Abstract Analytic Function Theory And Hardy Algebras

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~R
and

real

u,v.

1-2Rcos(u-v)+R 2 some

immediate

properties,

1-1zl O < - -

iii)

6~Itl~

for e a c h

< - -

=

P ( R , e i V ) < p ( R , e iu) P(R,eit)~o

i) P is c o n t i n u o u s

on DxS

and

1+Lzl < P(z,s)

l+Iz I = ii)

theo-

functions

in Harm(G)

the c l o s u r e

the r e s p e c t i v e

denote

of b o u n d e d

functions

Here G means

so t h a t ~ 6 G

a n d CHoI(G)

boundary

We l i s t

the

up to the p o i n t

G~¢.

D = {z6~: Izl
in the U n i t

theory.

for w h i c h

of the c o m p l e x

functions

extensions

HoI~(G)

functions

the c o n c r e t e

c a n be p u t

and CHarm(G)

continuous

to the

Functions

abstract

the r e a s o n s

subset

of h a r m o n i c

admit

theory

Theory

Functions

F o r G an o p e n class

describes

subsequent

illuminate

1. H a r m o n i c

Value

and H o l o m o r p h i c

chapter

for

I

for r e a l

for R+I 6>0.

V z6D a n d

s6S.

1_[z I u,v with

pointwise

lu!~ivi~

in O< itl=<~ and

and O~R
uniformly

in

disk

iv) P ( z e , s e ) = P ( z , s ) v) P ( R ~ , B ) = P ( R B , ~ ) A n d we r e c a l l sentation

for z6D a n d s,~6S. for e , B 6 S and O
from elementary

theorem.

Here

w i t h the n o r m a l i z a t i o n

1.1 R E P R E S E N T A T I O N f(Rz)

analytic

i denotes

THEOREM:

= I ~-Szf(Rs)dl(s) S

for f£Harm(D)

f(Rz)

For f£HoI(D)

= iImf(O)

we have

and O~R
1.2 REMARK:

on S

we have

+ / S+ZRef(Rs)dl(s) s - z S

V z6D and O~R
f P(z,s)dl(s) S

=I for z6D.

behaviour

we put f R : f R ( s ) = f ( R s ) Let

repre-

measure

w e have

We t u r n to the b o u n d a r y f:D~

the b a s i c

Lebesgue

V z6D and O~R
= ] P(z,s)f(Rs)dl(s) S

In p a r t i c u l a r

theory

I(S)=I.

= ~ P(z,s)f(Rs)dl(s) S Hence

function

one-dimensional

1~p~.

of the f u n c t i o n s

in H a r m ( D ) .

For

Vs£S.

For f£Harm(D)

then I

I (/If(Rs) IPdl(s)) p for

llfRll

:=

LP(1) is m o n o t o n e Proof:

increasing

For O
For

[

Max[f(Rs) I s6S

we o b t a i n

: f(R~z)

from

1 < q < ~ the c o n j u g a t e

1.1

r = / P(~z,s)f(Rs)dl(s) S 1
and p = ~ are a l m o s t o b v i o u s ,

flf(rz) IPdl(z) S

for p = ~

in O
F r o m this the c a s e s p=1, c a s e s p=1

1~<~

s

exponent

V z6S. separate

so we r e s t r i c t

treatment.

ourselves

we o b t a i n

< f(/ P ( ~rz , s ) I f ( R s ) I d l ( s ) ) P d l ( z ) S S

= /I/PLf~PdZ S S

The

to 1
_<

f(f P(-~z,s)d~(s))P/q(/ p(~z,s)If(Rs)I Pd~<s)]d~(z) S S

S

= f(/ P(--~z,s)If(Rs)IPd~{s)]d~{zl = /If(Rs) SPd~(s), S S

S

w h e r e v) above has been applied.

For 1~p<~ we define HarmP(D)

QED.

to consist of the functions f6Harm(D)

with Npf

:= lim IlfRl P(1) R+I

=

sup JlfRil < ~. O
For p=~ this coincides with the earlier definition.

N1f ~ Npf ~ N f we see that Harm~(D)

for f6Harm(D)

c HarmP(D)

behaviour of the functions Here ca(S)

From

c Harml(D).

in HarmP(D)

We formulate the b o u n d a r y

in the subsequent propositions.

denotes the class of c o m p l e x - v a l u e d Baire measures on S.

1.3 PROPOSITION:

i) For e6ca(S)

define the function

<8>:<8>(z) = / P(z,s)dS(s) S Then <8>6Harm1(D)

V z6D.

and N 1 < 8 > = I H ] : = t o t a l

R+I we have Convergence

<8>RI+@

variation of 8. F u r t h e r m o r e

in the weak • topology o(ca(S),C(S))

for =

~(c(s)~c(s) ~. ii) Let I ~ < ~. For F£LP(1)

consider the function

f = :f(z) = / P(z,s)F(s)dl(s) S Then f6HarmP(D)

V z6D.

and Npf= IIF I~ _P(1) . F u r t h e r m o r e

gence fR÷F in L P ( 1 ) - n o r m if I ~ < ~

for R+I we have conver-

and in the weak • topology o(L~(1),LI(1))

= a(LI(1)',LI(1)) if p=~. iii) For F6C(S) convergence Proof:

we have f=6CHarm(D).

Furthermore

for R+I we have

fR÷F uniformly on S.

I) <8>6Harm(D)

8 the d e f i n i t i o n

for 86ca(S)

represents

2) In order to prove iii) fR÷F for R+I. For O~R
is obvious since for real-valued

<8> as the real part of a function in HoI(D).

it suffices to show the u n i f o r m c o n v e r g e n c e

and z6S we have

fR(Z)-F(z)

= / P(Rz,s)(F(s>-F(z))dl(s) S

: / P(R,~)(F(s)-F(z)]dl(s) S

- 2~ f P(R'eit)(F(zelt)-F(z))dt' and hence for O<~
into

!tl~6 and d ~ I t I ~

IfR(Z)-F(z) i ~2~_~6 p(R, eit)~(6)dt + 2 ]IFII P(R,e i6) =< ~(~) + 2 IIFII P(R,e i6) ,

where ~ is the modulus of continuity lim sup R+I

Hf R- F H ~ ~(6>

SO that l{fR-F{l+o

of the function F6C(S).

Therefore

for each o<6<~,

for R+I.

3) We next prove i). For f=<e>6Harm(D)

and O
/{f(Rz) {dl(z) <_ /(~ P(Rz,s)d{0] (s))dl(z) S S S

=f(f

P(Rs,z)dl(z))dl% I (s) = ll81l,

s s therefore f6Harm I (D) and Nlf< lle]l. The weak* convergence to be shown means that [HfRdl ÷ fHd8 for R+I for each H6C(S). But this is true since S S for h = we know from iii) that / HfRdl = /(/ H(z)P(Rz,s)d@(s))dl(z) S S S = ]'(f H(z)P(Rs,z)dI(z))dO(s) S S And then

= f hR(S)de(s ) + / Hd8 for R%1. S S

IfHfRdll__< liHll flfRldl_<_ IIH]INIf for O
IfHdSI__< IIHIINIf for each H6C(S). This means S

implies that

IIsIl < N1f, so that we obtain =

Nlf= {{8 I{. 4) In order to prove ii) we obtain

I{fR{{ Lp(l) ~

for O~R
I{F{I Lp(%)

as in the proof of 1 •2 • Thus fEHarmP(D) case 1<__p<~ we use the fact that C(S)

and Npf~ IIFI~ P(%) . Then in the

is dense in LP(I). Thus for HEC(S)

and h=

we have

< II (f-h)Rl I + IIF-HII + IIhR-HII llfR -FII L p ( l ) = LP(I) LP(I) LP(I)

2 IIF-HII ~p(~) + IIhR-HII

for O
lim sup IIfR-FII < 2 IIF-HII R+I LP(%) = LP(%) and h e n c e

fR÷F in LP(1)-norm.

From

= lim

IIFII LP(1) In the case p : ~ the weak~ + fHFdl S

for R+I

HfRdl

LP(I)

= N

P

that

f.

to be shown m e a n s

But this

in L I (1)-norm

= f hRFdl S

S

is true

since

that

/HfRdl S for h = < H l > 6

and hence

+ / HFdl S

for R+I.

then I/S HfRd%l

implies

that

[IFII ~ ~N L (I)

=< IIHIILI(%) [IfRII L ~(l)

IfS NFdll

with

ii) Let with

% IIHII L I (l) N~f

f, so that we o b t a i n

1.4 PROPOSITION: 8 6 ca(S)

i) For each

=< IIHII L 1 (l)N~f

for O
for each H6L I (X). This means

N f= IIF!I ~ . QED. L (l)

f6Harm1(D)

there

exists

a unique

f=<8>.

1
f£HarmP(D)

there

exists

a unique

F6LP(1)

f=.

iii)

For each

Proof:

f6CHarm(D)

i) The m e a s u r e s

IIfR~II = Let

convergence

that hR+H

it follows

llfRI I

R+I

for each H6LI(1).

6Harm I (D) we know

And

this

8 6 ca(S)

exists

fRl6Ca(S)

there

f(Rz)

is a s e q u e n c e

a unique

with

f=.

V O <=R < I .

Nlf< ~

of these m e a s u r e s R(n)+l

with

= ] P(z,s)fR(S)dl(s) S

z6D we take H=P(z,.)

F6C(S)

fulfill

IIf~ll ~I (~)=<

be a w e a k • limit p o i n t

for each H6C(S) 1.1 we have

For fixed

there

and a s u i t a b l e

for R+I.

[HfR(n)dl÷[HdS. S S

Thus Now from

V z6D and 0£R
R=R(n)+I

to o b t a i n

f(z)

ii)

For

IIfRI[L p the

1~q<~

the c o n j u g a t e


(x)=

fR for

p

R+I.

exponent

for O
there

=

Then

of iii)

is s i m i l a r

needed.

The

we o b t a i n

but

(HERGLOTZ):

the n o n n e g a t i v e

since

assertions

V z6D.

L P ( x ) = L q ( I ) ".

a weak • limit

the c o m p a c t n e s s immediate

formula

f=<8>

fEHarm(D)

and

In v i e w

point

as in the p r o o f

are

The

functions

= <8>(z)

we h a v e

exists

f=

simpler

uniqueness

1.5 C O R O L L A R Y ween

= / P(z,s)de(s) S

of i).

The

argument

from

1.3.

defines

of

F6LP(1)

of

proof

is not

QED.

a bijection

the n o n n e g a t i v e

bet-

measures

86 P o s ( S ) . Proof:

For

f6Harm(D) f(O)

so t h a t and

Loomis. dition will

Convergence:

abelian-type As u s u a l

Baire

the

functions

and

the

assertions

is not

be v a l u a b l e

are

variation

= ½(@(t+)+@(t-))

for

follow

from

1.3

f Fd0 S We c a n e x t e n d

= / F(eit)d@(t) -7

@:[-~,~]÷¢

true

Fatou

unless

with

that

abstract

functions

in b i j e c t i v e @:[-~,~]÷~

the c o r r e s p o n d e n c e

its C o n v e r s e

convergence

famous

to a s s u m e

Itl
and

its t a u b e r i a n

for the

for us to w o r k

of b o u n d e d

@(0)=0:

is the

we p r o v e

converse

86ca(S)

Theorem

the p o i n t w i s e

answer

h e r e we h a v e

V.5

measures

for

The

theorem

is s a t i s f i e d :

in S e c t i o n

@(t)

Then

The Fatou

We ask

fR for R+I.

It is c o n v e n i e n t The

for O
: [IfRl [ LI (l)

and N 1 f = f ( O ) .

f6Harml(D).

functions of t h i s

we h a v e

QED.

2. P o i n t w i s e

Let

: SI f(Rs)dl(s)

f6Harm1(D)

1.4.

nonnegative

behaviour

theorem. converse

an e x t r a f>O.

to

tauberian

con-

theorem

theory.

of b o u n d e d

the

@(z) - O(~-)

due

The Loomis

correspondence

with

of the

Besides

variation. with

the

normalization

= @(-z+) - @(-z),

is V F6C(S).

to a u n i q u e

function

@1~÷~

with

the p e r i o d i c i t y

property bounded and

@(t+2~)-@(t)=const variation

@(0)=0.

Equivalent

The

with

V real

the n o r m a l i z a t i o n

above

correspondence

f Fde S

= ] F(elt)dO(t) T-~

In p a r t i c u l a r

2.1

8 is n o n n e g a t i v e

FATOU Let

THEOREM:

Let

f £ Harm1(D)

the

÷

iff

V real

t

and e a c h

real ~.

V real

u
@ is r e a l - v a l u e d

and mono-

above

we

@({e

2~

l({e

~dS, te

corollary.

limit

is = ~de 6

result We

and

t+O.

Let

iu

is,) for

: e - t < u < e + t}) <

exists

<

V

O
:s-t u ~ + t}) t+O

for

f6Harml (D) w i t h

f(Rs)

~ iu < < 8 (ie :~-t u ~+t})

for

l-almost

all

corresponding

l-almost

all

radial

limits

j

=

s6S,

el~6S.

06ca(S). and

the

Thus

we

Then

the

limit

func-

L I (1).

c a n be e x t e n d e d

shall

2.3 L O O M I S 86ca(S)

lim R+I

I 2t

iu

subsequent

2.2 C O R O L L A R Y :

for

_

I

tends ÷ 2~

limits.

86ca(S)

see t h a t

the

The

corresponding

f o r R+I.

this

radial

with

A 2z

-

ding

V F6C(S)

6 Pos(S)

@ ( a + t - ) - @(s-t+) 2t

tion

local

~6~ with

f(Re ie) ÷ A

From

and

8(t)=~(O(t+)+O(t-))

of

reads

= @(v-)-@(u+)

0(e+t) - @(s-t) 2t

have

is a f u n c t i o n

increasing.

G:~÷~.

Then

then

is

e({eit:u
tone

t, w h i c h

come

back

THEOREM: a n d 0:~÷~.

from

to t h i s

point

to n o n - t a n g e n t i a l

in S e c t i o n

Let f 6 H a r m I (D) be n o n n e g a t i v e Let

e6~ with

f(Rel~)÷A

4.

with

for R+I.

correspon-

Then

O(e+t)-e(~-t) 2t The

proofs

f(z)

have

to be b a s e d

. P(z,elt)d@(t)

=

transformation

convenient

to t r a n s f e r

A={s6~:Re S~O}via up the

allows

relation

V z6D.

us to a s s u m e

the p r o b l e m

an a p p r o p r i a t e

1-s h:h(s)=1--~ for

function

It m a p s

from

the

that

unit

~=O.

disk

fractional-linear

But

then

it is

D to the h a l f p l a n e

map.

L e t us w r i t e

this map

corresponds

which

(and is e q u a l

of b o u n d e d

of a n o r m a l i z e d

= -¢(X)

s6A a n d

e e

it it

number

to its

inver -

@(-'rr+)

1-ixs s-ix

-z

We

=

x6~ and

= cRes-

= eRes

In p a r t i c u l a r

lim ~+O

+

--~+6 -tan-2

increasing

for c o r r e s p o n d i n g

Re

1-ixs S--lX

/ Re , 1 - i x s d~(x) s-ix

for O < s < ~

,

Ttl<~,

iff ~ is so a n d

x6~ and

~-~ f P(z,eit)dO(t)

-~+6

d~(x)

°

and -l
is

find

= cRe.11+z + l i m 6+O

Fn ?

of b o u n -

c ~.

we h a v e

= f P(z,eit)d@(t) -~

@:[-~,~]÷~

function

the c o r r e s p o n d e n c e

-9(-~)

and monotone

z=h(s)6D.

+z

c:

[t[
variation

for c o r r e s p o n d i n g

=

@ is r e a ~ v a l u e d

c is >=O. L e t n o w

and

function

¢('~)-9(~-)

In p a r t i c u l a r

V x6~

consists

~:]~÷~ a n d a c o m p l e x

@(t)

f(z)

A÷D

t = e it ~ x = - t a n 2

a normalized

to a p a i r

ded variation

therefore

s6~ m a p s

i~÷S with

h(ix)

Under

on the

transition.

The se).

for t+O.

~ it = / Re e it +z d @(t) -~ e -z

-~ An o b v i o u s

A + 2~

we have

Itl
f(z)

= CS + -~

I+x2-~ S -~-~d~(x) S +x

with ~:~(x)=~(x)-~(-x) normalization

= CS+

I+x2~ S --~---~(X), S +x

for x~O a function of bounded variation

~(x)=½(~(x+)+~(x-))

monotoneincreasing

i

Vx>O and ~(O)=0,

if ~ is so. It follows

with the

and ~ real-valued

and

that f(z)÷A for z+1 is equiva-

lent to 1+x 2 S --~--~ d~(x)÷A o s +x

for s+O

and hence to / - - ~ 2 o s +x

d~(x)÷A

for s+O.

On the other hand we have

(x) _ ~ (x) -+ (-x) x x

_-

@(t)-@(-t) t tan

for corresponding

x>O and O
so that @ (t)-@ (-t) ÷~ 2t We can therefore

for t+O is equivalent reformulate

2.4 FATOU THEOREM:

~(x) x

to

our assertions

L e t ~ :[0,~[÷~

2 ÷ ~A for

as follows.

be of bounded

variation with ~(0)=0

and F: F(s) Then ~(x) ÷ a x

for x+O implies

2.5 LOOMIS THEOREM: with ~(0)=0

2 ~ ~ d~(x) = ~ s2+x 2 that F(s)÷a

for s>O.

for s+O.

Let ~ :[O,~[÷~ be monotone increasing

(and F as above)

Then F(s)÷a

for s%O implies

and bounded that ~(x) ÷ a

•

x

for x+O. Proof of 2.4: This

is a typical

proof of an abelian theorem.

Fix ~>O.

Then for s>O we have

F(S)

- 2_aa~arctan ~~ = ~2 of s +x

2 _

~2 S2+6 2~

(~(~ -a~) + ~ o F(s)

d(~(x)-ax)

al 2sx2

7

+ ~ 6

2

(s2+x2) 2ax + ~

s +x

s 2 i~(6 ) _ a6 I _ 2a~ arctan ~I =< ~2 s2+6

2Sup{i~(X)_a]:O<x<6} x =

+ 2 s ~ s2+~-----2var(~)

'

10

and hence

For

l i m s u p l F ( s ) - a I _<_ 2 Sup{I~---~)-al :O<x<6}. s+O

6+O the

The

assertion

proof

of

follows.

2.5 w i l l

2.6 U N I Q U E N E S S

QED.

be b a s e d

REMARK:

Let

Proof

for L e b e s g u e - a l m o s t

of 2.6:

The

an odd c o n t i n o u s

for O<s
and hence

it -e 7 +z d0(t) -~ e -z

= 0

If

x>O.

V

= ~ H(t~dt O J+t 2

function

for all

s6£.

X6~

of b o u n d e d

variation

dx = 2s ~f ~ H(x) O S +x

H(Ixl) 1+X 2

Under

~:~÷~.

the transition

We h a v e dx = O

A÷D d e s c r i b e d

@: [-~,~] ÷ 9

From QED.

Alternative WeierstraB spanned

proof

theorem.

1.3.i)

ha:he(x) identity

P(z,eit)d0(t)

of 2.6:

This

that

=0

I - e2+x2

shows

with

with

a~s

rr

H (x___Z


and hence

~=0 and hence

based linear

on the

Stone-

subspace

o<~
t h a t A is an a l g e b r a .

of S t o n e - W e i e r s t r a B . 2I 2 e -s

of b o u n -

functions

v O~x~ h6

proof

be the c l o s e d

Now o~

_ f h a ( x ) _ H(X) dx o s2+x 2

for all O < ~ , s < 1

V z6D,

function

0=0 and h e n c e

is an a b - o v o

C([O,~])

a n d by the

h -h 6 = (62-~2)h in v i e w

odd c o n t i n u o u s

we c o n c l u d e

Let AcRe

by the c o n s t a n t s

A=ReC([O,~])

f

and hence

is the c o r r e s p o n d i n g

the

assertion.

Vz6D

-~

ded variation.

The

function.

we o b t a i n

f

where

all

Baire

remark.

for all O<s<1,

= 0

= 7J s ( 1 + x 2 )2+ i x2( 1 - s 2 ) -~ S +x

1-ixs d~(x) s-ix

-~

uniqueness

formula : ~(x)

defines

subsequent

H : [ O , ~ [÷ ~ be a b o u n d e d

H(x)_ ~ a x o s +x then H(x)=O

on the

dx

-

]" H(x------Z d x ) 0

a2+x 2

= 0

Therefore

11

/ h(x) o

H(x____~)dx : 0 s2+x2

F r o m this the a s s e r t i o n Proof x>O.

of 2.5:i)

V h6A = R e C ( [ O , ~ ] )

is obvious.

There

exists

and O<s<1

QED.

an M > 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)<_Mx for all 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 O<_~t(x)<Mx

2sx (s2+x2)2

for t>O 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
measure and

enters

/ F(s)G(s)P(z,s)dI(s) S

the scene:

<(FG)~>

an(D)

t h a t HI(D) I=an(D)

w e e n the HP(D)

implies FG 6A(D)

for 1 < p ! ~,

cHolP(D)

relative

HoI(D)

classes

: 6 H o l ( D ) } .

It w i l l be a m a i n

A new aspect

tiplicative

On the side of

:= {@6ca(S) : <8> 6 H o I ( D ) } ,

CHoI(D) c H o l ~ ( D )

cat~vity:

w i t h the e a r l i e r d e -

v a l u e s we i n t r o d u c e

1.3 and 1.4 c o n t a i n

1
HolP(D) :={f6Hol(D) :Npf< ~} =

CHoI(D) : = H o I ( D ) D C H a r m ( D ) .

the c l a s s of a n a l y t i c m e a s u r e s ,

with

QED.

F o r p = ~ this c o i n c i d e s

We had a l s o d e f i n e d

A(D) c

Hence

we have in fact @ ( x ) = a x

c = l i m ~t(n) (1)=e(1)=a.

the f u n c t i o n

an(D)

Then

8 is m o n o t o n e

all x>O.

Functions

= HOI(D) AHarmP(D)

the b o u n d a r y

for t>O

for L e b e s g u e - a l m o s t

and s i n c e

In p a r t i c u l a r

3. H o l o m o r p h i c

finition.

e(x)=ax

x = 0

A(D)

situation

a n d H~(D)

multiplicative

the u n e x p e c t e d

The obvious

=

(F.and M . R I E S Z ) .

means

is m u l t i p l i -

are a l g e b r a s , relations

fact t h a t F , G 6 A ( D )

that

= CS F ( s ) P ( z , s ) d l ( s ) l < ] G ( s ) P ( z , s ) d l ( s ) l S

bet-

n o t i o n of a m u l -

S

V z6D,

13

w h i c h means that for each z6D the m e a s u r e P(z,-)l for z=O the m e a s u r e

I itself)

(and in p a r t i c u l a r

is m u l t i p l i c a t i v e on A(D). The same is

true for H~(D). 3.1 REMARK:

Proof:

Let 86an(D)

and F6A(D).

Let h=<e> and f=. f(z)h(Rz)

Then F86an(D)

For O~R
and =<8>.

then FhR6A(D)

= / P(z,s)F(s) hR(S)dl(s) S

and

V z6D.

For R+I it follows that

f(z)h(z)

3.2 REMARK:

= / P(z,s)F(s)dS(s) S

For Q6ca(S)

V z6D.

QED.

the subsequent p r o p e r t i e s are equivalent.

i) 8 6an(D) , that is '<8> 6HoI(D) . ii)

~ snd0(s) = 0 (n=I,2 .... ). S iii) / Hd0 = H(O) / do for all p o l y n o m i a l s H ( i n o n e complex variable). S S iv) / Ha0 = h(O) / dO for all H6A(D) with h(O)=(O)=/Hdl. S S S

In this case we have the C a u c h y formula <0>(z)

Proof:

i) ~ i v )

= / s d@(s) S s-z

is 3.1 for z=O.

tains the p o l y n o m i a l s . i i i ) ~ i i )

¥ zED.

iv) ~ i i i )

is obvious

is trivial,

ii)~i)

since A(D)

con-

and the last asser-

tion follow from

<0>Iz) = f PIz,s)d0<s)

= / (~s

S

S

+

S~)d0Cs ) I -sz

oo

= / S__~z 0(S ) + ~ ~n f sndo(s) S n=1 S 3.3 PROPOSITION:

i) A(D)cC(S)

bra of the p d y n o m i a l s . ReC(S)

V z6D.

