) ± then 8C(x)=O
Fix f6B(X,Baire)
(fL) C is defined
and extend
and continuous
phic on K ° after 2.4. Thus
X
it to f6B(~,Baire)
(fL)CIK6A(K)
that
We conclude 3.12 EXAMPLE
K
8C=o Lebesgue-almost
everywhere
(The Swiss Cheese):
iii) K := (DUS)
-
To this end consider
ii)
subset KC~
starts with the
an6D and radii O0 is such
lemma
that
(see C A R A T H ~ O D O R Y
Yu6V(a,6).
so that V ( a , 6 ) c G l ( R , a ) .
G is c o n n e c t e d
depend
on the
F o r a6K and m6M(R,a)
If z 6 C S ( ~ a - m ) then after
and the G l e a s o n
rela-
fundamental
lemma
4.1.
we have C S ( 6 a - m ) c G l ( R , a ) .
4.1.ii)
there
a c>I. Then i({Iz-al :z6~nV(a,T)})