QED.

is the supnorm closure of the subalge-

Furthermore

ReA(D)cReC(S)

is supnorm dense in

(the D I R I C H L E T property).

ii) Let 1~p<~. Then HP(D)cLP(1)

is the L P ( 1 ) - n o r m closure of A(D)

(and hence of the subalgebra of the polynomials). iii) H~(D)cL~(I)

is the weak* closure of A(D)

(and hence of the

14 subalgebra of the polynomials). exist functions Fn6A(D) L=(1)

sense

Proof:

We have more: For each FEH

(and hence the F

n

w h i c h annihilate

06ca(S)

in the

can be chosen to be polynomials).

i) It is clear that A(D)cC(S)

of 3.2 the m e a s u r e s

(D) there

with !IFnlI < llFIi and F ÷F pointwise = L~(1) n

is supnorm closed. And in view

w h i c h annihilate A(D)

the polynomials.

are the same as those

The d e n s i t y of the subalgebra of the

p o l y n o m i a l s also has a simple direct proof:

For F6A(D)

and f= we

have llfR-Fll+Ofor R+I, and in view of the Taylor series e x p a n s i o n each fR for O~R
is the u n i f o r m limit of polynomials.

Dirichlet property all F£A(D).

let @6ca(S)

annihilate ReA(D)

In order to prove the and hence F and F for

It follows that /sUdS(s)= 0 Vn6$ and hence e=O in view of S theorem.

the W e i e r s t r a S ii)iii)

In view of 3.2 HP(D)

= 0(n=I,2,...) F6HP(D) norm

consists of the F6LP(I) with /snF(s)dl(s) S and is therefore closed in the respective sense. For

and f= we have fR6A(D)

if

I~<~,

for O~R
w h e r e a s IIfRl[ ~ IIFII

sense if p = ~ after 2.1. QED.

and fR+F for R+I in LP(~) -

and fR÷F pointwise in the L~(1) L~(~)

We conclude this section with the formulation of the most fundamental classical

theorems.

We present almost no proofs. All these theorems

will be put into broad and natural context and appear as simplest special cases in the abstract theories of Chapters 3.4

F. and M.RIESZ THEOREM:

II and III.

Each analytic m e a s u r e

is a b s o l u t e l y con-

tinuous with respect to ~, that is an (D) =HI(D)~. In view of the future a b s t r a c t i o n we deduce this t h e o r e m from a certain m o d i f i e d version.

3.5 M O D I F I E D F.and M.RIESZ THEOREM:

A s s u m e that e6ca(S)

A(D),

that is f FdS=O VF£A(D). If @=~+6 with l-continous S lar 6, then e and ~ likewise a n n i h i l a t e A(D).

annihilates

~ and l-singu-

3.6 LEMMA: Let 0 6 an(D) be ~ O. Then there exists an n(n=O,1,2...) such that

1 Zn e 6 an(D) and where Z:Z(s)=s

Sf ~dn Z e + O,

is the identity function.

15 Thus in view

Proof of 3.6: From 3.2 we know that jsnd@(s)=O ~ Vn~1. S of the WeierstraB for a smallest

theorem we have /s-ndQ(s)40 S

for some n>=O and hence

n>=O. Then 3.2 shows that this n~O

fulfills

the asser-

tion. QED° Proof of 3 . 5 ~ 3.4: l-singular

i) Let @£an(D)

~. Then @-@(S)I=(e-@(S)I)+~

3.5 we see that

~ likewise

that a l-singular

But if now the above analytic measure with

3.7 JENSEN

A(D).

ii)

A(D)

~ and

so that from

In particular

analytic measure must annihilate

l-singular

8 must be = O .

annihilates

annihilates

integral=O. that

and @=~+~ with l-continuous

A(D),

i) shows

that is has an

B were 40, then 3.6 would lead to a integral

40. This contradiction

shows

QED.

INEQUALITY:

Let F6A(D)

and f=.

Then V z6D.

log]f(z) I ~ / P(z,s)logIF(s) Idl(s) s Hence the same is true for F6HI(D). 3.8 COROLLARY:

If F6HI(D)

Proof of A(D) ~ H I ( D )

is 4 ° then logIFI6L1(1).

in 3.7: Let F6HI(D).

For O~R
we have

loglf(Rz) I ~ / P(z,s)loglfR(s) Idl(s) S ~ P(z,s)log(IfR(s) l+£)dl(s) S Now IIfR-FII L1(1)÷O

V z6O and e>O.

for R+I so that for a suitable

have fR(n)+F pointwise

and under an Ll(1)-majorant.

for log(IfR(n) I+c) ÷ log(IFl+e).

It follows

follows

DP:DP(o)

= Inf{/IFIPd~:F£A(D) S

Then the same is true

V z6D and e>O,

from Beppo Levi.

then f(z)40 for some z6D. Thus the assertion For the last theorem we introduce

we

that

loglf(z) I ~ f P(z,s)log(IF(s) l+e)dl(s) S and for s%O the result

sequence R(n)+1

Proof of 3.8: If F40

is obvious.

QED.

the functionals with f(O)=1}

V ~ 6 Pos(S),

16

where

1
3.9 SZEG0-KOLMOGOROV-KREIN

THEOREM:

For

1<=p<~ we have

= exp(f ( l o gd~ ~ ) dl) S

DP(o)

V a6Pos(S).

In particular V O<_F£LI(1),

DP(FI) = exp(f(logF)dl) S which

is the Szeg~ theorem.

4. The Function

Classes

HoI#(D)

In the future abstract

and H#(D)

theory the function

class which corresponds

to the class H#(D)

to be defined

important

function classes which correspond

than the

in the present

section will be far more to the HP(D)

finite p~1. As before our main concern will be the transition

for

from D

to S. For G an open subset of ~ we define a function HoI#(G)

iff there exists a sequence

on G, fn ÷ I pointwise

on G

of functions

(and hence uniformly

f:G÷~

to be of class

fn6HOl~(G)

with

on each compact

Ifnl~1 subset

of G),

and f f6Hol~(G) for all n>1. We list some immediate consequences. n = i) HoI~(G) c H o l # ( G ) c H o l ( G ) , and HoI#(G) is an algebra, ii) HoI#(G) con-

tains the class HOI+(G) of the functions f£Hol(G) with Re f >=0. In fact, n we can take fn:=n--~, iii) If U,Vc(~ are open and @:U+V a holomorphic map, then f6Hol#(V)

implies

that fo@6Hol#(U).

Let us now turn to the unit disk situation. F6L(1) F

to be of class H#(D)

n6H ~ (D) with

FnF6H~(D)

iff there exists

[Fnl ~< I, Fn÷1 pointwise

(as usual

for all n~1. Then H~(D)cH#(D),

4.1 PROPOSITION:

For f6Hol#(D)

F(s) := lim fR(s) R+I

a function of functions

in the L(1)-sense)

and H#(D)

the radial

exists

We define

a sequence

is an algebra.

limit

for l-almost

all s6S,

t

and

17 and produces

an

element F6H#(D).

The map f ~ F

thus defined

is a bijec-

tion HOI#(D)÷H#(D). Proof:i)

Let f£Hol#(D),

the definition

and take functions

with gn:=fnf6Hol~(D).

dary functions.

Then

IF

f 6HoI~(D) n

Let Fn,Gn6H~(D)

as required

the respective

in boun-


=

f IFn-I 12dl < 2-2Re / Fnd~. = 2{1-ReFn(O)~÷O. S S Therefore

after transition

to a suitable

subsequence

we can assume

that

F +I pointwise, ii) We choose a Baire set NcS with ~ (N)=O such that in n each point s6S-N I) radial convergence fn (Rs)÷Fn(s) and gn(RS)÷Gn(S) for R+I takes place the Fn6H~(D)

for each n>1,

thus obtained

now the equation

and 2) the representatives

on S-N

fulfills Fn(S)÷1

fn(Rs)f(Rs)=gn(RS)

choose an n> 9 with Fn(S) tO.

for s6S-N and O~R
Then fn(RS)+O

S÷Fn(S)

of

for n÷ ~. Consider For fixed s6S-N

for R sufficiently

close to I.

Thus the limit F(s):=limf(Rs) exists in each point s6S-N. The element R+I F6L(I) thus produced fulfills FnF=G n for all n>__1. Therefore F6H#(D). iii)

In case F=O we have Gn=O and hence gn=O for all n>1.

fn÷1 this implies

that f=O. Therefore

the above map f~F is injective.

iv) Let us now start with a function F6H #(D), as required

in the definition

and take functions

F n 6H~(D) And put fn=, gn =

with Gn:=FnF6H~(D).

=£HoI~(D) • Then

IfnI<1 and fn ÷I on D. Now flgn=fngl

since the difference

is in HOI~(D)

FIGn-FnGI--O.

In view of

and possesses

for all 1 'n>1 =

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<~

[z-sicos e}

(and of course each time is equal to the radial

18

limit obtained Proof: proof

in 4.1).

i) A n i m m e d i a t e

shows

that we can r e s t r i c t

fix a p o i n t

assume

for z÷s on w(s,~)

transition

we h a v e a f u n c t i o n

e a c h O<~<~.

that f(z)÷c

for e a c h O<e<}.

to t r a n s f e r

[1950]

VoI.I

p.186),

fn:fn(Z)=f(~)

Vz6A

in e a c h p o i n t

z>O. T h e r e f o r e

exists.

is e q u i b o u n d e d

from D to the

Vz6¢.

such that f ( z ) + c

the aid of the V i t a l i ii) T h e s e q u e n c e

After

the

for z+O,

and for

theorem

(see

of the f u n c t i o n s

in A and s a t i s f i e s

in v i e w of V i t a l i

Then

We shall

We can of c o u r s e

the p r o b l e m

map h:h(z)=~

f6Hol~(&)

f6Hol~(D).

lim f ( R s ) = : c R+I

for z÷O o n ~ ( ~ ) : = { z 6 A : R e z ~ I z l c o s ~ }

T h i s w i l l be d o n e w i t h

CARATHEODORY

i) and ii) of the a b o v e

to the case

limit

A v i a the f r a c t i o n a l - l i n e a r

we h a v e to p r o v e

compact

ourselves

s=1. T h e n it is c o n v e n i e n t

half p l a n e

of p a r t s

s6S such that the r a d i a l

that f ( z ) ÷ c

prove

adaptation

fn(Z)÷C

for n÷ ~

f ÷c u n i f o r m l y n

s u b s e t of A. Thus for f i x e d O < e < ~ we have a s e q u e n c e

on e a c h of ~n %0

such that

Ifn(Z)-CI=If(~)-c This

implies

that

3.4-3.5

contain

important

Ho!#(D) and H#(D). N o t e

lity 3.7 and 3.8.

w i t h ½!IzI~1

I f ( z ) - c l ~ Sn V z6~(~)

The n e x t r e s u l t s classes

!e n v Z[~(~)

with

Assume

that f 6 H o l + ( D ) .

until

From

QED.

on the r i c h n e s s

d e p e n d u p o n the J e n s e n

of t h e s e r e s u l t s

and 3 . 7 - 3 . 9 w i l l n o t be c o m p l e t e

4.3 LEMMA:

IzI<~ and n~1.

information

that 4 . 5 - 4 . 7

So the p r o o f s

a n d n~1.

of the inequa-

as w e l l as t h o s e of

the end of C h a p t e r

1.5 we h a v e

e6Pos(S)

II.

such that

Re f =<@> and h e n c e f(z)

Then ef6Hol#(D)

Proof:

Then

i) A s s u m e

f 6HoI(D) n

+ / S+Zde(s) sS-Z

V z6D.

iff @ is a b s o l u t e l y c o n t i n u o u s w i t h r e s p e c t

P u t Fn: = M i n ( F , n ) fn:fn(Z)

= ilmf(O)

that @ is l - c o n t i n u o u s

to i.

a n d 8=FI w i t h O < F £ L I ( 1 ) .

and

= i Imf(O)

with

+ S/ ss+z JzFn(S)d~(s)

V z6D and n~1.

f ÷f and Re f = 6 H a r m n n n

(D) w i t h Re f < Re f . n=

19

Thus hn:=exp(fn-f)6Hol~(D) Vn~1. Therefore

with

e f 6 HoI#(D).

lhnI~1 and h n +I and h n ef=exp(fn)6Hol~(D)

ii) Let @=~+8 with l-continuous

~ and l-sin-

gular 8. Then ~ 8>=0. And f=u+v with u:u(z)

= J i m f(O) + / S-~d~(s) S s-z

v:v(z)

s+z.~(s) / ~c{a~

=

V z6D.

S Assume that ef6Hol#(D). functions

fn6HOl~(D)

and let Fn,Gn6H~(D)

Then eV:ef-U6Hol#(D)

be the respective

boundary

for Rfl for l-almost all s6S after 2.2. Thus = IgnJ=IfnleReV<1

JGnJ=JFnJ
and hence

fore Re v = O

and hence B=O. QED.

4.4 COROLLARY:

since

Let F6Re L1(1)

Now Re v(Rs)÷O

= and hence eReV<1

since fn +1 " There-

and V z6D.

and is in fact an invertible element of the algebra

4.5 COROLLARY:

We have H I (D)cH#(D).

Proof: Let F6HI(D) loglFI6L1(1).

functions.

Vn>1 = '

JgnJ=Jfnle Rev implies that

f:f(z) = S/ ~/~ s+z F(s)dl(s) Then ef6Hol#(D), HoI#(D).

le-Ul_<_1. Take

Jfnl<1 and fn +I and gn:=f n eV£Hol~(D)

with

be~O

and f=6Hol1(D).

From 3.8 we know that

Thus for S+Zlo g iF(s) Jdl(s) h:h(z) = ] ~c~

V z6D

S we have eh6Hol#(D) IfJ£jehj.

Therefore

in view of 4.4. Now the Jensen inequality f6Hol#(D)

and hence F6H#(D).

4.6 JENSEN INEQUALITY: Let f£Hol#(D) In case f#O we have logjFl6L1(1) and

3.7 reads

QED.

with boundary function F£H#(D).

logJf(z) j £ ] P(z,s)logjF(s)ldl(s)

v

zeD.

S Proof: Take functions 6HoI~(D)

fn£HOl

(D) with

Vn~1, and let Fn,Gn6H~(D)

JfnJ<1 = and fn +I and gn:=fnf6

be the respective boundary functions.

20 Then Gn=FnF.

We can of course assume

logIGnlfL1(~)

from 3.8 and hence

fn#O and hence gn~O.

loglFl6Ll(~).

Thus loglFnl,

And from 3.7 we have for

z6D l°glgn(Z) i ~ / P(z,s)logIGn(S) Idl(s) ~ f P(z,s)loglF(s) Idl(s) S S in view of

IFnl~1,

from which the result

4 • 7 PROPOSITION: algebra

HoI#(D)

of the functions

s+z f:f(z) = / ~L~ F(s)dl(s) S and c6~ with

Proof:

for n÷~. QED.

(HoI#(D)) x := the set of invertible

consists

IcI=1. Thus

which corresponds

follows

V z6D

We have to prove

The boundary

function H£(H#(D)) ×

!HI=e F.

that each h6(Hol#(D)) x is of the above

which has already been exhibited 6L1(I) and

a constant

as defined

of modulus

HOI+(D).

consist

Thus

lhe-f]=l

= Re f ( z ) V

z6D,

and hence h=ce f with c

IcI=1. QED.

To end the section H+(D)cH#(D)

above.

form

in 4.4. From 4.6 we obtain F:=logIH I

log]h(z) ] = / P(z,s)loglH(s ) ]dl(s)= / P(z,s)F(s)dl(s) S S with f6Hol(D)

of the

with F£ReLI(I),

lhI=exp().

to h satisfies

elements

h=ce f, where

let us consider

of the

boundary

the subclass functions

We ask for a direct characterization

HOI+(D)cHoI#(D).

of the functions of H+(D).

Let

in

Of course H+(D)c

c{F6H#(D):ReF_>_O}, but here ~ holds true. In fact, the function f:f(z)= I-Z + (D) has ReF=O. = ~ z z Vz6D is in HoI+(D) and its boundary function F=I-~-~6H So-F6H#(D) -f6Hol#(D)

has Re(-F)>O which

To explain exists

as well,

is ~HoI+(D)

the phenomenon

a measure

~£Pos(S)

recall

real part ReF suffers

to the function

from 4.3 that to f6Hol+(D)

there

such that Re f =<8> and hence

f(z) = iImf(O) From 2.2 we have R e F = ~

but it corresponds

so that -F~H+(D).

+ / S+ZdO(s) S s-z

for the boundary from an essential

V z£D.

function F£H+(D).

Thus the

lack of information

and in

particular

does

So the ~ a b o v e

not determine becomes

4.8 P R O P O S I T I O N :

quite

the

entire

clear.

F o r FEL(~.)

F up to an a d d i t i v e

constant.

B u t w e c a n p r o v e ~a p o s i t i v e

with

ReF>O

the

subsequent

result.

properties

are

equivalent. i) F 6 H + (D) . ii)

F~6H~(D)

for all

I ~6H

iii)

Proof:

(D)

i)~ii)

for

not

the c o n s t a n t

that

f has

some

and

= ss+F + s _ 16H ~ (D) w i t h

s6A:=

-I.

and

are o b v i o u s , hence

Re s > O .

iii)~i)

g:=6HoI~(D)

~-~c~-

Therefore

the b o u n d a r y

halfplane

s6A.

ii)~iii)

IGI
the o p e n

f:=~1--~6Hol+(D),

function

F. T h u s

We h a v e with

Igl<1,

and

s-f

and

F6H+(D).

s-F=

G : = s+F

from g=s-~

we

g is see

QED.

Notes

The

concrete

functional DUREN

theory

analysis

[1970].

In H O F F M A N

(the D i r i c h l e t treatises

additional

The

[1970] KONIG

here

[1953]

is p r e s e n t e d

books

also

the

initial

is w i t h i n

Notes

we

in the

of H O F F M A N

step

[1956]

restrict

spirit

[1962a]

of

and

of a b s t r a c t i o n

consideration.

and P R I W A L O W

In t h e s e

theorem

[1953] simpler

p.6.

The

than

the

function For

the

is in L O O M I S [1957].

former class

unit

[1956]

different,

can be t r a n s f e r r e d in the

2.3

and GEHRING

D in P R I W A L O W is q u i t e

ferred

disk

The

are m o r e ourselves

older

in the to

spiri

some

remarks.

[1970a].

named

[1962a]

situation)

analysis.

Loomis

is m u c h

algebra

of N E V A N L I N N A

of c l a s s i c a l

and KERR

in the u n i t

in the b e a u t i f u l

to the

ones. HOI#(G)

disk

[1943].

Our

proof

The

proof

for o p e n

it c o i n c i d e s

a n d N + in D U R E N

and

in t h e

abstract

f o r m of H+(D).

We a l s o

form

of 4.2 Gc~

with

[1970].

of H#(D)

theory.

refer

is f r o m K O N I G

Also

to A L L E N [1960]

and

is f r o m D U R E N

is i n t r o d u c e d

in

the

Smirnov

class

But

the d e f i n i t i o n

it is the o n e w h i c h

Hol+(D)

will

be t r a n s -

Chapter

Function

Algebras:

In the p r e s e n t subalgebra consider

chapter

AcB(X,Z)

the

the

:= {@6ca(X,Z) :~fd@

= 0 Vf6A},

c ca(X,Z)

spectrum

of the m e a s u r e s

Z(A)

functionals

:= o ( B ( X , Z ) , c a ( X , Z ) ) ,

~(f)

= Sfde

Yf£A

supnorm

even

be p r o b a b i l i t y

M(~)

Then

the

The

decisive

soon

and

a complex then

to

of the M(A,~)

is d e v o t e d F.

theorem The

of Z(A)

for the

to two

with

F.

topolo-

are of

~6E(A)

II~II=I, b u t

clear

that

each

in v i e w

Thus

the

= Sfd0

the

form

converse

~6E(A)

of ~ ( 1 ) = I

c a n be remust

then

set

Vf£A}

will

consists

be

~ #.

of all

on A.

fundamental theorem,

a subsequent

and M.Riesz

and of ca(X,Z)

direction

in the w e a k

functionals

for all ~6Z(A)

and M . R i e s z

Furthermore multiplica-

in t h e

which

0 6 Prob(X,Z).

A.

nonzero

c a n be r e p r e s e n t e d

of n o r m

become

measures

annihilate

continuous

The

are m u l t i p l i c a t i v e

inequality.

decompositions

are

:= { e 6 P r o b ( X , Z ) : ~ ( f )

abstract

Kolmogorov-Krein Jensen

and hence

It w i l l

M(A)

which

chapter

~6Z(A) : T h e

(X,Z)

It is n a t u r a l

of t h o s e

is w h i c h

0£ca(X,~).

by positive measures

which

that

continuous

= M(A,~)

union

86Prob(X,E)

~:A+~

for s o m e

n o t be true.

which

of A to c o n s i s t

linear

need

contains

Situation

space

constants.

tive

presented

fix a m e a s u r a b l e the

gy ~

course

Bounded-Measurable

annihilator

subspace

define

we

which

A± the

The

II

into which

theorems, and the

universal

theorem

leads

a n d A ±. T h e the abstract

both

for a f i x e d

abstract version

Szeg~of the

to the G l e a s o n - P a r t

latter

of t h e s e

theory

had

is

to be de-

veloped.

I. S z e g ~

Functional

In the p r e s e n t

and Fundamental

section

1<__p<~ it is n a t u r a l on the

subspace

we

Lemma

fix ~ 6 Z ( A ) .

to ask w h e t h e r

A m o d ~ c LP(o)

the

For

a measure

functional

and is L P ( o ) - n o r m

~:A÷~

a6Pos(X,E)

and

is w e l l - d e f i n e d

continuous

there,

that

23

is w h e t h e r := Sup{I~(f) l :f6A w i t h IIfll

lt p<~) Instead Szeg~

of this n o r m w h i c h

LP
< I}

is < ~ or = ~?

=

can be = ~ it is a d v i s a b l e

to i n t r o d u c e

the

functional

DP:DP(~)

=

Inf{~l flPdo

T h e n 0 ~ DP(~) ~ a(X).

: f6A w i t h ~ ( f ) = 1 }

One verifies

Vo6Pos(X,Z).

that

I [I~I[Lp(~

=

(DP(o)

P

V c6Pos(X,Z)

and 1~p<~,

) so that the L P ( ~ ) - n o r m DP(o)

= a(X)

t h a t ~(X)

in q u e s t i o n

also d e s e r v e s

special

is = ~ iff DP(c) attention.

= O. The

l i m i t case

F o r this we can a s s u m e

= I.

1.1 REMARK:

Proof:

Let oEProb(X,Z)

i) A s s u m e

t h a t o£M(~).

IlflPdo ~(~Ifld~) p ~

that DP(~)=I.

~11+tflPdo

~

or

1~p<~.

For fEA w i t h ~(f)=l

I

then

I and h e n c e

DP(o)=I.

F o r t>O then

I (1+2tRef+t21fl2)P-1

ll+tflP+1

2 p R e f + terms

= 1 iff oEM(~).

~ O. F o r t+O w e o b t a i n

I (11+tfl2)P-1 = ~

~

L e t us fix fEA w i t h ~ ( f ) = O .

f~(I1+tflP-1)do

~(ll+tflP-1)

T h e n DP(o)

IIfdal p = I, so that DP(o)

ii) A s s u m e

I

and

= ~

11+tf P+I

in t,t2,...

=

> pRef

.

11+tflP+1 And the convergence

II+zlP-ll

I

= Ip

~(I 1+tf[P-1)

Thus shows

it f o l l o w s

is m a j o r i z e d

by a c o n s t a n t

11~Zltp-ldt] ~ 1

p(l+l~l)P-llzl

vze¢ ,

=< p(1+l fl)P-11fl

VO
t h a t f R e f do ~ O. T h e n c o n s i d e r a t i o n

that ffdo = O for fEA w i t h ~(f)=O.

= 5(~(f)-f)d~

since

= 0 for all f6A,

Therefore

so t h a t ~6M(~).

QED.

of cf for c6¢

~(f)

- ffdo =

24

The duce

next

the

lemma

case

1.2 LEMMA: from

the

Proof:

hand

for

For

right

Fix

lim sup DS(o) s÷p

will

sometimes

be u s e f u l

in t h a t

in

fixed

oEPos(X,E)

the

function

I~<~.

f6A with

~(f)=1

For

~ ~iflPdo.

Therefore

obtain

t h e n DS(o)

and O~Q6Lq(c)

llflSd~ a n d

~

~ DP(o).

On

hence

the o t h e r

1-~

P D p ( a ) <= (DS(o))s(o(X)) < lim inf DS(o). s+p o£Pos(X,Z)

Vf6A,

I_P s,

QED.

=

Let

is c o n t i n u o u s

from H~ider P

DP(a)

p ~-> DP(o)

lim sup DS(~) s÷p

#iflPdo < (#ifl~do)S(o(x)) s

1.3 LEMMA:

to re-

1<=p<~.

1<__p<s<~ w e

and h e n c e

it p e r m i t s

1<=p<~ to 1
1
and

then

#PQdo = IIP11Lp fIQILq (~) iff P P , Q q Proof:

6 L I (~)

(o)

are

linearly

dependent.

i) We q u o t e

a simple

calculus

up 1 vq uv ~ t p - - + - - - p tq q with

equality

b:=IIQllLq(o ) be

iff

tPu p = t - q v q.

>O.

Choose

PP I Qq PQ ~ t p - - + - - - p tq q Thus

SPQda

= ab iff

Assume such

that

that

t>O w i t h

LEMMA:

DP(o)>O.

Then

We

can

u , v ~ 0 we h a v e

assume

that

t P a p = t-qb q. T h e n

SPQd~

we

QED.

Let

o£Pos(X,Z) exist

a:=~PIILp(o ) and integrate

ap I bq ~ tp -- + - ab. P tq q

integrated

i).

there

For

VO
to o b t a i n

the i n e q u a l i t y

is i f f tPP p = t - q Q q a f t e r

1.4 F U N D A M E N T A L

ii)

fact:

and

was

1
functions

an e q u a l i t y ,

with

P6LP(o)

that

! + ! = I. p q and Q6Lq(~)

25

i)

P is in the L P ( ~ ) - n o r m

ii)

~(f)

iii)

IPI p = PQ =

Proof: VuEA}

= II~ULp(d)~fQdd

I) On the

c Lq(d)

closure

of A m o d ~,

for all f6A,

IQI q =: F6L1(o)

and F ~ £ M ( ~ ) .

linear

T:=

we c o n s i d e r

subspace the l i n e a r

{h6Lq(d):fuhdd

functional

= ~(u)~hdd

9:8(h)

= ~hdo Vh6T.

Then

l@(h) I =

I~uhdol

~ IIUIILp(d)llhIILq(o )

V h6T and u6A with ~(u)=1,

I

II011 ~ c := (DP(a))P

SO t h a t

LP(d) e(h)

=

(Lq(~)) " we o b t a i n

= C[hPdd

for all hET.

of A m o d o. O t h e r w i s e with

~Pfdd

=

(ll~llLp(o)

a function 2) T h i s

there were

* O. B u t t h e n

)-I

.

From Hahn-Banach

P6LP(~)

function

an fELq(o)

with

and

~ I P l P d ~ £ I and

P is in the L P ( d ) - c l o s u r e with

~ufdo = O V u E A and

fET a n d O = [fdd = e(f)

= c~fPdo which

is a

contradiction. l~(u) I ~ ~n uILp(o)

3) We k n o w that Lq(o) ~(u)

=

(LP(d)) " we o b t a i n

= ~I ~uQd~ VuEA.

a function

Vu6A.

Thus

Q6Lq(~)

4) We p u t F : = I Q I q 6 L I ( g )

In v i e w of 1.1 a n d ~Fdo ~ I it s u f f i c e s

from H a h n - B a n a c h

with

and

~IQlqdo ~ I and

and c l a i m that Fd6M(~)

to show t h a t D q ( F O ) ~ I .

B u t we

have 1 I = ~(uv)

and h e n c e 5) F r o m Q6T. F r o m

I ~ flvlqFdo 3) we h a v e

Vv6A with ~(v)=1, ~uQdd = C~(u)

I) t h e r e f o r e

so that all t h e s e

course

1.5 T H E O R E M : exists

an m6M(~)

Let O6Pos(X,Z which

is <
In p a r t i c u l a r

= C~PQdo.

< IIPllLp
1.3 that

t h e n m u s t be =I. Thus

V u,v6A with ~(u)=~(v)=1

so that in fact D q ( F O ) ~ I .

=

are e q u a l i t i e s .

= I, and f r o m

Vu6A.

C = ~Qd~ = @(Q)

I = fPQd~ =< flPQld~

= flelqdq

I

= ~I ~ u v Q d o =< ~ ( ~ l u l P d c ) P ( f l v l q F d d ) q

We o b t a i n PQ =

~Qd~ = C. Thus

Thus

=<

I,

IPQI

and f l P I P d d

=

Iel q = clPl p for some real c w h i c h

P I p = PQ =

of

[QI q = F. QED.

with DP(d)>O

for some

1~<~.

Then there

26

Proof:

Combine

1.4 w i t h

1.6 C O R O L L A R Y : exists

an m6M(~)

Proof:

DI(IsI) The the

next

F.

THEOREM:

Proof:

flfldlSl

9.

with

~(f)

= ffd8

In p a r t i c u l a r

_>_ IffdSl

and hence

Assume

Zn v i e w

DP(o+[)

the

>

Vf6A.

Therefore

2. M e a s u r e

of

t h a t ~,T

1.2 we

< DP(~)

implies

lemma will

in S e c t i o n

6 Pos(X,Z) . Then

can

and can

be the

assume

therefore

function

that

such

t h a t T is s i n g u l a r

= DP(o)

1
assume

. We h a v e

that

in 1.4.

Q = O modulo

V1
to p r o v e

cP: = D P ( a + T )

Then

T. F o r

IQlq(o+~)

> O. 6 M(~0)

f6A w i t h ~0(f)=1

that

Cp < DP(~).

Theory:

be the r o o t of

3.

are

DP(o+~)

obtained

to T. T h u s

= ffQd(~+~)

= ~fQdc <

I

I

(~If]Pd~)P(~IQlqd~) q =

(~IflPd~) p

QED.

Prebands

PROPOSITION:

properties i)

there

M(~)~.

I

C = C~(f)

2.1

Then

I V f6A w i t h <0(f) = I a n d h e n c e

fundamental

theorem

are << C+T

is s i n g u l a r

1.4.ii)

of

and M . R i e s z

which

L e t Q £ Lq(o+T)

thus

is <<

consequence

to all m£M(~0)

that

8 £ ca(X,Z)

which

We h a v e

QED.

> 0. QED.

abstract

1.7

Let

1.2.

For

and B a n d s

a nonvoid

subset

K c ca(X,Z)

the

subsequent

are e q u i v a l e n t .

For each

countable

subset

T c K there

exists

an m 6 K

such that

• < < m VTET. ii)

Each

absolutely

In t h i s in ii)

8 6 ca(X,E)

continuous

situation

is u n i q u e .

admits

for e a c h

Furthermore

t h a t ll@ml[ is m a x i m a l ,

a decomposition

to s o m e m £ K

8 6 ca(X,Z) for e a c h

and we h a v e

8 = a+8 w i t h

a n d B £ ca(X,Z)

singular

the d e c o m p o s i t i o n

8 6 ca(X,E)

~ = 8 m and

there

e 6 ca(X,Z)

to all m6K.

0 = a+8

exist

m6K

B = @m for all t h e s e

such m6K.

27

Proof:

We

countable

0 :=

Then

start with

subset.

ii)

I Ira(l) I 1=1 21 l+tlm(1)ll

L e t T = {m(1) : i = 1 , 2 , . . . }

6 POS(X,~)

0` << m for s o m e m6K.

0`,6 ~ O. N o w

~ i).

c K be a

Decompose

Since

6 is s i n g u l a r

0 < 6 < % it f o l l o w s

that

into

e =

6 is s i n g u l a r

to e a c h m(1)

ii).

to m it f o l l o w s

and hence

6 = O. T h e r e f o r e

after

o,+8

singular

that

to

~. S i n c e

e=0`<<m so t h a t m ( 1 ) < < m

(i=1,2,...).

For exist m6K}

the m6K

rest such

of t h e p r o o f w e

~ IIell a n d t a k e m ( 1 ) 6 K

m£K with

m(1)<<m

VI~I.

0 =

we h a v e second

assume

t h a t IIemlI is m a x i m a l .

am(1) =

to m.

le£

8 6 ca(X,E)

there

c := sup{II@mII: And

let

from

era(l) + em(1)

em =

(am-am(1))

is s i n g u l a r

I) F o r e a c h

see t h i s

(1=1,2 .... ) w i t h l i e m ( 1 ) If+ c for I÷~.

Then

8m +

i).

To

+ @~

It

, where

follows

the

that

first

term

is

<<m a n d the

ll@m(1)[I = N@m-@m(1)I]

+ II@~H

and hence

llemllTherefore

= lie~/l)

Ilem(1)[I

llemH

~ c so t h a t

t h a t ll@mlI is m a x i m a l .

Then

d6K

~,m

and take

T6K w i t h

e =

em +

0m =

ll-

Ile~ll = Ilem-em(1)

llemlI = c. 2)

Let

@m is s i n g u l a r << ~. W e

e

+

e~

e: = (oT-e m) + e~ ,

e 6 ca(X,Z)

to all

obtain

as in

and hence

and

8 T : @m or

is a l s o

2) w e h a v e

sition

a

~

= em

Idl-null

in p a r t i c u l a r

8 = 0`+6 as c l a i m e d

decomposition.

set

~6K.

II e,~il = It e~-%lI

such

see t h i s

fix

I)

+ It o.~ il • •

. Thus

@~ l i v e s

so t h a t

e~ is s i n g u l a r

proved

in ii).

for e a c h

What

on s o m e

0 £ ca(X,Z)

remains

set

3) W i t h the

to all m6K}.

I)

decompo-

is the u n i q u e n e s s

some m6K},

K ^ := { 6 6 c a ( X , Z ) :6 << s i n g u l a r

ITl-null

to a.

Consider

K v := { ~ E c a ( X , Z ) :0` <<

and m 6 K

To

,

tl e~ I1 - It emil = tl emil - II e~ II = II e~-emll

which

ll ~ o.

of this

28

It is o b v i o u s once

that

f r o m the direct

definitions.

subspace

Therefore

sum decomposition

A nonvoid called the

t h a t K ^ is a l i n e a r

K ~ is a l i n e a r

subset

a preband.

linear

direct

c ca(X,Z).

as well.

K ~ N K ^ = {0}

ca(X,~)

ca(X,Z)

= K ~ + K ^ means

with

the

be c a l l e d

the p r e b e n d

ii) o<<e

e~

at

in fact

the

ca(X,Z)

i) w i l l

be

= K ~ • K ^ into

singular

decomposition

with

and to all m6K}

respect

to K a n d w i l l

be

@6ca(X,Z).

i) T h e

simplest

e6ca(X,Z)

a fixed

Then

prebends

are K = { m }

is the u s u a l

there

~£ca(X,Z)

for m 6 K we

for

Lebesgue

fixed m6ca(X,Z).

decomposition

exist

m6K

and

claim:

such

that

a prebend

K c ca(X,E)

llemlI is m a x i m a l c<<m

V~6K.

For

such

~ o < < m V~6K. such

an m 6 K

that

In p a r -

it is

that

llSmll is m a x i m a l

Proof

~: F o r

and

~6K we h a v e

as in the p r o o f

of

As

llaTlI - l [ e m l

above we

such

shows

.

Consider

Vo6K.

ticular clear

for

e = 8 K + e~ for

= em +

property

:= { a E c a ( X , Z ) :~ << s o m e m 6 K }

e = eK + ~

2.2 E X A M P L E S : Here

above

sum d e c o m p o s i t i o n

K ^ := { 5 £ c a ( X , Z ) :B

will

i)

is i m m e d i a t e

subspaces

K~

written

And

= K ~ @ K ^. QED.

K c ca(X,~)

The

subspace

that

see

ITI (X-T)

2.1.

@K = em

o<<m

Proof

=:

and h e n c e Fix

o6K

We t a k e

a B£E w i t h

it s u f f i c e s

since

Iml (B)= O,

iii)

ll~mlI - II~oI] = llam-~al I ~ 0

and t a k e

T6K w i t h

a T = am"

o , m <<

Now

T.

choose

T6E

= 0 and

Im[ (B)= O a n d h a v e to p r o v e

and h e n c e

IT[ (X-T)= O s h o w s

ve6ca(x,z).

= II~-~ml I and h e n c e

a T (B) = am(B ) = ~(BDT)

which

' @K = 8~

that

In t h e a l g e b r a

e a c h ~EZ(A)

the

for m l £ M ( ~ )

(1=1,2,...)

that

to p r o v e

iT[ (B)= O. N o w

IT[ (BDT)= O s i n c e

in f a c t

that

}oI (B)= O,

I~[ (BNT)

T<<e.

This

=

for

l~ml (B) = O

combined

with

IT[ (B)= O. QED.

situation

set M(~)

VB6E.

= M(A,~)

considered

in t h e p r e s e n t

c Prob(X,E)

we have m:=

is a p r e b e n d .

~ 2-1m16M(~) 1=I

and ml<<m

chapter

for

In fact, VI~I.

29

A

linear

subspace

B c

ca(X,Z)

is

called

a band

iff

it

satisfies

the

conditions i)

if

ii)

if O~nEB

Under

the

ii')

In

that

and

~<
is

of

then

i)

implication

ll~-On[l =

~(X)

ii')

ii)

is

2.3 c

in

But

case

after

as

K ~ need ii)

Let

ca(X,Z)

÷ O.

For

ii)

I

t%1 1.11%11

2-n

~

i)

not B c

For

6

II~-~nll÷O w e

For

above.

= B ~

iii)

see

a nonvoid

Then

even

be

a

ca(X,Z) B^

K c

K^

be

. This

is

ca(X,Z)

the

It

is

more

K c K ~ c K ^^,

and

we

have

the

K must

K ^^ i s

in

2.4 for

hand

ca(X,E)

We

be

fact

ii)

K ^^

Kv =

are

are

K ^ with

ii')

On<<e

O~On+O

observe

Vn~1

that

K A^

we

the

in

smallest

The

F.Riesz K ^^

implies

that

an6B

~<<e

and

define

K v,

singular

2)

hence

K^ c

band

of

B ~

band

bands

are

in

iff

K

For

nonvoid

2.1.

which

and

t o K.

B v = B.

is

For

For

K.

smallest Further-

ca(X,Z)

a preband to

K

obtain

{m} v =

in-

combine K v = K ^^.

K c

B A^ =

K.

the

On

= K ^ ~ K v so

nonvoid

K ^^ c

contains

B =

K c

= K ^ @ K ^^

3)

the

by

a preband.

{0}

B^ ~

Thus

theorem.

fact

generated

ca(X,Z)

K^ m

and

decomposition

K v n K^^=

that

view

a preband

band

as w e l l .

and

K c

is

a band

the

obvious.

simplest

B

K y = K ^^

K ^~

the

band

is

called

implies

have

K c ca(X,Z) called

Then

obvious

KVc

a preband

a band

EXAMPLE:

fixed

and

Kv c

= Kv @

other

that B c

i)

K c

~

since

subspace.

nonvoid

K.

ca(X,Z)

to

,

from

a band.

contains

clusions

B

a band

linear

which

I)

obvious

subset

is

band

Proof:

equivalent

i).

REMARK:

ca(X,Z)

is

that

n=l

~6B

ii)

~

eo

that

~6B.

condition

- 0n(X)

implies

e:=

Therefore

~6B,

norm-closed.

the

(n=I,2,...)

then

an+q

assumption

B

fact,

66B

ca(X,Z)

Bv =

B.

and

Thus

QED.

{66ca(X,~)

:@<<m}

m6ca(X,E).

conclude

the

Gleason-Part

let



the

: ca(X,E)

section

with

decompositions ÷

K v denote

some in

the

results

Section projection

which 4.

For 8 ~

will

be

a preband 0K

in

the

needed

for

K c

ca(X,Z)

direct

sum

30

decomposition

2.5

ca(X,E)

REMARK:

Let

c

ca(X,Z)

. Then

be

of

is

a band

and

<ENF>

=

<E>

ii)

E+F

is

a band

and

<E+F>

=

<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

<(E ^ A F^)^> = 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

<(EDF)^>@.

(E ^ N F ^ ) ^ =

equivalent,

<E>9

that

<E^>9

<(EQF)^>

follows have

+

6 ENF +

.

have

<E>0

<EDF>8

<E^>

obvious

thus

=

Then =

+



F ^.

<<

ca(X,Z)

with

E^ +

course

bands.

EAF

Proof:

=

E,F

K^

i)

<E>e

=

= Kv ~

we

obtain

e

=

QED.

c

ca(X,Z)

B ^=

s B~ ~I 6 -

with And

Ve6ca(X,Z).

BsQB t we

=

have

31

Now

B s c B E for s * t

we obtain

from

from

2.5

Id =

(

2.6.

For pairwise

+ )

n

(

+ )

Bs(1) > + i~1

that n

n

11oli = I1<~ B~(l~>Oli + 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

<

summable

is 6 B. A n d

is t r u e

operators

s(1),...,s(n)6I

n

= < i=I f]

It f o l l o w s

...

different

Theorem

situation Then

we have

it is o b v i o u s is a p r e b a n d

to be c o n s i d e r e d

that

the p r e b a n d for

as well,

any b a n d

provided

in the p r e s e n t

M(~)

and

the b a n d

B c ca(X,Z)

of c o u r s e

the

that

B N M (~) *~.

3.1 A B S T R A C T B D M(~)

. ~. T h e n

The most

and M.RIESZ

the b a n d ±

F.

THEOREM:

@6BDA ± implies

prominent

3.2 A B S T R A C T Thus A±÷A

F.

special

and M . R I E S Z

projection

case

that

Let

B c ca(X,Z)

@BNM(q0)

be a b a n d w i t h

6 A ±.

is B = c a ( X , Z ) .

THEOREM:

86A I implies

<M(~0)> = < 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<<@

we h a v e

measures

Assume

that

with

= m(f)~d@

Vf6A},

for ~. T h u s A ± c o n s i s t s

let us f o r m u l a t e

another

06an(~)

with

such that o < < m Vo6M(~)

em 6 an(~)

the p a r t i c u l a r

:= { 0 6 c a ( X , Z ) : m f d 0

~d~ m = ~de

special

of the 06an(~)

case of the theorem.

fd0 # O. T h e n t h e r e e x i s t

with

~<<@.

F o r such an m6M(~)

(and @m is in fact i n d e p e n d e n t

P r o o f of 3.1 ~ 3.3: The b a n d B := {0} v = {o6ca(X,Z) :o<<8} BnM(~)

• ~ in v i e w of 1.6. T h u s BNM(~)

can be applied. claimed, Thus

from

m).

It f o l l o w s

and t h a t

@m -

0BDM(~)

fulfills

upon which

that there exist measures

for t h e s e m we have

from 3.1 we o b t a i n

is a p r e b a n d

= @m" N o w

(~de)m 6 A ± w h i c h m e a n s

2.2.ii)

m 6 BNM(~)

as

9 - ( ~ d S ) m 6 B D A ±.

that

@m 6 an(~)

and Sd6 m = ~de. QED. The p r o o f

of 3.1 w i l l be b a s e d on 1.7 v i a the s u b s e q u e n t

technical

lemma. 3.4 LEMMA: D2(m+to) ~(Un)=1

Vt>O.

Let m6M(~)

and g , m 6 P o s ( X , Z )

Then there exists

such that u n +I

in L2(m+~)

a sequence

Then

I) S l u t l 2 d ( m + t o )

u t + O in L2(T)

SI utl 2d (m+to) Slut-ll2d~ and h e n c e

ut÷1

flutl2dm ~

Slut-ll2d

<

of f u n c t i o n s

Un6A with

I = ~(ut) + t2

implies

= J u t d m and Vt>O.

that

Slutl2dm < t 2.

Thus

2) W e have < D2(m+to)

I + to(X)

in L2(m)

(flutldm) 2 ~

<=

D2(m+tO)

~ D2(m+t~)

for t+O.

and D 2 ( m + t O + m )

and u n+ O in L2(m).

P r o o f of 3.4: We c h o o s e u t 6 A w i t h

SiutI2d(m+to+m) <

with o<<m

+ t 2 __< I + to(X)

+ t 2 - 2Refutdm

for t+O. I we h a v e

+ t 2,

+ I = to(X)

3) F r o m the a b o v e

flutl2dO £ o(x) +

2ReSutdo +

= 2/

+ t 2,

and t and h e n c e

X -ReSutdo > + t.

=

33

Thus

ut+l

~utdo

in L2(~)

+ o(X)

t(n)+O

for t + O

for t+O.

such

that Sut(n)da

Slutl2d(m+~)

is b o u n d e d

Compactness

there

subsequence

we h a v e

particular 2) a n d

therefore

Proof from

3.1:

the

of m,8

Therefore Hence

Let

from

after

in L2(I~I)

9 6 B N A ±.

it f o l l o w s

We c o n c l u d e

Since

sequential

selection

Hence

It f o l l o w s

of a In

P mod m = I after

that

Sut(n)dO

~ 6 B N M(~)

+ Sdo

exists

= SUUnd~

section

and hence

a sequence Now

of

is s i n g u l a r

functions

in v i e w

the

we h a v e

with

= D2(m+tI~I)

+ SUUnd~

with

=: ~+B

~ << some

~ << m + t i e I + I ~ I . T h e n

D2(m+tleI+lSIl

in L 2 ( I ~ [ ) .

is a p r e b e n d

+ 9~NM(~)

with

that 0 = Suds Vu£A

the

that

of w e a k

after

B N M(~)

9 = eBNM(~)

t>O a n d ~6M(~)

3.4 t h e r e

show

that

exist

for all h 6 L 2 ( m + o ) .

of o << m.

0 = SUUnd@

F o r n +~

that

proved

there

QED.

1.7 we o b t a i n

and Un+O

such

÷ SPhd(m+~)

in v i e w

6 B. T h u s

Then

in v i e w

for all h £ L2(m).

now

if w e h a v e

Our estimations

÷ SPhdm

decomposition

m 6 B A M(~). in v i e w

Let

result

is false.

Therefore

P £ L2(m+a)

Sut(n)hd(m+a)

P=I

the

this

÷ I • o(x).

a contradiction.

of

2.1

that

for O
exists

Sut(n)hdm

and h e n c e

and hence

Assume

to

~6B 181.

for all t>O.

u n 6 A with

Un÷1

of 9 6 A ± we h a v e

Vu6A

and h e n c e

discussion

.

@BAM(~)

= ~ 6 A ±. QED.

of i m p o r t a n t

special

situations.

3.5 C O R O L L A R Y : equivalent,

i)

c a s e m is c a l l e d plies

that

Proof: so t h a t

For

a f i x e d m6M(M)

@6A ± i m p l i e s dominant).

that

the

0 m £ A ±.

subsequent ii)

In p a r t i c u l a r :

properties

~ << m

are

Vo6M(~)

if M ( ~ ) = { m }

then

(in w h i c h @ 6 A ± im-

9 m 6 A I.

ii) ~ i) We h a v e

the

result

o - m 6 A ± and h e n c e

M ( ~ ) V = { m } v and h e n c e

follows

from

O m - m6Al"

3.2.

i) ~ ii)

In p a r t i c u l a r

0M(~)

For

= 9m

o£M(~)

V@6ca(X,E),

we h a v e

Sdd m = 1 so t h a t ~ = Om << m.

QED.

3.6 R E T U R N c B(S,Baire) theory

to the U N I T defined

is a f f e c t e d

i)

In v i e w

A(D) ± c ca(S) ~£Z(A(D))

DISK:

in S e c t i o n

in the

We

1.3.

subsequent

of the D i r i c h l e t

contains

no n o n z e r o

is r e p r e s e n t e d

consider The

the

connection

simple

property

A(D)

with

the

c C(S)

c

abstract

remarks.

I°3.3.i)

real-valued

by a u n i q u e

algebra

the

measures.

probability

annihilator Therefore

measure:

M(~)

each

= {m(~)}.

34

ii)

The

spectrum

E(A(D))

uations

in the

points

= F(a)

VF£A(D)

has

ure

a. A n d

for

for

has

the

The

fact,

Jail1, ~(F)

point

and we

In view theorem

of

4. G l e a s o n

In

the

start

remarks

with

for each

that

REMARK: either

Proof:

Let

point

= 6a

eval-

~a:~a(F)

:= D i r a c

=

meas-

exhaust

for

all

Thus

the

VF6A(D)

In particular

m ( ~ o)

the whole

is a c o m p l e x

number

of

= I.

Z(A(D)).

of modulus

polynomials

F and

hence

~ = ~a

concrete

consequence

ii)

whole

A band

of

of

B c

E(A)

Modified the

F.

above

is u n d e r

ca(X,E)

Thus

Now

there

be

3.2

is

says

a reducing

or M(~)cB",

positive

£ A ±,

= m ~ ( X ) m B 6 A ~.

4.2

Ya£DUS

= 0 B 6 A ±.

fix m£M(~)

and mB

= fvudm

(u-~(u))m B 6 A ~ and

otherwise

= P(a,.)l.

~(Z)

ii)

B c ca(X,Z)

M(~)cB

i) W e

m B - mB(X)m

6 B^.

m ( ~ a)

= ~F(s)P(a,s)dl(s) S

= ~a(F)

the

remark. 8

fv(u-~(u))dm

m B = O.

and

the

evaluation

and M.Riesz

3.5.

consideration.

called that

reducing

M(~) v

is

We

iff

a reducing

~6Z(A).

m = m B + m~ with means

i)

measure

1.3.3.i).

immediate

section

a basic

band

have

an

present

implies

4.1

is

~a a:=

= F(a)

members:

point

evaluation

m ( ~ a)

after

the

Parts

86A ±

we

~(F)

obvious

a6S

= (a)

then

VF6A(D)

1.3.5

point

evaluations

have

= ~a(F)

the

measure

if ~6Z(A(D))

some

For

representing

= f(a)

representing

iii) In

the a6D

<0a: <0a(F)

has

o f DUS.

and

prove

, m~

that

. Now

- ~(u)~vdm

in particular

and

hence

so once

In particular

either

Vv6A.

from

the

m~(X)mB(X)

m6M(~)

in B

were

some

~£M(~)

m6B

(u-~(u))m

= O

Then

for each

M(~)~cB

It

Vu6A,

follows

forces

assumption

= 0 so t h a t

all

in B ^ and

other then

have

since

this

that

Thus m B - mB(X)m B =

either

a6M(~) ~(m+~)

~6Z(A)

o r M ( ~ ) V c B ^.

o r m 6 B ^. W e

6 A±

~ u d m B - ~ ( u ) ~ d m B = O.

more

one

band. either

~

to be neither

= 0 or 6 B,

since

6 B nor

QED.

THEOREM:

For

~,~

£ Z(A)

tical or mutually singular M(m) v n S ( ~ ) v = { O } ) .

the

(which

bands after

M(~) v , M(~) v 2.6

are

is e q u i v a l e n t

either to

iden-

35

Proof:

From

M ( ~ ) V c M ( 9 ) ^, M(~)~cM(~) ~

4.1

and

means

alternative

We h a v e relation

thus

E(A)

the G l e a s o n which

defined

parts

To each

For

Gleason Then

Therefore

each

T6M(~),

o6M(~)

an e q u i v a l e n c e

combination

either

M(~)~=M(~) v

is s i n g u l a r

denote

denote

versa.

In v i e w

to e a c h

T6M(~).

on E(A).

classes

GI(~)=GI(A,~)

let F(A)

iff M ( ~ ) V = M ( ~ ) v .

and vice

relation

into equivalence

and

part our

different

after

iii)

occur.

is << some

for A. L e t

~£Z(A),

2.6

P£F(A)

above

we

P,Q6F(A)

which

Under will

this

be c a l l e d

the G l e a s o n

the c o l l e c t i o n

associate

the b a n d

can be r e f o r m u l a t e d

P6F(A)

the bands

is e q u i v a l e n t

If B c c a ( X , E )

can

results

i) F o r e a c h

4.3 R E F O R M U L A T I O N :

ii)

M(~)VcM(~)Vor

part

of all G l e a -

for A.

for any ~6P.

(which

either

~ , ~ 6 ~ (A) to be e q u i v a l e n t

is t h a t

decomposes

contains

son p a r t s

to d e f i n e

t h a t e a c h o6M(~)

the

3.2 w e h a v e

or M ( 9 ) ~ c M ( ~ ) ^. T h e

QED.

It is n a t u r a l

of 4.2

with

M(~)~cM(~) ~

and M ( ~ ) ~ c M ( ~ ) ^ c a n n o t

or M ( ~ ) ~ c M ( ~ ) ^.

This

in c o n n e c t i o n

likewise

the b a n d b(P),b(Q)

b(P)

b(P) :=M(~) v as follows.

is r e d u c i n g .

are m u t u a l l y

singular

to b ( P ) n b ( Q ) = { O } ) .

is a r e d u c i n g

band

then

for e a c h

P6F(A)

either

b(P)c_B or b(P)c-B ^.

N o w we a p p l y led to

2.7

to the

family

of b a n d s

(b(P))p6F(A).

Then

we are

~ 6 Z (A) which

thus

which

are m u l t i p l i c a t i v e

which

are

called

is the b a n d

singular

completely

4.4 T H E O R E M :

M(A) ^^ =

in t h e

sense

generated on A.

to all

these

by the

And

set M(A)

of all m 6 P r o b ( X , E )

B^=M(A) ^ c o n s i s t s

m6M(A).

The measures

of the

86ca(X,E)

86M(A) ^ w i l l

singular.

We h a v e @ b(P) P6F(A)

that

for e a c h

[ P£F (A)

and h e n c e

ca(X,Z)

@6ca(X,Z)

we have

llell ~ llell < ~ ,

=

@ b(P) P6F(A)

~ M(A) ^,

be

36

<M(A)^^>0

=

[ % P6F(A)

In p a r t i c u l a r

Here

the

decisive

last

effect

bitrary

the b a s i c which

band

with

ones,

4.6

Chapter

closure

RETURN

theorem:

to those are

existence

to be the

It r e d u c e s

which

<< some m 6 M ( A ) ,

of s u c h

of the H a r d y

ar-

are e i t h e r or e l s e

a reduction

is

algebra

situation

after

the b i p o l a r

IV. h6B(X,Z)

is in A II

of A in the

to the U N I T

= {~a:a6DUS}.

topology

are e i t h e r

are

{~z:Z£D}

of B a n a c h

for A(D)

are

DISK:

From

and

the

identify ~ a = a for a6DUS theory

be c o n s i d e r e d

(which

~=o(B(X,Z),ca(X,Z))

<< s o m e m6M(A)

iff

or c o m p l e t e l y

£M(A) ^.

Z(A(D)) these

must

namely

+ <M(A)^>%.

< M ( A ) ^ > @ 6 A I.

M. R i e s z

consideration

= O f o r all e £ A i w h i c h

singular

and

what

F.and

6M(A) ^. T h e

A function

is the

[ @ P£F(A)

b(P)=M(~) ~ and hence

for the

4.5 T H E O R E M : theorem

expresses

special

singular

reason

starts

fhd%

sentence

in some

completely

@ =

< b ( P ) > 8 £ A ± VP6F(A)

of the a b s t r a c t

86A ± to v e r y

contained are

86A ± iff

and hence

interior

we h a v e

the G l e a s o n

parts

{~a } for the p o i n t s

(see III.1

algebras).

the

In 3.6.iii)

ii)

Then D and

for

the

Z(A(D))

seen

a6S.

= DUS.

are

immediate:

It is u s u a l

connection

the o n e - p o i n t

that

for A(D)

And

with

the

to Gelfand

the G l e a s o n

parts

{a}

parts

of the p o i n t s

a£S.

5. T h e

abstract

The the

present

chapter.

to p a v e

Chapter

IV,

into

the that

We r e t a i n abstract mental algebra

5.1 define

section

While

purpose

duced

Szeg~-Kolmogorov-Krein

the

the

is d e v o t e d abstract

road

abstract theory

the

1.4.

We n e e d

and M.Riesz algebra

in f a c t

situation

two

is b o u n d

of

the

theorem simple

fundamental

theorem

theorem

the m a i n

theory

Szeg6-Kolmogorov-Krein

Szeg~-Kolmogorov-Krein

lemma

to the o t h e r

F.

to the H a r d y

and

algebra

Theorem

lemmata

which

theorem

to b e c o m e

chapter

will

has

be

are

with

intro-

core.

fix ~ 6 E ( A ) .

be d e d u c e d which

will

its

and

starts

of

from

The

the

funda-

unrelated

to the

situation.

GEOMETRIC

MEAN

LEMMA:

Assume

that m6Prob(X,~)

a n d O~f6L1 (m) a n d

37 I

0: 8(t)

Then

i) 8 is m o n o t o n e

when

f is c o n s t a n t ,

called

from

8(O+)~I(f) 1 + tflog Thus

for O
and in fact s t r i c t l y

8(0+)

= exp(flog

we obtain

from H~ider

f m u s t be c o n s t a n t

@(O+)~I(f).

for t>O, h e n c e

(I + t f l o g

3) The p r o o f

fix O<8~I

and o b t a i n

V x 6 ~ we ob-

f dm) I/t for t>O s u f f i c i e n t l y

with

can in v i e w of

e>O i n s t e a d of f, so that

even

f~1. T h e n

log f ~ O. We

for O
--

ft

1+x

fftdm ~

that 8(O+)~I(f)

f ~ some e>O and h e n c e

and de-

2) In o r d e r to p r o v e

t h a t f l o g f d m > -~. F r o m e x ~

B e p p o Levi be c a r r i e d o u t for max(f,s) that

except

(which is

t h a t 8(s)~8(t)

if 8 ( s ) = @ ( t ) .

log f) ~ I + t log f

f d m > O and 8(t) ~

we can a s s u m e

increasing

f dm) =: I(f)

m e a n of f).

w e can a s s u m e

t a i n ft = e x p ( t

small.

increasing

I) For O<s
(fftdm)[

and ii)

the g e o m e t r i c

Proof: duce

=

tl

exp(t

log f) = 1 + t log f +

~T(log

f)l

1=2 oo

=< I + t log

0(t)

and h e n c e

5.2

< (I + t ( f l o g

[ ~1. ( l o g 1=2

MEAN

f)l =< I + t ( l o g

VALUE

Assume

a point

follows.

that -~
T6[a,b]

QED.

and

such t h a t

e (b) - e ( a )

0 ( t ) -69 (T)

der sup e(T) b-a Proof:

6f),

f

F o r 6+0 the r e s u l t s

THEOREM:

Then there exists

f +

I

f d m + 6ffdm)) ~

@(O+)~I(f)exp(6ffdm).

EMBRYONIC

@:[a,b]÷~.

f + t6

:= lim sup t+T

t-T

i) F o r a~u
e(v)-e(c)

v-c

e(c)-e(u)

c-u

+ V-U

V--C

V--U

<____m a x ( @ (v)-@ (c) V--C

ii)

F r o m i) we o b t a i n

w i t h b n - a n = 2 -n(b-a)

C--U

, @(u)-8(c)>

.

U--C

a c h a i n of i n t e r v a l s and

V--U

[ a , b ] = [ a o , b o ] m . . . D [ a n , b n ]=.."

38

@ (b)-0 (a)

0 (bo)-@ (ao) -

b-a iii)

9 (bn)-e (an ) <

...

<

L e t T 6 [ a n , b n] Vn~1.

Then

..~

<

bo-a °

bn-a n

for each n we have,

whatever

an
or

T=a n or T=b n , f r o m i) that

8(bn)-8(a

n)

8(tn)-8(T)

<

with

bn-a n For n +~ the r e s u l t 5.3 A B S T R A C T and O<_F6L I(~) Then

tn=a n

or

=b n

and

tn#T.

tn-T follows.

QED.

SZEG~-KOLMOGOROV-KREIN

and

there e x i s t m e a s u r e s

in the e x t e n d e d

THEOREM:

1
sense,

m6M(~)

Assume

a n d DP(Fo)

w i t h m<
that 0 6 P o s ( X , Z )

are n o t b o t h = O.

such that Slog F d m e x i s t s

that is such t h a t S ( l o g F ) ± d m are n o t b o t h = ~.

And

Inf{exp(flog

DP (Fc) : all t h e s e m} < - <

Fdm)

=

Sup{exp(Slog The main

that

: all these m}.

step w i l l be the p r o o f of the s u b s e q u e n t

5.4 S P E C I A L F£LI(o)

Fdm)

Dp(o)

CASE:

Assume

that a £ P o s ( X , Z )

w i t h F ~ some s>O. T h e n t h e r e e x i s t s

log F 6 L1(m)

special

case.

and lO, an m6M(~)

with m<
and

such

and D p (Fg) exp(Slog

Fdm)

P r o o f of 5.4: W e can a s s u m e DP(Fto)

~ DP(~)

and Qt6Lq(Ft~)

> O. T h e r e f o r e

!nf{SIPt-ulPFtd~

ii)

SuQtFtd~

iii)

IPt Ip = PtQ t =

to p r o v e

that

-

=

Dp(o)

-

that F~I.

For O~t~1

from 1.4 we o b t a i n

then

I~Ft~F and h e n c e

functions

Pt6LP(Fto)

with

i)

Let us d e f i n e

<

: u6A}

= O, I

= ~(u) (DP(Ft~)) p

Vu6A,

IQt lq =: F t w i t h F t F t o 6 M ( ~ ) .

the f u n c t i o n

@:e(t)

= - l o g DP(Ft~)

for O~t~1.

O u r a i m is

39 (*)

der sup 8 (t) ~ -f (log F) FtFtdo

VO~t~1.

Then 5.2 furnishes a point O~T~I such that m := FTFTo 6 M(~) fulfills the assertion. The proof of (*) requires several steps. I) Fix O<s
t

f lPt-Un IpFtd~ = S1PtFP-UnFPlPdo + O. From ii) we obtain SUnQtFtdo = ~(Un)eXp(-19(t))

and hence ~(Un) ~ exp(18(t)) t

~(~n)eXp(-18(s))

t-s

s

= lUnQsFSd~ = I
and therefore t exp ~(0 (t)-0 (s)) =

t-s

Pt F

s Qs F

d~.

Here we apply the H~lder inequality in two different ways,

in that we

combine the middle factor once with the first one and once with the last one. It follows that I

8 (t)-@ (s) /f/l\ t-s t \t-s exp t-s =< ~ J ~ ) FtF d~)

,

q(t-s) P I exp @(t)-@(s)
I log (I) ~ FtF t d~

for O
And in the other one we fix O<s<1 and let t+s. Then from 5.1 we have =

lim sup @ (t)-@(s) < t+s t-s =

Ilog(~)FsFSdo

Now combine these results to obtain

for O<s<1 = "

(*). QED.

Proof of 5.3: In view of 1.2 we can assume that 1O. Then we can apply 5.4 to the function F+s for

40 e>O

and

thus

obtain

f(log(F+e))+dme appropriate

m6M(~)

well-defined.

an m EM(~)

< ~ we see

And

with

that

in 5.3 is

me<
the

fulfilled,

it f o l l o w s

From

~(log

assertion so t h a t

the

F dm

ii)

Next

DP(Fo) V u6A

that

represents

T

DP(Fo)>O.

F6LI(o)

and hence

Inf

: g(x)

:= FO 6 P o s ( X , Z ) .

Choose

a fixed

function

=

Then

such

flog

G dm e x i s t s

I f(x)

if f ( x ) ~ O

0

if f ( x ) = O

0 ~ G:=g

i) to T a n d G. T h u s

there

in the

modT

exist

6 LI(T)

extended

sense,

whenever

f(x)~O.

GF=I

log G + log F = 0 in L(Fo)

so t h a t

L e t us r e f o r m u l a t e : flog

Fdm exists

and

There

in the

these

exist

extended

Inf{exp(-flog

with

We

m<
DP (GT) DP(o) < - < J = Dp(T) = DP(Fo)

g(x) f(x)=1

: all

m6M(~) and

G dm)

m<
> 0

and GT=gT=gfo~o.

measures

f Inf~exp(flog L Now

f:x~f(x)

and d e f i n e

can a p p l y that

~(u)=1,

= Dp(o)

g

Put

with

DP(o) assume

is

DP(a)

f luTPFd~ Inf <

which

of

in q u e s t i o n

IlulP(F+e)do < =

=<

< flog(F+e)dm£

Dp(~)

=

existence

Inf

that

DP((F+e)~) Inf < f l o g

F)+dmE

on the

m~

Thus

gf=1

measures sense,

F dm)

modulo

m6M(~)

FJ

and hence and hence

with

m<
modulo in L(m).

such

that

and

: all t h e s e

DP(~) < ~ = DP(F~)

m}

,

DP(F~) < Sup{exp(flog =

DP(o) NOW observe taken exists 5.3.

In fact,

defined.

DP(o)>O.

the

in any Now

in case

proved

the

last

larger

in ii) QED.

the

inequality

sense,

assertion case, Inf

DP(o)=O

iii)

: all

remains

set of the m6M(~)

in the e x t e n d e d

fulfilled

vial

that

f r o m the

F dm)

with

From

i)

the

Inf

is p r o v e d

in i)

DP(F~)>O.

And

a n d is t r i v i a l

allow

that

for DP(~)>0

the S u p

in c a s e

the e n t i r e m6M(~)

is

are w e l l a n d is tri-

inequality

DP(F~)=O

m to be

f l o g F dm

we o b t a i n

in q u e s t i o n

since

for D P ( F o ) > O

such

of a p p r o p r i a t e

and S u p

inequality then

m}.

true w h e n w e m<
a n d ii)

on t h e e x i s t e n c e

so t h a t

these

since

is then

41

The s p e c i a l O~F6LI(o), to d e r i v e

case o6M(~)

an e s t i m a t i o n

5.5 T H E O R E M :

with

log ~

do

~<<~.

log ~

B of c o u r s e m e a n s

sense.

If D P ( ~ ) > O

Vo£M(~)

From

that

m£BNM(~)

and 1~p<~.

with o<<m

: o6M(~)

2.2.ii)

flog

Assume

that m6M(~)

is such

and B

~ DP(e)

(a m)

=

}

de do is to e x i s t in the e x t e n d e d ~-~

e x i s t m£M(~)

fulfills

with m<<~

such that o < < m

the a b o v e

inequalities the b a n d

so that BnM(~)

It f o l l o w s

is a p r e b a n d

that t h e r e e x i s t m e a s u r e s

QED.

JENSEN Vo£M(~)

INEQUALITY: w i t h 0<<8.

log,Sd8 , ~ S u p { f l o g If S d @ # O t h e n t h e r e

Thus

In the case D P ( e ) > O

BNM(~)%~

can be applied.

as claimed.

5.6 U N I V E R S A L

We p r o c e e d

w i t h o < < m and B ,

1.7 we h a v e D P ( e ) = D P ( e m ) .

such that o < < m

for

for all e 6 P o s ( X , Z ) .

f r o m 5.3 in v i e w of D P ( m ) = I .

upon w h i c h

of DP(Fo)

<< some o 6 M ( ~ ) .

as required.

B : = { e } v = { o 6 c a ( X , Z ) :o<<e}

0<<8

)

then t h e r e

w i t h o<<s

Proof: follow

do

measures

Then

: o6M(~)

{ (;

Sup exp

where

to an e s t i m a t i o n

of DP(e)

Let e 6 P o s ( X , E )

that o < < m Vo£M(~)

Inf exp

leads

that is for the p o s i t i v e

d~m do

e x i s t m6M(~)

L e t @6an(u).

Assume

that m6M(~)

is

Then

: o6M(~)with w i t h m<<8

o < < m and B} .

such t h a t o < < m

Vo6M(~)

with

as required.

Proof:

We have

can be applied.

lud8 = ~ ( u ) I d 8

V u 6 A and h e n c e

5.7 C O R O L L A R Y :

F o r u 6 A and m6M(~)

Take

8 := um £ an(~)

The corollary that it is m u c h

Now

we h a v e

logI~(u) j ~ S u p { ~ l o g J u J d o Proof:

DI (10J)~Jld@J.

QED.

5.7 looks

in 5.6.

at least

less p o w e r f u l .

: o6M(~)

w i t h o<<m}.

QED.

as nice as 5.6. B u t we s h a l l see

5.5

42

5.8 T H E O R E M :

L e t m6M(~)

and

1~p< ~.

Then

the

subsequent

properties

are e q u i v a l e n t . i)

M(~) ={m} . for all o 6 P o s ( X , E ) .

iii)

DP(e) ~ exp

iv)

loglIdSl

In this

iii)

5.1. iv)

see

i) ~ ii)

from

~ ~(X)

5.7

Then

= I from

for all

from

5.5

I = DP(o)

5.1.

86M(~)can(~)

@6an(~).

tlhat logic(u) I ~ I l o g l u l d m

and

~ exp

8=m

for all

= iii)

log

i) ~ iv)

and d e d u c e

ii)

u6A.

is trivial.

dm

dm =

do and ~ = const

Tinus 0m=O,

d~ ~-~ = I and O=Om=m.

Therefore ~ i) L e t

for all o 6 P o s ( X , Z ) .

is i m m e d i a t e

~ i) L e t o 6 M ( ~ ) .

= Om(X)

dm

=< ;log ~d@ dm

case we

Proof:

log ~

once

is i m m e d i a t e

more

from

as in the p r o o f

from

5.6.

of iii)

~ i).

QED.

5.9

RETURN

~ a 6 Z(A(D)) Therefore 1.3.9

to the

5.8 c o n t a i n s

(for a=O)

closed

UNIT

DISK:

in the p o i n t s

all gaps

a6D.

Consider From

the c l a s s i c a l

evaluations

we k n o w

that M(~a)={P(a,.)l}.

Szeg~-Kolmogorov-Krein

as w e l l

as the J e n s e n

we

in C h a p t e r

left

the p o i n t

3.6.ii)

inequality

1.3.7.

theorem

Thus

we h a v e

I.

Notes

In H O F F M A N

[1962a]

Chapter

a n d the

departure

into

central

character

of the F.

rems

becomes

appeared the

abstract

BROWDER none

F.

[1969],

of t h e m

restriction situation

quite

around

evident.

1970

all

a n d M. GAMELIN

covers

the

to u n i f o r m of the

4 the c o n c r e t e

abstraction

and M.Riesz The

contain

Riesz

algebras,

subsequent

or

LEIBOWITZ

extent

on the less

abstract

III.

disk that

theory

comprehensive

[1970],

of e i t h e r

the m o r e

Chapter

unit

a way

and S z e g ~ - K o l m o g o r o v - K r e i n

treatises more

in the

in s u c h

and S z e g ~ - K o l m o g o r o v - K r e i n

[1969], full

theory

are p r e s e n t e d

special

which

versions

of

theorems:

and STOUT

theorem

the theo-

[1971].

- in p a r t

due

compact-continuous

But the

43

The

abstract

F.

and M.Riesz

LOWDENSLAGER

[1958]

AHERN

where

[1965]

[1969].

All

these

pact-continuous III.2

will

tinuous that

papers

situation

which

spaces.

impl~ed

Also

except

[1967] that

identical,

required

uses

the

extra

The basic

version sults

algebra 3.1

Section

abstract

dual

type.

it is of i n t e r e s t

Gleason that

But

this

parts

were

[1964].

We

procedure

reduction

but

that

defined the one

situation) come

theory

(in the

The extended

is c l o s e

to the

in G L I C K S B E R G

result

re-

[1972]

the b a n d

algebras

complicated

[1957].

in S e c t i o n

is c o n t a i n e d back

in

decom-

of s e c o n d

new machinery.

3.1

can be o b t a i n e d

[1969].

in G L E A S O N used

measure

is due

in w h i c h

full

are

application

[1963].

theorem

rather

was

[1969]~

involved its

in e x t e n d e d

the

on c o m p a c t 4.4-4.5

or as in G A R N E T T - G L I C K S -

summarized

requires

situation,

b y the R A I N W A T E R

prodecure

in an a p p r o a c h

com-

Section

K~NIG-SEEVER

discovered

[1968],

idempotents

to k n o w

shall

of t h a t measures

supplemented

and M . R i e s z

that

methods

as the s i m u l t a n e o u s

of K O N I G - S E E V E R

with

compact-continuous

via

approach

method

definition

BISHOP

F.

[1973]

are p e r f o r m e d

simple

the

point

to the

on B a i r e

to G L I C K S B E R G - W E R M E R

Cole - unpublished

Thus

by the

such

[1963],

and KONIG-SEEVER

decompositions

had been

HELSON-

3.2 in t h e c o m p a c t - c o n -

before

to be band

to L U M E R

i d e a of

3 - and SEEVER

positions

due

situation)

of the

of B r i a n

this

efforts

BERG

[1967].

had

of

reduction

situations

different

of m e a s u r e s

at this

theorem

steps FORELLI

are r e s t r i c t e d

proof

the p a r t i c u l a r

version

the

for a s y s t e m

Dirichlet

us a n n o u n c e

in the

[1962b], [1967],

last one

fundamental

and before

certain

the

independent

in s p e c i a l

[1969]

theorem

Let

HOFFMAN

GLICKSBERG

representation

The GLICKSBERG

fact

except

another

3.2 e v o l v e d

[1959],

proved,

situation.

is the F . R i e s z

Hausdorff not

3.5 w a s

contain

theorem

and B O C H N E R

to the

4

The

equivalence

(which

is true

in a f u n d a m e n t a l

Gleason

part

of

in the

result

theme

of

in S e c t i o n s

III.3-4.

The

abstract

Szeg~-Kolmogorov-Krein

HELSON-LOWDENSLAGER [1964],

[1958]

and HOFFMAN-ROSSI

of u n i q u e n e s s

situations

f o r m of an e q u a t i o n . the important cases

note

and GAMELIN

[1969].

LUMER

The

[1965]

[1964].

full

evolved

in the

[1959],

HOFFMAN

[1962b],

[1965]-

where

is a p u r i f i e d

[1967a],

theorem

and K ~ N I G

is c o n c e r n e d

5.8

in A H E R N - S A R A S O N

and BOCHNER

Beyond

version this

GAMELIN-LUMER

theorem

the

is due

LUMER

as far as the

theorem

retains

of t h e m a i n

frame

steps

there

[1968], to K O N I G

result

are

LUMER

frame the of

special [1968],

[1967b][1967c].

Chapter

Function

In the p r e s e n t of C h a p t e r tinuous

ter

We

supremum

the p o w e r f u l

versions

pendix

to o b t a i n facts

the

sentative

measures

and M . R i e s z

sition

of

Section tions

seen

continuous

linear bra

us

and

to use

in the A p -

of t h e

funda-

of r e p r e -

for the

ab-

approach

metrics,

to the G l e a s o n

It p r o d u c e s

f o r m the

situation

a decompo-

one o b t a i n e d

the

in

two d e c o m p o s i -

to C h a p t e r

II

on A,

Measures

linear

and h e n c e

= o(C(X) ,ca(X))

functionals

for s o m e

by a w e l l - k n o w n

multiplicative

(with I~I!=~(I)=I)

Introduction

described

measures

of C h a p -

compact-conti-

enables

for the e x i s t e n c e

different

and J e n s e n

of A i m p l i e s

rem that each nonzero

o(C(X) , (C(X))")

situation

identical.

Measures

completeness

proofs

another

and H a r n a c k

compact-continuous

to be

theorem

Jansen

presents

is a p r i o r i

In the

I. R e p r e s e n t a t i v e

The

chapter

which

in the

theorem.

the G l e a s o n

Z(A)

II.4.

are

the

via

space X and a n d is c l o s e d

theorem

and natural

situation

compact-con-

constants

in the p r e s e n t

in p a r t i c u l a r

and of s o - c a l l e d

special

topological

the are

of X. B u t

of the H a h n - B a n a c h

theory:

stract

theme

Hausdorff

contains

representation

independent

of

Furthermore

sets

the F . R i e s z

mental

part

which

to the m o r e

(= supnorm) ° T h e n w e

~:= t h e B a i r e

situation

F.

fix a c o m p a c t

Situation

the b o u n d e d - m e a s u r a b l e

ourselves

AcC(X)

norm

The Compact-Continuous

we a b a n d o n

restrict

subalgebra

II w i t h

nuous

chapter

II and

situation:

a complex in the

Algebras:

III

Banach

functional

continuous

= ~IC(X) . T h u s

is the

theo-

is n o r m

in t h e w e a k

Z(A)

topology

as d e f i n e d

is the s e t of all n o n z e r o

that

algebra ~:A÷~

in the

multiplicative

spectru/n of A in the B a n a c h

alge-

sense.

In the p r e s e n t the

fined u6A.

section

set of r e p r e s e n t a t i v e to be

a J~nsen

L e t MJ(~)

Exponentiation

measure

= MJ(A,~) shows

we

fix ~ 6 Z ( A ) .

measures

for ~ iff

denote

the

Recall

M(~)

for ~. A m e a s u r e

= M(A,~) s6Pos(X)

logl~(u) ] ~ f l o g i u l d ~

set of all J a n s e n

that MJ(~)cM(~) : For

u6A we have

c Prob(X) is de-

for all

measures eU6A with

for ~.

45

~(e u) = e ~(u)

so t h a t

a6MJ(~)

= logl~(eU) I ~ ~ l o g l e U l d ~ ~(u)

= ludo

of J e n s e n sures

measures

(which

1.1

Follows

~(u)%0}

The

also known

has

situation

We

thus

of the

from

1.1

obtain

the

list

which

some

is c o m p a c t with

o(K)=I.

S : = { l o g l u I - logl~(u) I : u £ A

The

and hence

latter

fact

If A has dense

in ReC(X),

a fast

representative

be a J a n s e n

the D i r i c h l e t then

measure.

weak.com-

is one of the

situation.

of a u n i q u e

that m must

important

f r o m the in turn

upper lower

measure

property,

mea-

(this that

it is o b v i o u s

Applied

and u n c o n v e n t i o n a l

functional

1.3 REMARK:

Let

c Prob(X)c(ReC(X))"

Proof:

In o r d e r

that QED.

estimation

is i m m e d i a t e

that

with

properties,

estimation

on ReC(X).

is f~g i m p l i e s

vious.

If K c X

a6MJ(~)

that

to the u n i t proof

is

is if each

disk

of the

classi-

1.3.7.

immediate

The

finite-valued

plies

mea-

%:USC(X)+[-~,~[.

It is de-

to be

Yf£USC(X). lows

to

exists

compact-continuous

(f) = Inf{Re~0(u):u6A

We

that

for the e x i s t e n c e

of r e p r e s e n t a t i v e

and weak.closed

representative

inequality

introduce

proof

a b i t more:

there

applied

are c o n v e x

is s u p n o r m

we

cal J a n s e n

then

In the c a s e M ( ~ ) = { m }

a unique

= logle~(U) I = it f o l l o w s

QED.

from II.5.8)°

Re A c ReC(X)

fined

from A.2.4

peculiarities

1.2 REMARK:

~£Z(A)

We c l a i m

of P r o b ( X ) c c a ( X ) = ( C ( X ) ) ' .

s u r e m it f o l l o w s

short

for the e x i s t e n c e

MJ(~)~.

sets M J ( ~ ) c M ( ~ )

prominent

Re~(u)

f r o m II.I.6).

c USC(X).

subsets

that

, from which

is a v e r y

t h a t Ilull = IlulKll V u 6 A

Proof:

pact

There

and hence

is k n o w n

PROPOSITION:

~ such

with

for a l l u£A.

implies

= ~Re uda

Re u > f]

i)

Min

The

% is s u b l i n e a r ,

%(f)~(g).

o6(ReC(X))*.

iv)

%(Re

Then

(via the F . R i e s z

~ note

o6Prob(X)c(ReC(X))"

a(f)~#(f)

< Max

Vf6ReC(X)

fol-

Re u

for all

Vu6A

~ is that

u6A.

~ o6M(~)

theorem).

o(f)~(f)~Max The

f

~ is isotone,

u) = Re~(u)

A.2.2.

~ Max

estimation

In p a r t i c u l a r iii)

representation

that

after

f ~ %(f) lower

Re u < Re~(u)

via exponentiation.

ii)

to p r o v e

-~ ~ Inf

is obvious. in

Y f 6 u s c (X).

other

f

Vf6ReC(X)

details

im-

are ob-

46 We can c o m b i n e sublinear tional,

1.3 w i t h

functional

to o b t a i n

another

measures.

But combination

much more

informative

1.4 THEOREM: Proof: exists

proof with

%(f)

the H a h n - B a n a c h

: o6M(~)}

In v i e w of 1.3 we h a v e to p r o v e

~6M(~)

such that ~(f)

in the a l t e r n a t i v e

that e a c h

some l i n e a r

func-

of r e p r e s e n t a t i v e

v e r s i o n A.I.1

for all

yields

a

a 06M(~)

w i t h g~f}.

f6USC(X).

that for e a c h

We a p p l y A.I.1

f6USC(X)

there

to the c o n v e x set

such that Inf{¢(g) :g6ReC(X)

B u t here the f i r s t m e m b e r

from the d e f i n i t i o n

= Ifdo per d e f i n i t i o n e m . with

= ~fd~.

to o b t a i n

as is i m m e d i a t e

we c o n c l u d e

version

dominates

for the e x i s t e n c e

= Max{ffdo

g ~ f } = I n f { ~ ( g ) :g6ReC(X)

= %(f)

Hahn-Banach

space

result.

{g6ReC(X) :g~f}cReC(X) with

the p r i m i t i v e

on a real v e c t o r

while

is

the s e c o n d m e m b e r

is

QED.

a useful modification

lemma w h i c h w i l l be a p p l i e d

p r o o f of the a b s t r a c t F . a n d M . R i e s z

theorem

in the

next section.

1.5 LEMMA: a sequence

Assume

that O ~ f n 6 U S C ( X )

of f u n c t i o n s

Vn6A with

with

IVnI~1

%(fn)~O.

T h e n there

and % ( l l - V n l ) ~ O

exists

such that

IIVnfnlI~o. Proof:

Choose

u n6A with

numbers

Re u n => f

and

tn>O w i t h ~(Un)

of $. T h e n v n := e x p ( - U n / t n ) 6 A

IVnl = e x p ( - R e

IVnfnl

It r e m a i n s

Un/tn)

= Re~(Un)

~ exp(-fn/tn)

< tn(fn/tn)exp(-fn/tn)

= exp(-~(Un)/tn)

< t n2

a f t e r the d e f i n i t i o n

satisfies

to show t h a t ~ ( [ 1 - V n l ) ~ O .

~(Vn)

~ ( l l - V n l ) ~ ( 2 t n ) I/2 a f t e r

~ I , I

__< t n ~

.

But

> exp(-tn)

(fll-VnldO) 2 ~ f l l - V n l 2 d o

and h e n c e

}(fn)
~ 1-t n ,

~ 2 - 2Re~(Vn)

1.4. QED.

< 2t n

V a £ M (q~) ,

47

2. R e t u r n

to the a b s t r a c t

The crucial M.Riesz

point

theorem

purposes

Let S,T~Pos(X)

with

to e a c h

It w i l l be u s e f u l

lUnI~l

If in p a r t i c u l a r

Proof:

be c o n v e x w e a k , c o m p a c t

T6T. T h e n

F.and

for o t h e r

Vg6S

S=M(A,~)

on ReC(X)

@S:

= M a x ffdO ~6S

The b i p o l a r

theorem A.I.7

and

= Gs(f)

functionals

vT6T.

functional

that the l i n e a r

the m e m b e r s

functionals

of S. ii)

the s u b l i n e a r

+ @T(g )

{(f,g)6V:f,g__>O and f+g=1}cV.

6(ReC(X)) ~

Consider

on the pro-

functional

V(f,g) 6V.

~6V* h a v e the f o r m ~ ( f , g ) = o ( f ) + T ( g )

w i t h O , T 6 ( R e C ( X ) ) ~. A n d i) s h o w s iii) W e a p p l y the H a h n - B a n a c h

that Un6A.

Vf6ReC(X).

duct vector space V:=ReC(X)×ReC(X)

linear

of f u n c t i o n s

then we can a c h i e v e

the s u b l i n e a r

shows

are ~ 0 S are p r e c i s e l y

0: @(f,g)

~ ~ such that each

a sequence

T([Un+O])=T(X)

for some ~6~(A)

i) D e f i n e

0s(f)

there exists

such that

g([Un+1])=~(X)

The

p r o o f of the a b s t r a c t

lemma.

as well.

2.1LEMMA:

which

Theorem

in the a l t e r n a t i v e

is the s u b s e q u e n t

~6S is s i n g u l a r Un6C(X)

F.and M.Riesz

that ~ @

is e q u i v a l e n t

v e r s i o n A.1.I It f o l l o w s

v(f,g)6V

to o6S and T6T.

to 0 and to the c o n v e x

t h a t there

set

e x i s t o6S and T6T

such t h a t

I := I n f { 0 s ( f ) + @ T ( g ) : O ~ f , g 6 R e C ( X ) = Inf{~(f)+T(g):O~f,g6ReC(X) We a s s e r t g(B)=O

t h a t I=O.

that T ( K ) > T ( X ) - e . fn+XK

In fact,

and T ( B ) = ~ ( X ) ,

a Baire

fn6ReC(X)

and T ( I - f n ) ÷ T ( X ) - ~ ( K ) < ~ .

I=O.

with

iv) We c o m p l e t e

f+g=l}

f+g=l}.

set BoX such that

to each ~>0 a c o m p a c t

We can find f u n c t i o n s

. Then ~(fn)÷O

all e>O and h e n c e

there e x i s t s

and h e n c e

with

G 6 - s e t K c B such

w i t h O
It f o l l o w s

the p r o o f

that I ~

for

of the f i r s t a s s e r -

48

tion.

F r o m iii) we o b t a i n

@S(1-Un)
has

functions

and 0 T ( U n ) < n - 2 .

full m e a s u r e

Then

Un6ReC(X)

w i t h O~Un~1

the B e p p o Levi

for all ~6S and that

[Un+O]

for all T6T. v) It r e m a i n s

to p r o v e

S = M(A,~)

for some ~£Z(A)

we h a v e @S = @ a f t e r

dification

lemma

functions

Vn6A with

can a s s u m e

the s e c o n d

1.5 to the f u n c t i o n s

while

has

such t h a t tells

full m e a s u r e

assertion.

In the case

1.4. We a p p l y

constructed

us t h a t

the mo-

in iv)

to o b t a i n

n and @ ( l l - V n l ) + O such that IlVn(1-Un) ll+O. We

IVnl~1

that @ ( I 1 - V n l ) < n -2. T h e n

~£S = M(A,~)

1-u

theorem

[Un÷O]C[Vn÷O]

[Vn+1]

shows

that

has

full m e a s u r e

[Vn÷O]

has

for all

full m e a s u r e

for all ~6T. QED.

B e f o r e w e come to the a b s t r a c t other

related

position

consequence

a f t e r II.2.1

2,2 REMARK: preband

Now

L e t McPos(X)

2.1

to M and

that

t h e o r e m we m e n t i o n

particular

of the p r e b a n d

T h u s e a c h 86ca(X)

• ~. T h e n M is a admits

a unique

%M << some m 6 M and 9~ s i n g u l a r

le~l y i e l d s

a much sharper

@~ lives on some B a i r e

an-

decom-

case.

be c o n v e x w e a k , c o m p a c t

8 = @M + 8~ w i t h

applied

it f o l l o w s

M.Riesz

an i m p r o v e m e n t

in an i m p o r t a n t

in the s e n s e of II.2.1.

composition

F.and

of 2.1:

set BoX w h i c h

de-

to all m6M.

statement

on @~:

is a c o m m o n null

set for all m6M.

We c o n c l u d e

with

the a l t e r n a t i v e

p r o o f of the a b s t r a c t

F.and M.Riesz

theorem.

2.3 A B S T R A C T that

Proof: with

F.and M.RIESZ

THEOREM:

Let ~ 6 Z ( A ) .

T h e n @6A ± i m p l i e s

@ M ( ~ ) 6 A ±.

We a p p l y

lUnl~1

[Un÷O]

has

2.1 to M(~)

such that

[Un÷1 ] has

full m e a s u r e

O = SUUnd6 so that SUd@M(~)

3. T h e G l e a s o n

We present

and

for

le~(~) I to o b t a i n

full m e a s u r e

functions

for all m6M(~)

Un6A

and

19~(~) Io For u 6 A then

= SUUndgM(~)

+ lUUnd~M(~0 ) + lUd@M(~)

,

= O. QED.

and H a r n a c k

another

Metrics

approach

to the G l e a s o n p a r t t h e m e v i a the G l e a -

49

son and H a r n a c k modulo

metrics.

the F . R i e s z

can be e s t a b l i s h e d X and

a complex

Gleason

where

under

and H a r n a c k

little

AcB(X)

functions

which

are d e f i n e d

= ~UP~F--~U) : F 6 R e A w i t h

REMARK:

i) G is a s e m i - m e t r i c

ii)

We h a v e

this

point

transitive

and

Vu,v,w6X

it is

that

far

the

in p a r t i c u l a r

G(u,v)<2

the

F>O

the

constants.

}

Vu,v6X

on X a n d a m e t r i c And

Vu,v6X.

relation

from evident

the

And

iff A s e p a r a t e s G(u,v)<2

in v i e w of

H ( u , v ) < ~ is t r a n s i t i v e .

that

the

relation

relation

H ( u , v ) < ~ is s y m m e t r i c . two

or H ( u , v ) < ~

set The

,

relation

that

the

they

on B(X).

Vu,v6X.

1~H(u,v)~ ~

that

fix a n o n v o i d

with llfll<=1} ,

supnorm

of X. We h a v e O ~ G ( u , v ) ~ 2

H(u,w)~H(u,v)H(v,w)

rem shows

the

the e s s e n t i a l s

to be

rE(v)

If-If d e n o t e s

We

contains

H:H(u,v)

_

contains

It is a s u r p r i s e

structure:

= Sup{Lf(u)-f(v)I:fCA

is sym/netric,

Thus

section

theorem.

G:O(u,v)

the points

At

very

subalgebra

of c o u r s e

3.1

The present

representation

relations

is an e q u i v a l e n c e

are

G(u,v)<2

The

in f a c t

relation

next

is

theo-

identical.

on X.

(2+G 2 3.2 T H E O R E M :

It w i l l functions

be

that

to c o n s i d e r

defined

besides

G and H the

family

of the

to be

llfll~1 and

If(u) l~t}

Vu,v6X.

t~St(u,v)~1.

For

~ H(u,v)<~

an e q u i v a l e n c e equivalence

I+S t

1+t

1-S t

1-t

We h a v e

3.4 C O R O L L A R Y :

for A.

.

= S u p { I f ( v ) I :f6A w i t h

3.3 T H E O R E M :

G(u,v)<2

H = \~/

convenient

S t for O~t
St(u,v)

Note

We h a v e

u,v6X

H

a n d O~t<1

~ St(u,v)<1.

relation

classes

- -

will

be

we have

In p a r t i c u l a r

on X. U n d e r

which

for e a c h O ~ t < 1 .

this

called

the

equivalence

the r e l a t i o n

relation

G(u,v)<2

X decomposes

the Gleason-Harnack

into

parts

is

50

3.5 C O R O L L A R Y : metrics that

on X

The

is p r o d u c e

the

In the p r o o f s closed. tine

same

of t h e

For we have

uniform

above

the

G and

extended

log H as w e l l

sense

structure

results

subsequent

we

as S O are

~ ~) w h i c h on X.

can assume

remark

semi-

are e q u i v a l e n t ,

which

t h a t A is s u p n o r m

admits

an o b v i o u s

rou-

proof.

3.6 R E M A R K :

The

of A c o i n c i d e

Furthermore mations

the

3.7 R E M A R K :

essential

the u n i t

z ~



disk

that

a,b6D

>

ii)

l>I=I<,z>l.

iii)

l

iv)

lal-lbl 1_lallbl

closure

These

linear

are

the

transforfunctions

a6D,

one.

We

list

some

simple

sequel.

and

z6DUS.

Then

= z.

=

of

fractional

fixed

of m o d u l u s

supnorm

G, H, S t for A.

itself.

for

in the

i)

Proofs

use of the D onto

factors

for the

functions

I-Zz

be n e e d e d

Assume

S t computed

a-z := - -

constant

will

G, H,

respective

map

with

which

functions

with

we make

which

multiplied facts

functions

(log H in the

< =

for all

il

3.2-3.5:

For

z=O in p a r t i c u l a r

Il=Il.

16S.

< laI+Ibl. = 1+lal Ibl

I) D e f i n e St(u,v)-t

ct(u,v)

for u , v E X

and

O~t<1.

1-tSt(u,v) We c l a i m

that

ll

< Ct(U,V )

V f6A

with

llfll<1,

V f[A

with

llfH<1.

c t ( u , v ) + l f (v) I If(u) I __<

l+ct (u,v) If(v) I To p r o v e Geometric

the

first

series

inequality

expansion

IgI=l<-te,>l

from

write

shows 3.7.ii).



that

g

Thus

=:

z =

Izle w i t h

:= < < f ( u ) , - t e > , f > from

lel=1.

£ A. A n d

Ig(u) l = [ < - t e , O > l = l - t e l = t

51

we o b t a i n Ig(v) l = l<-ts,lzls>i

= l<-t, lzl>I = ~ < St(u,v) 1+tl zl =

w h i c h is at once t r a n s f o r m e d the s e c o n d i n e q u a l i t y

into the a s s e r t i o n

it suffices

to assume

3.7.iv) w i t h the first i n e q u a l i t y

to o b t a i n

Izl~ct(u,v).

To prove

that nfIl<1. Then combine

If(u) l-lf(v)I < I I < ct(u,v) , 1-1f(u) ilf(v)l = ' = w h i c h is at once t r a n s f o r m e d

into the assertion.

2) We c l a i m that S t ( v , u ) = St(u,v) fices to prove that St(v,u) we o b t a i n from I)

~ St(u,v).

ct(u,v)+If(v)I If(u) l ~

for u,v6X and O~t
If(v) l~t

ct(u,v)+t ~

1+ct(u,v) If(v) I It follows that St(v,u)

It sufand

~ St(u,v).

St(u,v). 1+tct(u,v) 3) L i k e w i s e we o b t a i n

from I)

ct(u,v)+St(w,v)

If(u) t

for f6A with llfll~1 and

If(w) l~t,

1+Ct(U,v)St(w,v) c t (u,v) +c t (w,v)

ct(u,v)+St(w,v) or

St(w,u)

ct(w,u)

~< I +C t (U,V) C t (W,V)

l+ct(u,v)St(w,v)

and hence ct(w,u) ~ c t ( u , v ) + c t ( w , v ) . In p a r t i c u l a r since Co=S O it follows that S o is a s e m i - m e t r i c on X as c l a i med in 3.5. 4) We prove

> in 3.3. F i r s t deduce

from 1) that

1-ct(u,v)

1-1f~u) I ~

(1-1f(v)t)

V f6A with I!fll~1.

I+Ct(U,V) If(v) I Fix now k 6 A with

K: = Re k > 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

llfll<1 a n d

If(v) I. T h e n

[I-fl 2

that

1+If(v) I

l+f (v)

1-1f(v) I

1-f (v)

= k(v)

= K(V)

I-I f(u)12 = H(U,V)

< H(u,v) K(u)

I-I f(u)t 2

I+I f(u) I

< H(u,v)

I1-f(u) 12 = The i n e q u a l i t y

which

If(u) I~t. It f o l l o w s

H(U,V)

(I-I f(u) I)2

results

remains

1+t < H(u,v)

1-1f(u) I =

true

for all f6A w i t h

.

1-t [fl~1

and

that

1+St(u,v) 1-St(u,v) 6) It r e m a i n s

=

to p r o v e

1+t < H(U,V)-1-t

3.2. We c l a i m t h a t 4G(u,v)

So(U,V)

¥u,v6X, 4+G2(u,v)

which

combines

let f6A w i t h I) w i t h

at once w i t h

3.3 for t=O to y i e l d

IIf~<1 and c o m b i n e

the e l e m e n t a r y

ll

the a s s e r t i o n .

~ Co(U,V)

To p r o v e

= So(U,V)

from

inequality

41q-~h I = 411-1al2+~(a-h) l ~ 4(I-Iai 2) + 2(21al)la-h i 4(1-1el 2) + 41al 2 + la-bl 2 = 4 + !a-b! 2 It f o l l o w s

V a,b6~ with

lal~1

.

that

4 If(u)-f(v) I 4+ If(u)-f(v)l 2 < So(U'V)

and h e n c e

the a s s e r t i o n .

a := f(v)

and d e f i n e

even

for all f6A w i t h

To p r o v e ~ let f6A w i t h

tlflI
llfll<1 and f(u)=O.

,

Put

53

2b

a

b

Then

h

:= I+ 7----~/1_lal z ,

:=

21b 1 =

so t h a t

£ A satisfies

h(u)

lh(u)-h(v) I ~ G(u,v).

2161

a

= b and h(v)

It f o l l o w s

3.8 R E T U R N c C(DUS)

f~flS.

I) T h e

sidered

above

I p.137)

So(O,z)

2) F o r

From

u6D

to the

3.7.i)

So(U,Z)

easiest

then

the

functions

functions

F

:= Re

and

for

z6S.

points

s>O the

fEA w i t h

ifl~1 ,

IIf~l

u,v6S

function

via

functions

are

=

Iz I

G , H , S t con-

in o n e - t o - o n e

It

follows

[1950]

Vz6DUS.

v i a h ( z ) = f ( < u , z >)

as well.

c

re-

(see C A R A T H £ O D O R Y

f(O)=O}

h(u)=O

A = CHoI(D)

c C(S)

corre-

Vz6DUS.

that

llhll<1 and h ( u ) = O }

ll

the

lemma

f(O)=O

Vz6DUS

algebra

to A(D)

of the

and

h£A with

I:f6A w i t h =

the

=

Ilfl
easiest

f: f(z)

f(O)=O} Vz6DUS.

to p r o v e

is t h a t

H(u,v)

= ~.

= I _ zu + £ is 6 A w i t h

f > 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')

3) F o r

for all

isometric

. F r o m the H . A . S c h w a r z o o b t a i n at o n c e

= Sup{If()

In fact,

We

= S u p { I f ( z ) I:f6A w i t h

fixed

spondence

DISK:

is s u p n o r m

is S we

we o b t a i n

QED.

UNIT

which

striction

Vol.

to the

Thus

.

~ 4+O2(u,v)

assertion.

c B(DUS)

= -b.

= -b

that

even 1+1512

the



4G(u,v)

If(v) I=lal

and hence

or

1+ibi2

~+O

=

~

v (I - Re ~ + e) <__ H ( u , v )

it f o l l o w s

for A = C H o I ( D )

t h a t H(u,v) =~.

are

seen

4) T h u s

,

the G l e a s o n - H a r n a c k

to be D a n d the o n e - p o i n t

parts

parts

{z} w i t h

54

4.

Comparison

The

Gleason

of the two G l e a s o n

Part

part

of S e c t i o n

decomposition

measurable

situation:

subalgebra

AcB(X,Z)

part

which

decomposition

it w i t h

the

put

latter

the

Gelfand

On a m e a s u r a b l e

took

contained

place

Gleason-Harnack theory

into

transformation:

the

(X,E)

spectrum

function

Z(A).

In o r d e r

of S e c t i o n This

is sent

a complex

and the G l e a s o n

position.

f6A

for the b o u n d e d -

we a s s u m e d

constants,

decomposition

a comparable

The

II.4 w a s

space

on the

part

Decompositions

into

to c o m p a r e

3 we have

is done the

by

to

the

function

^

f6B(~(A))

defined

11~II

:=

via evaluation

I~(~)l

s~p

=

sup

~6~ (A) since

Z(A)

contains

metric.

That means

theory

to c o n s i d e r

the p o i n t

on Z(A)

3 t h e n has the

~Re, (f) = SUP
and

for O=
St

: St(~,~)

the

4.1

is s i m p l e

THEOREM:

(M(~)) v =

Proof:

Fix

T = ho w i t h

in one

iso-

to A c B ( Z ( A ) ) .

llk0-911,

} Re f > O

,

to Re f > O on

the

llfIl~1 and

two e q u i v a l e n c e

l~(f) l~t}

relations

V~,~6E(A).

on Z(A)

in

direction.

the b o u n d e d - m e a s u r a b l e

(M(~)) v i m p l i e s

T6M(~).

f is

is s u p n o r m

functions

between

Assume

llfl[__<1} =

f > O on X is e q u i v a l e n t

= S u p { l ~ ( f ) I :f6A w i t h

comparison

question

the

f~

=

functions

: H(~,~)

Z(A),

which

to be a p p l i e d

H

Re

~x:~f(x)

transformation

c B(Z(A))

= Sup{l~0(f)-@(f ) l:f6A w i t h

that

evaluations

the G e l f a n d

:= {f:f6A}

: G(~0,@)

f6A we n o t e

Then

l~(f) I = llfIl,

G

for

then

Thus

A ~ A

of S e c t i o n

where

Now

x£X.

isomorphism

The

V~6Z(A).

~6Z (A)

in p a r t i c u l a r

= ffd@ x in the p o i n t s an a l g e b r a

f(~)=~(f)

Then

O~h£L1 (~). T h u s

that

there

G(~,@)

exists

~(f)-@(f)

situation.

F o r ~,~6Z(A)

= ll~-~II < 2.

o6M(~)

such

= ff(1-h)d~

that Vf6A.

T<
or

It f o l l o w s

that I

~-~II

- 2 ! ;11-hldo

- 2 = ;(11-h1-(1+h))do

=

4h do

-

l l-hl+(1+h)

,

55

which

is < O. QED.

In the b o u n d e d - m e a s u r a b l e be true. simple

We shall present

the c o n v e r s e

4.2 LEMMA:

Let

~ , ~ : A ÷ ~ be m u l t i p l i c a t i v e

a l g e b r a A.

then

Proof:

If

linear

n e e d not

the s u b s e q u e n t

functionals

( I - c ) ~ + c@ is m u l t i p l i c a t i v e

on a

on A for some

~=~. For u 6 A we h a v e

(1-c) (~(u)) 2 + c(~(u)) 2 =

(1-c)~(u 2) + c ~ ( u 2) =

=

(((1-c)~0+c~) {u)) 2 =

=

(I-c) 2(~0(u)) 2 + c2(~(u)) 2 + 2(1-c)c~0(u)~(u),

and h e n c e

((1-c)~0(u)

(~0(u)-~(u))2(1-c)c

4.3 E X A M P L E : countable

dense

subset.

And

striction

H e n c e A has

as CHoI(D),

I) We a s s e r t

which

with

different

points

a , b 6 D and TcS a

of all s u b s e t s

By the m a x i m u m isomorphism

the same n o n z e r o

modulus

of X. D e f i n e

principle

CHoI(D)+A

which

multiplicative

are the p o i n t e v a l u a t i o n s

the re-

is s u p n o r m

linear

functio-

~z at the p o i n t s

In p a r t i c u l a r

=

@ if z~X } {~z } if z6X

Z ( A ) = { ~ z : Z 6 X }. T h e

and w r i t e

inclusion

a({x})=:Cx~O

Vx6X.

Vz6DUS.

m is trivial.

For

z6DuS

To see c let

t h e n ~ 6 M ( A , ~ z) m e a n s

that

f(z) In case

and t h e n

= Caf(a)

zED we see

P(z,.)l

z£DUS.

that

M(A'~z)

a£Prob(X,E)

+ c~(u)) 2 =

let Z c o n s i s t

f ~ f l X is an a l g e b r a

isometric.

( ( 1 - c ) ~ + c ~ ) ( u 2) =

= O. QED.

Let X={a,b}UT

A : = C H o I ( D ) Ix c B ( X , E ) = B ( X ) .

nals

statement

It r e q u i r e s

lemma.

complex O
situation

a counter-example.

+ Cb~(b)

f r o m II.3.6

= CaP(a,.)%

+ tiT ctf(t) that

+ CbP(b,.)l

from 4.2 that e i t h e r

V f 6 C H o l (D).

+

[ ct6 t , so t h a t ct=O t6T

z=a and a = 6 a or z=b and a=6 b.

Vt6T,

56

And

in case

z6S we

see

that

6z = CaP(a,-)l

so t h a t

z6T

supnorm

isometric

it f o l l o w s

G(~a,~ b)

= I[~a-~bllA"

But

and

I) tells

~ =

+ CbP(b,.)l

us t h a t

to e a c h

other.

indeed.

QED.

Thus

In c o n t r a s t

The

4.4 T H E O R E M : the

Assume

i)

G(R0,g;)

= [l<0-~[]<2.

ii)

H(~,@)<~.

iii)

H(~,9)<~. , and

~<_H($,~)

.

~

iv)

(M(~))~

Proof: 3.2.

And

=

E(A)

the

converse

compact-continuous

in q u e s t i o n

3 lead

singular

4.1

is false

situation

c a n be p r o v e d

to a p l e a s a n t

compact-continuous

to

are

equivalence

situation.

the

to be identheorem.

F o r q0,96Z(A)

there

exist

~6M(<0)

and

T6M(<0)

such

that

to e a c h

We

can a s s u m e

definition

from A.2.6

T6M(~)

to e a c h

there

o6M(<0)

exists

there

o6M(~)

such

exists

T£M(~)

that

such

that

(M(~)) ~.

Re(H(~0,~)q0(f)-~(f)),

Thus

< 2 .

are e q u i v a l e n t .

likewise

i) ~ ii) The

is

T
T
from

And

CHol(D)+A

(M(A,~b)) v = ¢@b

example

in the

on

the

restriction

3.8 t h a t

CHoI(D)"

of S e c t i o n

assertions

and

the

,

= Ii~a-~b[ I

above,

subsequent

~
2) S i n c e

from

in the p r e s e n t

relations

results

X ct6 t t6T

(M(A,~a)) v = ~6 a a n d

to the

two e q u i v a l e n c e tical.

6 z as well.

+

of the

t h a t ~%~. function

Re(H(~0,@)@(f)-~(f))

we o b t a i n

measures

~,B

H(~,~)~(f)-~(f)

= ~fd~ ,

H(~o,9)~(f)-qo(f)

= ~fdB

Then

H shows

~ 0

6 Pos(X)

I
= H(~,~0)< ~

that

V f6A w i t h

such

Vf6A,

that

Re

f > 0

•

57 I

~(f)

Ifd (H (q0,~) ~+8) H 2 (~0,~) -1 I

~(f)

~fd (~+H (~0,~) ~)

¥f£A.

H2 (~,~)-I It f o l l o w s

that H(~,~)~+B

£ M(~)

and

T

~+H(~,~)B

:

:= H 2 ( ~ , ~ ) _ I

Furthermore tions

H(~,~)o

required

be r e p r e s e n t a t i v e

so t h a t

o6M(~)

is 4.1.

QED.

We r e t a i n

the

a look

E(A).

Gelfand

gy in w h i c h

closed

in the

4.5 R E M A R K : son p a r t

Proof: that

at the

Gleason

topology

in ii).

iii)

of A'.

Gelfand

Assume

the

~ iv)

is o b v i o u s ,

Let

a n d T°6M(~)

then

and

iv)

us c o n c l u d e

to be

transform

the

the w e a k e s t

restricted

theorem

implies

secof

topolois c o n -

to Z(A) that

c

Z(A)

is

topology.

the

compact-continuous union

c Z(A)

situation.

of c o m p a c t

be the

subsets

Gleason

part

Then of

each

of ~ £ Z ( A ) .

Note

set

in the G e l f a n d

topology.

Thus

the

assertion -

n= I

=

1

Glea-

Z(A).

f6A is c l o s e d

~ i)

topology

f6B(~(A)):~(f)

~(A',A)

The Banach-Alaoglu

Let P = GI(A,~) c>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

1<__p<~ are m u c h

that ~ admits

is the s i t u a t i o n

diate.

functions

B•

complex

to be the a p p r o p r i a t e

will be the i n e q u a l i t y

This

the p o i n t

of u n b o u n d e d

H p of H for

chapter

~appears

for a t r a n s p a r e n t

r e m of the c h a p t e r

after

the p r e s e n t

algebra

would

compact-continuous

be to work

cated a r t i f i c i a l the former

of u n c o m p l e t e

construction

= {c(U-V) :U,V6M and c>O}.

61 Likewise we introduce MJ :={O~FELI (m) :logl~(u) l~/(loglul)Fdm the class of Jansen tiation

functions

Vu6H},

(=densities),

shows that MJc_M. At last we define

and NJ as above.

Exponen-

for 1<_p<~ the Szeg~ func-

tional dP:dP(F)

= Inf{f~IPFdm:uEH

with ~(u)=1}

V O<=F6L 1 (m).

We list some simple properties. 1.1 REMARK:

i) HKcK.

={f6K:/fdm=O}.

iii)

1.2 REMARK:

ii) Hi: = the annihilator

plicativity from ii)

i)I~dml~

d1(Ifl)

for all f6K.

i) has an obvious

of ~. ii)

is immediate

in view of the bipolar

from the definitions, on the linear

ii) From

subspace

is well-defined with

IPI~1

=fFPdm.

It follows

1.3 THE DIRECT

ii) For each O~FELI (m)

and /fdm=dl(Ifl)=d1(F). one-line

proof based on the multi-

from the definition,

theorem.

l~(u)Id1(F)~/lu[Fdm

{uF:uEH} eLl(m)

that f:= FPEK fulfills IMAGE CONSTRUCTION:

VuEH we conclude

which

the constants

m6Pos(X,Z)

such that there are measures

that

uF~(u)d1(F)

there exists a function Vu6H.

In particular

the assertion.

Assume

subalgebra

follows

is immediate

the linear functional

such that ~(u)d1(F)=fuFPdm

contains

and iii)

Proof of 1.2:i)

and of norm
P6L~(m)

=

H={u6L~(m) :/ufdm=O Vf6K with /fdm=O}.

there exists an f6K such that IfI~F Proof of 1.1:

of H in L1(m)

QED.

that AcB(X,Z)

and that ~6Z(A). in M(A,~)

d1(F) =

is a complex And fix an

which are <<m.

i) Then A m := A modulo mweak*

is a weak*

closed complex

subalgebra which contains

there is a unique weak* continuous ~m(umodm)=~(u)

c L~(m)

linear

Vu6A. The functional

(Am,~ m) is a Hardy algebra

situation,

m

functional

the constants.

is ~0 and multiplicative. ii) For

K = { f E L l ( m ) : f m 6 an(A,~p)}, M = {O~F6LI(m):Fm6M(A,~)}, N c {f6ReLl(m) :fm E N(A,~)},

And

~m:Am+~ with

(Am,~ m) we have

Thus

62 with equality

w h e n M ( A , ~ ) c { m } v. H e r e of c o u r s e N(A,~)

s p a n ( M ( A , ~ ) - M ( A , ~ ) ) = { c ( o - ~ ) :O,T6M(A,~)

{f6ReL1(m):fm6N(A,~)}

which

and c>O}.

c {f6KNReL1(m):ffdm=O},

in V I . 4 . 4 w i l l be seen to be the L 1 ( m ) - n o r m

t h a t m is c h o s e n

so that

(Am,~ m)

is r e d u c e d

MJ(A,~):={o6Pos(X,Z):logl~(u)l!flogluldo as above.

:= r e a l - l i n e a r

Furthermore

It w i l l at o n c e be c l e a r

closure

(see below),

Yu6A}

of N p r o v i d e d iii)

as in III.1,

that MJ(A,~)c-M(A,~)

We d e f i n e

and NJ(A,~)

as b e f o r e .

We

claim that MJ = {O~F6LI{m):Fm6MJ(A,~)}, NJ c {f6ReL1 (m) : f m 6 N J ( A , ~ ) } , with

equality

all O~F6LI(m)

Proof:

w h e n M J ( A , ~ ) c { m } v. iv) At last we h a v e d P ( F ) = D P ( F m )

i) a n d ii)

the a p p r o x i m a t i o n

follow

suffice:

norm closure an h6Lq(Fm)

as in 1.3.7. and

In fact,

/vhFdm=O

1.4 T H E

algebra

INVERSE

situation

if this w e r e

(X,I,m).

is a c o m p l e x 4-

subalgebra

~:~(u) = ~ ( u m o d m )

of B(X,Z)

4-

Vu6H defines 4-

annihilates

The f u r t h e r

details

Assume

that

(H,~)

is a H a r d y

i) T h e n

:= {f6B(X,Z) :f m o d m

4-

which

6 H} contains

a functional

+

the c o n s t a n t s . 4-

~6E(H).

And

ii) We h a v e

4-

an(H,~0) = {fm:f6K}, 4-

4-

M(H,~) 4-

and

I~< ~ .

M +~.

{Fm:F6M},

=

{fm:fEN}.

4-

N(H,~)

In p a r t i c u l a r

=

re-

f a l s e then t h e r e w e r e

T h u s hF6Ll(m)

QED.

IMAGE CONSTRUCTION: on

attention.

In b o t h c a s e s the s u b s e q u e n t

V v 6 A and /uhFdm~O.

and can be omitted.

and iv)

1<__p<~ e a c h u 6 A m is in the LP(Fm) -

A and h e n c e A m so t h a t we h a v e a c o n t r a d i c t i o n . immediate

In iii)

u £ A m b y m e m b e r s of A r e q u i r e s m o r e

For O ~ F 6 L I ( m )

of A m o d Fm. with

from direct verification.

of f u n c t i o n s

In iii) o n e has to p r o c e e d mark will

for

and 1<_i0<~.

iii) We h a v e D P ( H , ~ , F m ) = d P ( F )

for all O ~ F 6 L I ( m )

are

83 Proof:

i) and iii) are obvious.

first assertion.

In ii) it suffices to prove the

Here the inclusion D is immediate

from the defini4-

4-

tions. The same is true for c once we have shown that each @6an(H,£0) must be <<m. To prove this let E6Z with m(E)=O. For all f6B(X,Z) fxEmOd m =O and hence fXE6H with ~(fXE)=O. that

then

It follows

I@I (E)=O. QED.

We retain the Hardy algebra situation present section we use 1.4 to transfer 1.5

Thus /fXEdS=O.

the main theoremsof Chapter II.

ABSTRACT SZEG0-KOLMOGOROV-KREIN

and O~F6L(m)

with FP£LI(m)

and I ~ < ~

(H,~). In the remainder of the

THEOREM:

Assume that O~P£LI(m)

such that dP(P)

and dP(FP)

not both = O. Then there exist functions V6M with [V>O]c[P>O] course modulo m-null

sets)

are (of

such that f(logF)Vdm exists in the exten-

ded sense. And Inflexp(f(logF)vdm) :all these V} < dP(FP) =

dP(p)

< =

Sup{exp (/ (log F) Vdm) : all these V}. 1.6 THEOREM: [V>O]c[F>O]

Let O~h6L I (m) and I ~ < ~ .

VV6M with

[V>O]c[h>O].

Assume that F6M is such that

Then

Inf~exp( f (log~)Vdm]:V6M with [V>O]c[F>O] IF>O]

and 9}

dP(h) = dP(hx[F>O]) 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 f
the

will

algebra

H.

then

all

that

into

that

each

of the H a r d y section

be o b t a i n e d

i)

Inf

functions

in the o p p o s i t e other.

theorem

f~e(f)!Sup

the c o n v e n t i o n

case The

algebra

contains

situa-

some

in S e c t i o n

Szeg6-Kolmogorow-Krein

n(tf)=t~(f)

implies

that and

The present

properties,

e (loglul)=logI~(u)I of

~(u)=O,

essentials

provided that

V f 6 R e L(m).

be t r a n s f o r m e d

the

theorem

e is s u b a d d i t i v e ,

B u t we do n o t

have

once

abstract

~(u)=1}

~ ( f ) = ~ in c a s e

consideration.

consequence

ii)

with

above

tion

(H,~)

to be

I :u6H w i t h - i o g l u I ~ f}

= Inf{Sup(f+loglul):u6H

the

is d e f i n e d

pre-

3 as a

1.5.

f V f6ReL(m)

.

~ + ( - ~ ) = ~ is a d o p t e d .

v f6Re L(m) a n d t>O.

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)<~

such

Then

/f+Vdm<~}~

that

are

~(f).

/fVdm~e(f)~

immediate,

ii)

Sup f V f 6 R e L ~ (m) follows

at o n c e

QED.

2.l.ii) 3.9 w h i c h

is a r a t h e r

weak

relation

is in the o p p o s i t e

com-

direction.

67

It is s o m e t i m e s s i o n of t h e

a:a(F)

which

more

functional

convenient ~. We

= S u p { I ~ ( u ) l:u6H w i t h

appears

to w o r k

introduce

to be a n a t u r a l

with

the

an e x p o n e n t i a t e d

ver-

functional

luI
V O <= F £ L(m),

definition.

The

connection

between

a

a n d e is

a(e -f)

Thus

the d o m a i n

portant ties.

= e -~(f)

of e is s o m e w h a t

for our p u r p o s e s .

Let

2.2

that

the

P~MARK:

Proof:

with

Let O~P6LI(m) the convention

with

0~=O

follows.

The

aim

o

next

,a :Re L ~ ( m ) ÷ ~ .

2.3 LEMMA:

some

ii)

but

this

of the

is u n i m -

above

a(F)a(G)~a(FG)

is a d o p t e d ,

and O~F6L(m)

.

iv)

with

proper-

VO~F,G6L(m),

a(lul)=l~(u)I

PF6LI (m). T h e n

v u 6 H x.

dl ( P ) a ( F ) ~

0~=O.

~(u)=1

and

=dl (P) I~(uv) ] ~ / l u v I P d m ~ / l u I P F d m assertion

restrictive,

VO~F6L(m).

convention

For u,v6H

more

us r e f o r m u l a t e

i) O ~ I n f F ~ a ( F ) ~ S u p F ~

provided

V f £ReL(m)

IvI~F we h a v e

and h e n c e

dl (P) l~(v) 1=

d1(p)]~(v)I

~d1(pF).

The

QED.

is to d e r i v e We n e e d

Assume

that

h(u+v) < h ( u ) + h ( v )

from

a the

sublinear

the

subsequent

the

function

V u,v>O

and Sup s>O

simple

h:]O,~[÷~

limit

functionals

lemma.

satisfies

h(s) < co. s

Then

Proof: sequence

Let

h(t) t

÷ Sup h(s) s s>O

for

h(t) t

÷ Inf h(s) s s>O

for

S and

s(£)+O

I denote

the

t+O ' t+ ~.

Sup and

Inf

in q u e s t i o n ,

i) T a k e

with

h(s(i)) s (Z)

+ liminf s+O

h(s) s

=:

c < lim sup h(s) = s+O s

< S. =

a

68 Fix

t>O.

For

Z sufficiently

large

t = m(1)s(Z)+u(1)

with m(Z)6~

< m(9~)h(s(Z))+h(u(i))
h(t) t

m(Z)s(Z) -<-t

and hence

for

h(s(Z)) s(~)

Z÷~ a n d

the a s s e r t i o n

for

h(t)

t%O.

The

above

h) (~

the assertion

lemma

permits

h(s) s

t>s

then

__ u h(s) + _Su _. t

s

t

h e n c e < I < lim inf h(t__~) = = t t+~

for t+ ~.

to f o r m

QED.

the

functionals

= lim ~s(tf) t+O

I = Sup ~ s ( t f ) , t>O

s~{f)

= lim ~e(tf) t+ ~

= Inf ~ s ( t f ) t>O

properties

REMARK:

A similar damental

~ ° , s ~ : R e L ~ ( m ) + ~,

Let V£ReL1(m).

result

fact which

(f) = Inf

Proof:

are

(

then

s° and ~

for the will

Then

i)

Inf

s u b l i n e a r . iii)

f~(f)~s(f)~°(f)~ s O and

~

are

isotone.

/fVdm~s~(f)

functional

V f£ReL~(m)~V6MJ.

a ° c a n be d e d u c e d

from the

fun-

n o w be e s t a b l i s h e d .

We h a v e

Re ~0(u):u6H w i t h

I) L e t

immediate,

are

V f£ReL~(m).

u) =Re ~(u) V u6H.

2.5 P R O P O S I T I O N : o

For

s°(f)

s ° ( R e u) = ~ ( R e

S

s>O.

to be

subsequent

2.4

Fix

S
and O < u ~ s ,

V s>O and

~ S u p f V f6Re L ~ ( m ) . ii) iv)

ii)

that

that

we obtain

defined

h ~ ) < c . It f o l l o w s --j--~

~ mh(s) + h ( u ) ~ mh(s) + Su,

l i m sup h _ ~ < t+~

The

Su (9~)

< m__ssh(s) ÷ Su _ t - u h(s) + S u = ~ t s t t s t

It f o l l o w s

,

+

S<~ we o b t a i n

t = ms + u w i t h m 6 ~

Thus

and O
h(t)

From m(Z)s(Z)+t

h(t) t

then

o(f)

denote

the

Reu>f Inf

}

V f £ R e L ~ (m).

in q u e s t i o n V f 6 R e L ~ (m).

Then

o:

69

ReL~(m)÷~

is a f u n c t i o n a l

<Sup f V f6ReL~(m), properties. In fact, >~(f).

ii)

with

3) We c l a i m

To see this that

immediate

and

iii)

properties

isotone.

2) ~(f)
for u6H w i t h

Re u > f

we have

i) Inf f
We need two more

s °(f)
Re%0(u)=a(Reu)>_~(f),

so that

a(f)>__

that o(-e -f)

such

the

sublinear,

let u6H w i t h

< -e -~(f)

V f6ReL~(m).

=

lul~e -f, and let c be a c o m p l e x

l~(u) l = c ~ ( u ) = ~ ( c u ) = R e ~ ( c u ) . Re(cu) < Icu I : lul < e -f

number

of Ici=I

Then or

Re(-cu) > -e -f

e l°gl<~(u)! = l~0(u)I= Req0(cu) =-Re<0(-cu) < - o ( - e -f) ,

and from this

the a s s e r t i o n

o(1-e -tf) < o ( I ) + o(f) = o

Now after

le -tf

I e -tf t l+tf + ~ - - - - I

and hence

o(f) < ~t c 2 e t C + o

and hence

o(f)=~°(f).

A function sequence L(m)

Classes

A function

t 2 tc ~ ~ c e

f6L(m)

when

(f) for all t>O.

so(f).

Ifl~c,

It follows

that ~ ( f ) ~ o(f)

Then

/fVdm~s°(f)

V f 6 R e L ~ (m) iff V6M.

H # and L #

is d e f i n e d

to be of class

~nU6H with

and U n f 6 L

Furthermore

+

term f o r m u l a

Let V 6 R e L I ( m ) .

f6L(m)

le-tf~1+tf)

QED.

of functions

sense),

algebra.

< o

remainder

e-tf-1+tf t

2.6 COROLLARY:

4) For t>O we can e s t i m a t e

o(-e -tf) < 1-e -e(tf) < ~ ( t f ) < s °(tf) = te °(f) ,

the T a y l o r

3. The F u n c t i o n

follows.

fUn I
(m) for all n>1.

F6L # and is d e f i n e d

Then

IfI
L # iff there

pointwise L that

to be of class

exists

(of course

(m)cL eL(m),

a in the

and L # is an

f6L #. H#iff

there

exists

a

70

sequence of f u n c t i O ~ U n 6 H

with

lUnI~1, Un÷1 pointwise,

and Unf6H for

all n~1. Then HcH#cL #, and H # is an algebra. 3.1 REMARK: H#AL~(m) =H. Proof: m is obvious. To prove c let f6H#n~(m) and take functions Un£H as required in the definition.

Then Unf+f pointwise and lUnfi~IfI~

gconst so that f6H. QED. 3.2 REMARK: For f6L(m) the subsequent properties are equivalent. i) f6H # . ii) There exist functions fn6H with fn ÷f and

!fnl~[f I

and f6L #.

iii) There exist functions fn6H # with fn÷f andlfnl ~ some F6L #. Proof: uz÷l and

It suffices to prove iii)~i). Take functions uz6H with luzI~1, # IuzIF~c i Vi~1. Then Ulfn6H AL (m)=H and lU£fnl~Ci, and Ulfn+Ulf

for n+ ~. Thus ulf6H V£~I. It follows that f6H #. QED. Let us remark that H # and L # do not depend on the functional ~. We prove that ~:H+~ possesses a unique continuation ~:H#+~ which is continuous in the adequate sense. 3.3 PROPOSITION:

There exists a unique continuation ~:H#÷~ of ~ which

is continuous under L#-majorized convergence: ~F£L # then

(f6H # from 3.2 and) ~(fn)÷~(f). ~ is a multiplicative

functional. Proof: ~(f)

If fn6H # with fn ÷f and

= Ifnl<

linear

In the sequel it will be named ~ as well.

i) For f6H # we define := ~(uf) (u)

for any u6H with uf6H and ~(u)#O

Such functionsu6H exist after the definition of H #, and it is obvious from the multiplicativity of ~ that the quotient in question does not depend on the particular u6H. The functional ~:H#÷~ thus defined is multiplicative

linear and 0]H=~. Also ~ is continuous as claimed:

If

fn' f£H # with fn ÷f and Ifnl IfI~F6L # then choose u6H with ~(u)+O such that uF6L~(m), and it follows at once that ~(fn ) -

~(uf n) ~(u) ~ ~

= ~(f)"

7~ ii) The uniqueness 3.4 REMARK:

assertion

Assume

is clear from 3.2. QED.

that F6LI (m) with ~(u)=/uFdm

Vu6H.

Then ~(u)=/uFdm

for all u6H # with uF6L1(m). The above remark

is obvious

from 3.2. But it is important

to note

that for u6H # there need not exist an F6M such that uF6L1(m). in V.3.5 we shall obtain examples that ~(u)

is not real

immediate

is that the versions

Assume

that u6H # with ~(u)#O.

are

V6M with

[V>O]c[F>O]

i) For F6MJ we have ii) For F6M there exist

such that /(loglul)-Vdm<~.

logic(u)I ~ sup{/(loglul)vdm:VeMwithEV>O]=[~>o]and to the main results.

a(F) =Sup{l~0(u) I :u6H # with This can be sharpened 3.6 PROPOSITION: ~P(f)=a(F).

inequality

to H #. The proofs

so that we can omit the details.

3.5 REMARK:

We turn

In fact,

u6H # such

of the Jensen

remain true when H is extended

f(loglu I)-Fdm <~ and logI~(u)I~f(loglul)Fdm, functions

functions

(even in the unit disk situation).

An other simple remark we met hitherto

of real-valued

From the above luI
for the functions

And

/Ixoglul

it is obvious

that

v O<=F6L(m). O
For each O_<_F6L# there exists an f6H # with

In particular

,'-v~<~}

IfI
a(F)<~.

3.7 PROPOSITION: For each O~F6L # there exist functions FP6LI(m) and dl(p)>o. And

a(F)

= Inf < -dI(Fp) ~(p)

O
with

:all these O
Proof of 3.6 and 3.7: i) Let us fix O~F6L #. We first prove the existence of functions

O~P6LI(m)

where G6M and u6H with ~cG6Ll(m)

with FP6LI(m)

and dl(p)=d1(luGl)~I/uGdml=I~(u)

Inf in 3.7 is now well-defined. a(F)~I.

and d1(p)>o.

luIF~c<~ and ~(u)#O

Then FP=FIulG

I>O in view of uG6K.

It will be called

We claim that there exists

both 3.6 and 3.7 will be proved,

In fact, put p:=luIG

as above.

an f6H # with

iii) We have

ii) The

I. From 2.2 we have

Ifl~F and ~(f)=I.

Then

72 Id I (P) < d I (FP) <__/FPdm

V O
with FP6LI(m),

II/hdml~Idl (lhl) ~ /lhlFdm V h6K with hF6LI (m). Thus on the linear subspace

{hF:h6K with hF6L1(m)} e L l ( m )

the linear

functional hF~I/hdm is well-defined and of n o r m ~ I. Therefore there exists a function QEL~(m) with

IQI~I such that V h6K with hF6L1(m).

I/hdm = /hFQdm Thus f:=FQ6L # with

!fI'_.F. we claim that f fulfills the assertion. To see

this take funct ons Un6H with then UnhEK with

fUn !I, u n +I and lUnlF < Cn<~. For each h£K

[Unh[F
/(Unf)hdm = /(Unh)fdm :

I /Unhdm = I~(u n)/hdm.

It follows that Unf6H and ~ ( U n f ) = I ~ ( U n ) .

Therefore

f6H # and ~ ( f ) = I .

QED. We combine 3.7 with 1.5 to obtain the main result of the present chapter. 3.8 THEOREM: For each O~F£L # there exist functions V6M such that f(logF)+Vdm<~.

And

Inf{exp(/(logF)Vdm) :V6M with f(logF)+Vdm<~} < a(F)< ~. For the subsequent reformulation define C to consist of the functions f6ReL(m)

such that e-f£L #. Then CcRe L(m) is a cone which in particular

contains the functions which are bounded below. From 3.8 we see that for each f6C there exist functions V6M such that /f-Vdm <~. We define 8:@(f) = Sup{/fVdm:V6M with ]f-Vdm< ~}

V f6C.

Thus -~<@(f)<~. We list some simple properties, t>O. And

tion that /f-vdm<~ VV£M. ii) e is isotone, @(If I) = O =

i)e(tf)=tS(f)

Vf6C and

e(f+g)j9(f)+ 8(g) Vf,g6C at least under the additional assumpiii) If (H,~) is reduced then

f=O for all f6Re L(m) .

3.9 REFORMULATION:

-~< ~(f)<0(f)

for all f6C.

73 3.10 CONSEQUENCE: Proof:

e°(f)=8(f)

for all f6Re ~ (m).

< follows from 3.9 and > from 2.6. QED.

Another consequence of 3.7 is the subsequent useful Fatou type theorem. 3.11 PROPOSITION:

Let O ~ F , F n ~ G 6 L #. Then

lira s u p F n:< F i m p l i e s

that

n~m

lim sup n+m

a ( F n) =< a ( F ) .

3.12 REFORMULATION: Assume that -g~f,fn6Re L(m) with eg6L #. Then f ~ lim inf fn implies that ~(f) ~ lim inf ~(fn ) . n÷~ n÷~ Proof of 3.11: In view of 3.7 it suffices to prove

limn÷~Sup a(Fn) ~

d1(Fp) d1(p ) VO~P6L1(m)

with FP6LI(m)

and d1(p)>o.

Let us fix such a function O~P6L1(m). And take functions uz6H with Iu£[~I, uz÷1 and

luzIG~c£. We can of course assume that W(ui)>O. Also

let v6H with ~(v)=1. Then

lw(u~)w(u)Id I (p) = l~(uzuv)

Id I (P) <

/luzuvIpdm /lu%vIPFndm

1 a(F n) ~ d1(p)~(uz) Now put Gn:=Sup{Fs:s~n} G +lim sup n

V u6H with

luI
/luzvlPFndm.

so that Fn~Sn~G and

luzIGn~lU£1S~c Z. Furthermore

Fn~F. We thus obtain I

a(F n) ~ d I (p)~(uz) I lim sup a(F n) < n+~ = d1(p)~(uz) lim sup a(F n) < /]v]PFdm n~ = d1(p)

flv]PluzIGndm, /ivlPlul]Fdm < f]v[PFdm = d ! (p)~(uz) ' ¥ v£H with ~(v)=1.

Now pass to the infimum over these v6H. QED. The next result is a related technical characterization of L #.

74 It is m u c h

simpler

and in fact a d i r e c t

consequence

of the d e f i n i t i o n s

involved.

3.13 REMARK:

For f 6 R e L ( m ) c o n s i d e r

the s u b s e q u e n t

properties.

i) e f 6 L # . ii) T h e r e e x i s t iii)

There

real c o n s t a n t s

exist

functions

t n such that ~((f-tn)+)÷O.

f n 6 R e L ( m ) w i t h exp fn 6 L # such that

~((f-fn)+)÷O. Then

i) ~ i i ) ~ iii),

Proof:

i) = i i )

and put t n : = l o g

Take

and iii) ~ i) w h e n

functions

c n. T h e n

we o b t a i n

iii) ~ i) We can find f u n c t i o n s

and

t h e r e are f u n c t i o n s IVnleXp f<=exp fn'

We can a s s u m e ~(Wn)÷1 have

and

sequence

version

p.429)

Wn:=UnVn6H

to p r o v e

is d e v o t e d

from Chapter

CONSEQUENCE:

if fn£S w i t h

of 3.14: ~ is obvious. to p r o v e

f6L~(m)

be in the w e a k • c l o s u r e

L2(m)-norm

then f u l f i l l

that Wn÷1.

[F>O],

lWnI~1,

But for F 6 M we sub-

and h e n c e on X if

(H,~)

on X. QED. to an a l t e r n a t i v e

idea of

case 3.10 of the m a i n r e s u l t

II b u t uses i n s t e a d theorem

a beautiful

(see D U N F O R D - S C H W A R T Z

For a c o n v e x

In o r d e r

of S(R).

= o(L2(m),L2(m))[L~(m) of S(R)

subset

ScL~(m)

IfnI~C<~ and fn÷f p o i n t w i s e

that S(R) : = { f 6 S : I f l ~ R }

closure

IVnl~l

so that for a s u i t a b l e

on

leads to the s p e c i a l

Smulian

o(L~(m),L2(m))

that is

par[1958]

in L ~ ( m ) - s p a c e s .

3.14 K R E I N - S M U L I A N S is w e a k ~ c l o s e d ~ Proof

occurs

of the K r e i n - S m u l i a n

valid

is obvious.

fUn I~I ' lUnleXp fn:
to be d o m i n a n t

of the s e c t i o n

at l e a s t

It is i n d e p e n d e n t

ticular VoI.I

products

convergence

and F 6 M is c h o s e n

The remainder

lUnlef~Cn

ii) ~ i i i )

IVnleXp((f-fn)+)~1,

lWnleX p f ~c n. It r e m a i n s

proof which

and

and w i t h - l o g l ~ ( V n ) ~ < a ( ( f - f n ) + ) + !n and h e n c e [~(v~l÷1.

that ~ ( V n ) ÷ 1 . T h e

the d e s i r e d

Un÷l

and h e n c e O ~ ( ( f - t n ) + ) ~

~((f-tn)+)÷O.

Un6H w i t h

Vn6H with

is reduced.

lUnl~1,

/lWn-ll2Fdm~2-2Re/WnFdm=2(1-Re~(Wn))÷O,

is r e d u c e d

3.9.

Un6H w i t h

lUn[eXp((f-tn)+)~1

~ - l o g l ~ ( U n ) I . F r o m ~(Un)÷1

Also

(H,~)

since

S(R)

to p r o v e ~ is s u f f i c e s is w e a k • c l o s e d of S(R)

is convex.

Thus

after Krein-

for e a c h R>O. L e t

T h e n it is a f o r t i o r i

closure

we have:

t h e n f6S.

and h e n c e

in the in the

there exist

f 6S(R) n

75 with fn÷f pointwise.

The assumption implies that f6S. QED.

Proof of 3.10 from 3.14

(and from 2.6 and 3.12): i) Let us consider

S:={f6ReL~(m) : ~ ° ( f ) < O } = { f 6 R e L ~ ( m ) :

~(tf)<0

Vt>O},

which is a cone in ReLY(m). We deduce from 3.14 that S is weak* closed. In fact, if fn6S with

= and fn ÷f then 3.12 implies that ~(tf) =< Ifnl
_<_lim inf ~(tfn)O and hence f£s. ii) We determine the dual cone n+~ D : = { h 6 R e L 1(m) : /hf dm <_0 Vf6S}. From 2.6 we know that McD and hence {cV:V6M and c>=O}cD. In order to prove the converse let h6D. Then h>O since S contains all functions f
Furthermore S contains

Re cu for all u6H with ~(u)=O and complex c, so that Re c ]uhdmO}. iii) Now we obtain from the bipolar theorem A. 1.7 S = s w e a k * = { v 6 R e L ~ ( m ) : f h v d m ~ S u p /hfdm for all h6ReLl(m)} f6S = {v6ReL~(m): /hvdm~O = {v6ReL'(m): /hvdm~OV -- { v 6 R e L ~ m )

for all h 6 R e L l ( m ) w i t h

/hfdm
h6D}

: /hvdm~OVh6M}

= {v6ReL'(m) ,@(v)
where 8(v) :=Sup{/hvdm:h6M}Vv6ReL~(m) as above. Thus for v£Re L ~ (m) we have 0(v-0(v))=0(v)-8(v)=O ~0(v). And e°(v)~0(v)

and hence a°(v)-0(v)=e°(v-8(v))~O or e°(v)~

is clear from 2.6. Thus ~°(v)=@(v) V v 6 R e L ~ ( m ) . QED.

3.15 RETURN to the UNIT DISK: Consider for fixed a6D the Hardy algebra situation

(H=(D),~a) on

see that H # is =H#(D)

(S,Baire,l). From 1.12 we know that Ma={P(a,-)}.We

as defined in Section 1.4, and it will result from

4.1 below that L#is =L°(P(A,.)I)=L°(1),

both independent from a6D as it

must be. Now what is the extended functional ~a on H#(D)? Recall from 1.4.1 the bijective correspondence H#(D~+HoI#(D) is

via

radial

which in the d i r e c t i o n ÷

limits. The restriction H~(D}++HoI~(D) is in the direction

÷ described by F~f=, tion f:f(a)=~a(F) V a 6 D .

that is to F6H~(D)

there corresponds the func-

It is almost obvious that this remains true on

H#(D) : for F6H#(D) we have ~a(F)=f(a) V a6D where f6Hol#(D) to F. In fact, there are functions u,v£H

corresponds

(D) with uF=v and ~a(U)~O. Then

76

f=,

so that ~a(U)f(a)=~a(V)=~a(U)~a(F)

we have incorporated H#(D) and HoI#(D)

and hence f(a)=~a(F).

So

into the abstract Hardy algebra

theory.

4. The Szeg~ Situation

4.1

THEOREM: For F£M and I ~ < ~

the subsequent properties are equiva-

lent. i) M={F}. ii) L#cL°(Fm)

and ~(f)=/fFdm V f6C.

iii) e(f)~/fFdm V f6Re L~(m). iv) dP(h)=exp(

/ (log~)Fdm)

for all O~h6L I (m).

IF>O] v) dP(h)~exp(

f (log~)Fdm)

for all O~h6L I (m).

IF>O] vi) log[/hdml~

/ (log F ~ ) F d m

for all h6K.

IF>O] If (H,~) is reduced then the equivalence extends to ii*) L#=L °(Fm), or C={f6Re L(m) : /f-Fdm<~}. And ~(f)=/fFdm Vf6C. In the subsequent chapters we shall exhibit several other equivalent conditions. Proof: We shall prove i) = i v ) ~ v ) ~ i ) i) and i) ~ i i ) ~ iii) ~ v ) iv) ~ v )

for each I ~ < ~ ,

is trivial, v) ~ i) For V6M we have

1=dP(v)~exp( f (log~)Fdm)~ f ~ F d m = / V d m ~ f V d m = [F>O] [F>O] [F>O] Thus V=O on [F=O], and V ~=const on [F>O] from II.5.1. i) ~ v i )

then i) ~ v i )

for p=1. i) ~ iv) is immediate from 1.6, and

is immediate from 1.7, and vi) ~ i )

i) ~ i i )

I. It follows that V=F.

follows as v) = i )

above.

For f6C we obtain from 3.8 and 3.9 that /f-Fdm<~ or e-f6L°(Fm)

and e(f)~/fFdm.

If e(f)=~ we are done. If ~(f)<~ then 2.1.ii)

that ff+Fdm<~ and / f F d m ~ ~(f). ii) ~ iii) is trivial, First we conclude from 3.12 that a(f)~ / f F d m

iii) ~ v )

implies for p=1:

for all f6Re L(m) which are

bounded from below. Fix now O
77 d1(h)>O. For 6>0 and R>O the function h+6F [ log ~

on

[F>O]

]

i

on

IF=O] j1

f:= R

is bounded from below so that e(f)=/ < fFdm. It follows that a(e -f) = e-e(f)>= exp(-/fFdm)

= exp(-/(log~Fd~ [F>O]

.

Thus from 2.2 we have exp[-/(log~)Fdm)[F>O]

~a(e-f) =<

dI dl(h)(he-f):<--11di (h)

d1~h) (/luIFdm+e-R/lulhdm)

/

lulhe-fdm

v u6H with ~(u):1,

since he-f=hh+--~F on IF>O]. Now let R+ ~ and then take the infimum over the u6H with ~(u)=1. In view of d1(F)=1 we obtain

Then for 6%0 the assertion follows from Beppo Levi. It remains to prove i) ~L°(Fm)cL # if (H,~) is reduced. Assume that f£ReL(m) with ef6L°(Fm), or /f+Fdm<~. Let fn:=min(f,n) so that O<(f-fn)+%O and exp(-(f-fn)+)6L~(m)cn #. Thus from 3.9 and i) we have ~((f-f~)+)~ ~/(f-fn)+Fdm,

and therefore /f+Fdm<~ implies e((f-fn)+)÷O.

Since exp fn

EL~(m)cL # we see from 3.13 that ef£L #. QED. In this connection we formulate an immediate consequence of 3.8. 4.2 REMARK: We have L#c U L°(Vm). V6M It is natural to ask for the inequalities which are opposite to iii) and v) in 4.1. The answer is as follows. 4.3 THEOREM: lent.

For F6M and I_<~<~ the subsequent properties

are equiva-

i) F6MJ. ii) For f6Re L(m) with ~(f)<~ we have /f+Fdm<~ and IfFdm~e(f).

78

iii) /fFdm~e(f) V f6Re L ~ m ) . iv) exp( / (log~)Fdm)~dP(h) v O £ h E L 1 (m). [F>O] Proof: i) ~ i i ) ~ i i i ) ~ i )

is as immediate as 2.1. i) ~iv)

For u6H with

~(u)=1 we have /JulPhdm~

/ l u l P ~h F d m >= expl / ( l o g ( i u J P ~ ) ) F d m 1 [F>O] [F>O]

= exp(pf(logluJ)Fdm)exp(

f (log~)Fdm I ~ e x p (

[F>O]

f (log~)Fdm),

IF>O]

from which the assertion is immediate, iv) ~i) 6>0 we put O~h:=F(IuI+6)-P6LI(m). Then

For u6H withw(u)#O and

jq0(u) Jexp<-f(log(juJ+~))Fdm I = Itp(u)Jexp{ 1

/ (logh)Fdm}

~[F>O] <

i~(u) J(dP(h)

< [/lu[Phdm)P= (/(

)PFdm)P
J~0(u) I<explf(log(luI+$))Fdm > V 6>0, from which the assertion follows for 6+O. QED. The Szeg~ situation is defined to be the case that M={F},

in which

case F6M will be called Szeg~ as well. The above results show that then F£MJ. But there are worlds between F6MJ and the Szeg0 situation M={F}. We present a simple example. 4.4 EXAMPLE:

In the unit disk algebra H~(D)c-L~(1) the functions f£H~(D)

with /fZdl=" (0)=O form a weak* closed subalgebra TcH~(D) of codimension one. One verifies that for the Hardy situation

(T,~ o) the set M con-

sists of the functions FC:=I-Re(cZ)~O for the complex numbers c of modulus cJ~l, and that on the other hand MJ={Fo}. We conclude the section with a first look at a special situation which is much less restrictive than the Szeg~ situation but still permits important additional conclusions.

We shall come back to it in Chapter VIII.

4.5 PROPOSITION: The subsequent properties of a Hardy algebra situation (H,~) are equivalent. i) M is compact in o ( R e L 1 ( m ) , R e L Y ( m ) ) . ii) Each linear functional

~E(ReL~(m)) * with ~ e O = e j R e L ~ ( m )

is of the

79

form ~ ( f ) = / f V d m V f 6 R e L ~ ( m ) for some V6M.

In this situation we have ~ ° ( f ) = ~ ( f ) = M a x { / f V d m : V 6 M } V f 6 R e L ~ ( m ) . F u r t h e r more M J # ~ and ~ ( f ) = M a x

{/fVdm:V6MJ} V f 6 R e L ~ ( m ) .

Proof: We c o n s i d e r the dual space o ( ( R e L ~ ( m ) ) * , R e L ~ ( m ) ) =:T. ~e}

is

McS(0)

T-compact.

(Re L~(m)) * w i t h the weak, topology

It is w e l l - k n o w n that S(9) :={~6(Re L~(m))*:

Under the usual e m b e d d i n g Re L 1 ( m ) c (ReLY(m)) * we have

from 2.6, and the bipolar theorem tells us that

~T = {~6(ReL~(m)),:

T h e r e f o r e M=S(@)

~ ( f ) < S u p / f V d m = 8 ( f ) V f6ReL~(m)} = S(@). V6M

iff M is

T-closed,

that is

T-compact,

and that is com-

pact in T I R e L I ( m ) =o(Re L1(m), Re L~(m)). Thus i) and ii) are in fact equivalent.

The remainder

is then immediate

each sublinear functional

from the H a h n - B a n a c h v e r s i o n that

is the p o i n t w i s e m a x i m u m of the linear functio-

nals w h i c h are b e l o w it, c o m b i n e d with 2.4. QED. M c R e L 1 ( m ) is always o ( R e L 1 ( m ) , R e L Y ( m ) ) closed and bounded. we are in the situation d e s c r i b e d

Therefore

in 4.5 w h e n e v e r dim N <~.

Notes

In 1965 the abstract Hardy algebra theory achieved complete clearness about the Szeg6 situation. [1964], and H O F F M A N - R O S S I KONIG

The last steps were HOFFMAN [1965] and KONIG

[1965] was the e q u i v a l e n c e of p r o p e r t i e s

and those in VI.4.9.

The first step

[1962b], L U M E R

[1965]. The main result of i) and iv)-vi)

beyond was A H E R N - S A R A S O N

in 4.1 [1967a]

in

a still rather special situation: d i m N <~ and an a d d i t i o n a l u n i q u e n e s s assumption.

The paper i n t r o d u c e d

important further ideas w h i c h will be

dealt w i t h in Sections VI.6 and IX.1. The u n i v e r s a l abstract H a r d y algebra situation was then e l a b o r a t e d and G A M E L I N

in G A M E L I N - L U M E R

[1969] as well as in KONIG

in parallel d e v e l o p m e n t s w h i c h of course p e n e t r a t e d Gamelin-Lumer

[1968], L U M E R

[1968],

[1967b][1967c][1969a][1969b][1970~ each other.

In the

approach a basic tool is the K r e i n - S m u l i a n consequence

3.14

and the resultant access to the core of the Hardy algebra theory as discovered in H O F F M A N N - R O S S I native proof of 3.10.

[1967] and described above in form of an alter-

In the p r e s e n t work we adopt the other approach.

80

The Szeg6 main

function situation,

ideas.

class w h i c h

poses,

coincide

introduced [1967c]

in K O N I G

[1966]

special

definition

introduced

in the G a m e l i n - L u m e r

in important

[1966b][1967a]

has the p r e s e n t

approach.

situations,

different.

In contrast

to the usual

and appears

to be superior

(:=invertible conjugation.

for s u b s t i t u t i o n

elements All

these

of H#~) topics

theorems,

and factorization, will

be dealt with

in the and the

his u n i v e r s a l

H # is an a l g e b r a

in p a r t i c u l a r

chapters.

KONIG

is fundamental

are quite

the class

H # was

while

At the same time L U M E R

tion classes tions

class

Hardy

The two func-

but the defini-

spaces

H p for l<__p<~

for several

for o u t e r

pur-

functions

and for the a b s t r a c t in the subsequent

Chapter v Elements

In the p r e s e n t

chapter

(X~).

We a s s u m e

develop

a basic portion

lection

is m o t i v a t e d

We start with tible elements the e a r l i e r to

the

strates

(H,~)

respective

situation

theorem.

HOI#(D).

Both theorems

functions

situation

fine an a p p r o p r i a t e on t h e i r

(except

functions

while

f6H w i t h

have a p p l i c a t i o n s

in H + p o s s e ~ c e r t a i n

extend

the r e s p e c t i v e

notion

in the t r i v i a l

(=functions

I. T h e M o d u l i

case H=~)

in H of m o d u i u s

of the i n v e r t i b l e

1.1 REMARK:

F o r u6H # w e h a v e

of the i n v e r functions

specializes

substitution of

one is

into f u n c t i o n s

of c l a s s

to the v a l u e d i s t r i b u t i o n the i m p o r t a n t

weak-~

of

at once

of its p r o o f d e m o n Then

the m o r e d e l i c a t e

IfI~1

classical

of s p e c t r u m

value distribution.

T h e se-

one is on the s u b s t i t u t i o n

in H and H #. N e x t we i n t r o d u c e

H #. The f u n c t i o n s

the r e s u l t

IV.3.8=3.9.

on

theorems.

the o u t e r

The e a s i n e s s

functions,

of f u n c t i o n s

theory.

the m o d u l i

replace

(H,~)

T h e aim is to

famous classical

H #, w h i c h

the s i m p l e r

f6H # into e n t i r e

situation

Hardy algebra

to c h a r a c t e r i z e

classical

are e s t a b l i s h e d :

functions

tion

of the a b s t r a c t

In the S z e g ~

Theory

from the start.

in p a r t by c e r t a i n

the p r o b l e m

on the s u b s t i t u t i o n

results

we fix a H a r d y a l g e b r a

to be r e d u c e d

of the a l g e b r a

theory.

Hardy Algebra

the p o w e r of our m a i n t h e o r e m

theorems

disk

of A b s t r a c t

subclass

properties

of

H + of

which in the unit

theorems.

At last we de-

for the f u n c t i o n s

in ~ to d e r i v e

In p a r t i c u l a r ,

in the S z e g ~

the e x i s t e n c e

of n o n c o n s t a n t

one)

Elements

situainner

is proved.

of H #

IV
O
The first assertion a with

is

obvious

f r o m the d e f i n i t i o n

the r e m a r k w h i c h p r e c e e d s

n o w u6(H#) x and v 6 H # w i t h u v = i. F r o m ~ ( u ) ~ ( v ) = ~ ( u v ) = 1 ~(u),~0(v)%0. leads QED.

Then multiplication

of the

IV. 3.6 and from IV. 3.6. L e t it f o l l o w s

that

of O
to 1=l~0(u) I l~(v)l < a ( i u ! ) a ( { v I) < a ( l u v l ) = a ( 1 ) = l .

Thus

I~(u) l =a(lul) .

82 1.2 LEMMA: Assume that 0~F,G6L(m) our convention Proof:

Take functions Un,Vn6H with

0<~(Vn)+a(G).

Then UnVn£H with

/lUnVn-112Vdm~2(1-~(UnVn))÷0 have UnVn+l pointwise Therefore and

with FG~I and a(F)a(G)=1

0~=0 after which 0
lUn[~F, IVnl~
lUnVnl ~ I and ~(UnVn)÷1.

Thus in view of

VV6M and of the reducedness

of (H,~) we

after transition to an appropriate

subsequence.

lUnVnI~FG shows that FG=I. Furthermore

]UnVnIG~[Vn[FG=IVn[

(recall

Then F,G6L # and FG=I.

from

]UnVnlF~lUnIFG=lUnl

we see that F,G6L #. QED.

1.3 THEOREM: Assume that 0
factor of modulus one. Thus there exists a unique FX6(H#) x

IFXI=F and ~(FX)=a(F).

Furthermore

u6H # with

lul~F and ~(u)=a(F)

implies that u=F ×. Proof: =a(F)a(~).

i) Let f6(H#) x with

fi=F. From 1.1 then 1=I~(f) I I~(~) I=

ii) Assume now that a(F)a(~)=1.

From 1.2 then F,1F--£L#. Thus

after IV.3.6 there exist functions u,v6H # with IuI~F and ~(u)=a(F), and ivl~ ~I and ~(v)=a(~) . Take any pair of such functions u,v£H #. Then uv6H # with

luvl~1 and ~(uv)=a(F)a(~)=1,

so that uv=1 since

(H,~) is reduced.

Thus u6(H#) x and luI=F, iii) we know from ii) that there exists a unique u6H # with luI~F and ~(u)=a(F). Then 1.1 implies that each f6(H#) x with Ifl=F must be = cu with complex c of m o d u ~ s Icl=1. Now all assertions are clear. QED. 1.4 COROLLARY:

For f6H # with m([f=0])=0

we have

iogI~(f) I ~-~(-loglfl) ~@(loglfl), -~ < logI~(f) I = ~(loglf!) ~ f6(H#) x Proof:

The first assertion means that i~(f) i < a(ifl) < I = =a(l~l)

which is immediate equivalent

to

from the definitions.

Thus -~
is

83

0
a(ifl) _

I a(ITI)

,

which in particular implies that 0
and a(Ifl)a(l}l)=1.

But after 1.3 this is in-

deed equivalent to fC(H#) x. QED. We combine

1.3 with IV.3.8 to obtain the subsequent result.

1.5 THEOREM: Assume that 0
satisfies

I # i) F,~6L and ii) the integral /(logF)Vdm has the same value cE[-~,~] VEM for which

I

Then c6~, and a(F)=e c and a(~)=e f6(H#) × with

for all those

it exists in the extended sense. -c

. Thus there exist

functions

Ifi=F.

Proof: From IV.3.8 applied to ~a(~)< ~. Hence the assertions.

I

F and ~ we obtain eC~a(F)< ~ and e-C~

QED.

For 0
Ifl=F, while this cannot be said about condition ii)

Assume now the Szeg~ situation M={V}. Then ii): is void, so that i) is a necessary and sufficient condition. But now i) means that F,~E L° (Vm) and hence that

logF6Ll(vm).

1.6 SPECIALIZATION:

Assume the Szeg~ situation M={V}. Let 0
Then there exist functions fE(H#) × with

IfI=F iff logF6Ll(vm).

1.7 COROLLARY: Assume the Szeg~ situation M={V}. For f6H # we have -~
V6MJ. ~) We obtain loglfiEL1(Vm)

and 0
from IV.4.1, so

that 1,6 and 1.3 imply that fE(H#) ×. QED. 1.8

FACTORIZATION THEOREM: Assume the Szeg~ situation M={V}. Each

function uEH # with logluIEL1(Vm) where PEH with

IPI=1

possesses a unique factorization u=Pf,

(an inner function)

fact f=lul ×. Furthermore logluI6L1(Vm)

~(u)+O.

and fE(H#) × with ~(f)>0. In

is fulfilled for uEH#wh~never

84

Proof: equivalent

We habe to prove the last assertion. to

lul6L°(Vm)=L #, while

But

(logiul)+£L1(Vm)

(loglul)-6L1(Vm)

results

is

from

IV. 3.5.i). QED.

1.9 RETURN

to the UNIT DISK:

In the special

see from 1.4.6 that logluI£L1(l) An immediate

consequence

is fulfilled

situation

(H~(D),~O)

we

for all nonzero u6H#(D).

of 1.8 is the subsequent

characterization

of H # . 1.10 PROPOSITION:

Assume

the Szeg~

situation M={V}.

Then

H # = {~:u,v6H with v6(H#)×}. Proof:

m is obvious.

uh6H and ~(u)#0 • Then

TO prove c let h6H #. Take a function luJXh6H as well.

tation of h as claimed.

2. Substitution

in

~

is a represen-

Functions

THEOREM:

F: F ( t ) =

x

rul

QED.

into entire

2.1 SUBSTITUTION

Therefore

u6H with

Let f: ~÷¢

be an entire

Max {If(z)i: z6~ with

function

and

Izl0.

For u6H # with F(lul)6L # then f(u)6H # and ~0(f(u))=f(~0(u)). Proof:

Take functions

each compact nomials

Un6H with lUnJ~1, u n +I and u n u6H. Since on subset of the complex plane f is the uniform limit of poly-

it is clear that f(UnU)6H

~F(Iul)6L #, and f(UnU)÷f(u)

pointwise

from IV.3.2 and ~(f(u))=f(~(u)) 2.2 SPECIAL CASE:

Assume

and ~(f(UnU))=f(~(UnU)) . Now and f(~(UnU~÷f(~(u)).

from IV.3.3.

If(UnU)J~

Thus f(u)6H #

QED.

that u6H # with eJUI6L #. Then eU6H # and ~(eU) =

=e ~(u) " The above

substitution

course be extended

theorem is simple but efficient.

to entire

functions

It can of

of several complex variables.

85

As a f i r s t stant

example

real-valued

ducedness

of

we

turn

to the q u e s t i o n

functions

(H,~)

means

2.3 P R O P O S I T I O N :

in H #. R e c a l l

that

Assume

all

whether

from

real-valued

there

Section

functions

that

u 6 H "~ is r e a l - v a l u e d

Vz6~

we h a v e

c a n be n o n c o n -

IV.I

that

in H are

and e,U~6L#.l I

the reconstant.

Then

u=const.

Proof:

For

f:f(z)=sin

F(iltul)~e tlul 6L # so t h a t hence

z

sin(tu)6H # from

sin(tu)=const=c(t)

so t h a t

u=const.

It is of plies

interest

For

For

t>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
is s h a r p

We

in the

as well.

and

will

the a b o v e

constant.

in 5.7.

f6Hol(D),

tution

M={V}

u 6 H # such

examples

sense

F(Itul)=F(tu)<e

f(tu)=cos t~6H#NL~(m)=H a n d h e n c e cos t ~ = I u Now ~(1-cos t~)÷~ for t%0 so t h a t u = c o n s t . QED.

are

functions such

therefore

But

situation

u6H#NLI(vm)

there

shall

since

2.1.

R
what

then

f6Hol(D).

of C l a s s

functions

functions without

HOI#(D)

u6H with

f ( u ) 6 H # and restrictions:

the r e s t r i c t i o n

[uI~1

the

into

formula

as in the

F(Iul)6L # had

functions ~(f(u))=f(~(u)). simple

substi-

to be imposed.

Let

c a n be e x p e c t e d .

it is c l e a r If

lul<1

t h a t we o b t a i n

then

f(u)6H

the d e f i n i t i o n

of

and ~ ( f ( u ) ) = f(u)6L(m)

is

86

still obvious, present

but f(u)6H#ceases

an e x a m p l e :

The f u n c t i o n

to be true for all f6Hol(D) o L e t us 1+z. g:g(z)=exp(-~-C-{) V z 6 D is in HOI~(D)

< . 1+Z ~ it Ig(z)I I Vz6D. Its b o u n d a r y f u n c t l o n G = e x p ( - ~ f ~ ) £ H (D) :G(e )= t I 1+z = e x p ( - i c o t a n ~) V t 6 ~ f u l f i l l s IGI=I. Let f=~:f(z)=exp(~-L- {) V z 6 D w l t h •

slnce

the r a d i a l

limit

ness a s s e r t i o n Take now

1 ,1+Z, F=~=exp[TL-~).

function

in 1.3. Thus

(H,~)=(H~(D),~o)

Then F~H#(D)

f~Hol#(D),

on

which

(S,Baire,l)

from the u n i q u e -

also follows

and u=½(1+Z).

f r o m 1.4.3.

Then

lul<1 b u t

.3+Z, 1+Z 2 ~ H# f(u)=f(½(l+Z))=exp~T/~z~=exp(1+2T~z)=eF 6L (i) is not in (D). Therefore functions u6H w i t h

even

for f u n c t i o n s

f6HoI(D) lul~1

is needed.

as well,

of X. This r e q u i r e s for e x a m p l e function function natural

(H,~)=(H~(D),~O)

not perfect, functions

S-N:=

~s6S: L

a Baire

a representative order

and u=Z:

behaviour

a fact w h i c h

u6H with

requires

luI~l.

limit

set NcS w i t h

lul~1

m-null

sets)

the c o n s t a n t

There u6H with

is a n o t h e r

a~f6Hol#(D).

Thus

In p a r t i c u l a r , cannot

reason

luI~1: W e w a n t

with

F£H#(D)cL(1).

once more

Therefore

for all f 6 H o l # ( D ) ,

for e a c h B a i r e (determined

Icl=1

set N c S w i t h of c o u r s e m o d u l o In p a r t i c u l a r ,

c a n n o t be a d m i t t e d .

on the p a r t of the f u n c t i o n s

as a b o v e

the c o n s t a n t

in

the f u n c -

f ( u ) 6 H #, b u t a l s o ~ ( f ( u ) ) = f ( ~ ( u ) )

the same r e a s o n

requires

functions

that

u=const=c

for

I~(u) I be
IcI=1

be admitted.

Let us n o w turn to p o s i t i v e have

function

for r e s t r i c t i o n

to h a v e

that

s÷f (s) on S-N is

that is h a v e m ( [ u 6 N ] ) = 0 .

u=const=c

is n i c e b u t

on the p a r t of the

and the f u n c t i o n

[u6N]:={x6X:u(x)6N}

functions

it is

f6HoI#(D).

f ( s ) : = l i m f (Rs) e x i s t s ~ J R+I

of f(u)£L(m)

can be m a d e void,

f6Hol#(D)

the

limit

to be 6H#(D)

w e k n o w f r o m 1.4.1

s h o u l d be such t h a t set

Take

radial

to the f u n c t i o n s

some r e s t r i c t i o n

l(N)=0,

of the b o u n d a r y

the i n v e r s e

behaviour.

to be a d m i t t e d

this f(u)=f(Z)

ourselves

on the

functions

luI=1 on the w h o l e

for f£Hol(D)

For f£Hol#(D)

the r a d i a l

I(N)=0

to a d m i t

those with

of the f u n c t i o n s

to a l l o w the d e f i n i t i o n

tion u£H with

some r e s t r i c t i o n

s h o u l d be at l e a s t the p o i n t w i s e

that we r e s t r i c t

N o w the b o u n d a r y

luI<1

to have a n i c e b o u n d a r y

of f on S. S i n c e we r e q u i r e to d e c i d e

defines

in p a r t i c u l a r

f£Hol(D)

f(u)=f(Z)6L(1)

u6H w i t h

But it is i m p o r t a n t

to e x c l u d e

other

functions

results. u6H w i t h

We f i r s t luI~l

show that we do not

t h a n the a b o v e c o n s t a n t s .

87

3.1

REMARK:

Proof:

We have

to p r o v e ~ .

=~(1-~u)=f(1-Su)Fdm provided Icul~1

that

F6M

we obtain

and

m([u6B])=0

Proof:

set

that

subsets

of S.

B is an o p e n

={eit:a
0~f~l. ther

2

for

t=a

0

for

6~t
1.2.1

sets

sets

BcS

some

is s e e n

that

real

and

from

and V V 6 M .

BcS w i t h

I(B)=0.

to be a m e a s u r e

m e a s u r e s are o u t e r

for

or Re c u = l ,

this

regular

we c a n a s s u m e

it is an o p e n

e. ii)

The

on the

interval

B=

function

for a < t < b

I

with

we h a v e

and

t=b

and b < t ~ e + 2 ~

0~F~I

and h e n c e

f(Rs)÷f(s)

f : = < F l > 6 R e Harm~(D)

for R+I

in all

points

with

s6S.

We

fur-

introduce

h:h(z)

so t h a t

For

Baire

0=1-~c=1-~(u)=

Re(1-cu)=0

X. F r o m

Ic]=1.

l~(u]l < I. T h e n

V Baire

B÷m([u6B])

that

Then

with

QED.

and

Baire

~c~ I=1.

over

or u=c.

of S and h e n c e

£
an F 6 R e L~(1) From

with

imply

lul~l

I (B)

function Since

I

defines

cu=1

for all

subset

with

- it)- = f (e

Thus

I t :(u) (u)

~

l~(u) I=1 ~ u = c o n s t = c

to be d o m i n a n t

Let u6H with

In p a r t i c u l a r

i) T h e

Re c u ~ I c u l = l u ~ l

I m c u =0.

(Vm) ([u6B])

Then

Let ~(u)=:c

is c h o s e n

3.2 P R O P O S I T I O N :

Baire

lul~.

Let u6H with

R+I

h6Hol(D)

and

f=Reh.

For

fh(Ru)Vdm

= ~(h(Ru)]

ff(Ru)Vdm

= f(R~(u))

it f o l l o w s

N o w we h a v e

= / s+z f (s)di(s) S s-z

that

0
V z6D

we c a n

form

h(Ru)6H

and o b t a i n

= h(R~(u)), V V6M.

/f(u)Vdm=f(~(u))

from dominated

convergence.

the e s t i m a t i o n s

f(~(u)) : /P(~(u),s)f(s)dX(s)

1+

~(u)

=

(Vm)

<_ ~_ ~(u)

k

(B)

,

S

If(u)Vdm ~

/ f(u)Vdm

--[u6B]

=

/ Vdm [u6B]

[u6B])

.

88

The

combination

The

next

desired

3.3

of all

result

implies

is the b a s i c

substitution

theorem.

PROPOSITION:

f=6Harm1(D).

these

The

f(u)6L(m)

/If(u) IVdm < ~ and

Proof:

We can

preparation idea

Let u 6 H w i t h

Then

assume

that

the a s s e r t i o n .

for

the

of p r o o f

]u]~1

and

F>0

I~(u)l
f>0

half

same

Let

after

= flqo(u))

so t h a t

second

is the

is w e l l - d e f i n e d

/f(u)Vdm

QED.

of the

as above.

F6LI(I)

3.2.

and

We c l a i m

that

V V6M.

as well.

Put

Fn:=min(F,n)

oo

and

f := 6 H a r m (D) so t h a t 0
lim R+I so t h a t F and

the

Fn,

f(Rs)=:f(s)

functioDss~f(s)

and

for n÷ ~. N o w

ii)

choose

which of 3.2

/f(Ru)Vdm

for R+I

equation

in v i e w

n÷ ~ thus

can be d o n e

= f(R~(u)]

of d o m i n a t e d

/f(u)Vdm=f(~(u))

f6Hol#(D).

S~fn(S)

3.2.

use.

From

convergence

Assume

f(u)6L(m)

that

fn(S)if(s)

of

Vs6S-N

of u 6 H w i t h ~(x)l~1

for

0
= fn(R~(u)]

the o t h e r

we o b t a i n

VV6M.

ones

/fn(U)Vdm=fn(~(u))

Levi.

we o b t a i n V V6M.

For

QED.

u6H with

is w e l l - d e f i n e d

f(u)6H # and ~(f(u))=f(~(u)).

Then

exist,

representatives

Thus

x~u(x)

/fn(RU)Vdm

from Beppo

THEOREM:

Then

Vn~1.

function

and

V n~1

on S-N are n)

after

is of no d i r e c t

3.4 S U B S T I T U T I O N

Let

and

fn(RS)=:fn(S)

a representative

and u(x)~N Vx6X

first

lim R+I

it is f n ( S ) = m i n ( f ( s ) ,

as in the p r o o f

The

and

lul~1

after

If f 6 ( H o l # ( D ) ) × t h e n

and

3.2.

l~(u) l <1.

We c l a i m

f(u)6(H#) x,

and

that fur-

thermore

l]loglf(u)

Proof: 6HoI~(D). limits

i) T a k e Then

II V d m

functions

choose

<~ and

l(loglf(u)l)

fn6HOl~(D)

a Baire

set NcS

with with

Vdm

= loglf(%0(u))I V V 6 M .

Ifnl~1, fn÷1 I(N)=0

such

and that

fnf=:gn 6 the r a d i a l

89 lim fn(RS)=:fn(S) R+I

and lim f(Rs)=:f(s) R+I

and hence

lim gn(RS) =: gn(S) = fn(S)f(s) R+I

exist for all s6S-N. Furthermore

in view of

/ Ifn(S)-1 12dl(s) < 2[1-Re fn(0))+0 S-N we can achieve after transition

to an appropriate

subsequence

and to a

larger N that fn(S)÷1 for n÷ ~ Vs6S-N. Let x~u(x) be a representative function of u6H with lu(x)l<__1and u(x)~N V x6X. Then for 0
gn(RU)6H with
and q)(gn(RU)] = gn[R~(u)].

For R+I we obtain in view of dominated convergence
fn(U), gn(U)£H and

and ~O(gn(U) ] = gn(~O(u)].

Furthermore [fn(U)l~1 and fn (U)÷I. Since fn(U) f(u)=gn(U) f(u)£H #. And fn[
=
= q)(fn(U)f(u)]

= gn[~(u)]

= fn(q)(u)]

it follows that

= f(q~(u)]

shows that ~(f(u))=f(~(u)). ii) If f6(Hol#(D)) × then f(u)6(H#) x is obvious. have f=ce h, where c6~ with h:h(z)

= S/ ~s+z G(s)dl(s)

V z6D

Then l o g l f I = R e h = < G l > = : g 6 H a r m 1 ( D ) . the substitutions last assertion

Now from 1.4.7 we

IcI=l and with G 6 Re LI(1). It is clear that loglf(u) I=g(u) for

f(u) 6H # after i) and g(u) 6L(m)

after 3.3. Thus the

follows from 3.3. QED.

In connection with the above theorem we recall from Section 1.4 that all functions f£Hol+(D) are in HOI#(D) and hence (except the constant 0) in (Hol # (D)) ×

90

3.5 E X A M P L E : inner will H=~).

We c o n s i d e r

functions.

In 6.9

be e s t a b l i s h e d Of c o u r s e

example

u=Z).

I~(u) I
3.1.

u-c

-

Then

(u-c)

is a n o n c o n s t a n t

u 6 H we c a n p r o d u c e ~(v)=c.

ii)

exist

for c 6 D the

function

~(v)=-c.

f:f(z)

(except

inner

in the

disk

so-called functions trivial

situation

function,

case

(for

so t h a t

~(u) -c %9(v) - - 1-c~(u) If c=~(u)

from each

inner

then

nonconstant

c6D a nonconstant

1-z

-

the inner

function

be a n o n c o n s t a n t

function

IuI=l,

in the u n i t

as well.

Thus

for e a c h

Let u6H

situations

~ (~u) ~ 6 H w i t h ~=0

inner

[9(u)=0 w e h a v e

u6H with

of n o n c o n s t a n t

u6H be a n o n c o n s t a n t

1-cu

case

Szeg6

functions

i) L e t

from

v:=

in all

such

functions

the e x i s t e n c e

inner

q0(v)=0, inner

function

function.

Note

and

in

function v6H with

that

the

2 V z6D

l+z 2 is in H o I + ( D ) c H o l f(u)-

I-u--2 1+u2 -

So we o b t a i n attains upper

all

An

Thus

I Imu ~-R--~EH

it f o l l o w s

from

(f(u))

# with

a nonconstant

that

I-(<°(u))2 l+(w(u)) 2 .

W

real-valued

3.4

function

~I m U 6 H #

. Since

~(u)

numbers

halfplane.

q~( I m u , 2 ) (R---e--~)

# (D).

in D the v a l u e ~ ( I m u ] a t t a i n s all n u m b e r s in the o p e n ~ I m u ] 2
attains

important

3.6 T H E O R E M : I={eit:a
all

complex

numbers

consequence

is the

Assume

EcIcS

an o p e n

that

interval.

m([u6E])

=

m([u6UND])

except

subsequent

where

the r e a l

value

distribution

E is a B a i r e

Let u6H with

luI~l

= >0.

numbers

set w i t h

and

theorem.

I(E)>0

I~(u) I
and

fulfill

O, = 0 for

some

open

Uc~ with

ICU.

Then m([u6I])=0.

Proof:

i) L e t us

fix V6M.

G: G(z)

is a h o l o m o r p h i c

=

function

Then

I u+z V d m [TU
V z6U

ReGis

a real-analytic

function

91

on I, that is t~Re G(e it) is real-analytic on a
ii) Let F6LI(1)

{ 1-1uI2 [lu
V z6I.

vanish outside of I and f=6Harml(D).

have flf(u) IVdm<~ and ff(u)Vdm=f(~(u)).

ff(u)Vdm =[lu{=l ]F(u)vdm + [ lu{
From 3.3 we

Now

~s_u 12

=[lu{=l]F(u)Vdm + IfIRe G(s)l F(s)dl(s),

f(~(u))

=

f i-i~(u) I2

I ]S-~(U) I2 F(s)dl(s),

where the Fuhini theorem has been applied.

If in particular F vanishes

outside of E then I

F(u)Vdm = 0 because of m([u6E])=0.

[Iu =11 It follows that Re G(s) = 1-1~(u) 12 for l-almost all s6E and hence for all s6I,

Es-~(u) l2

where the real-analyticity of Re G on I has been applied. But this implies that [lu{=1]F(u)Vdm = 0 V F6LI(1) which vanish outside of I. For F=XI we obtain since

(Vm) ([u6I])=0 for all V6M. Therefore m([u£I])=0

(H,~) is reduced. QED.

3.7 EXAMPLE: Let u6H be a nonconstant inner function. For each Baire set EcS with l(E)>0 then m([u6E])>0.

This follows at once from 3.6

applied to I=S and U=~. Thus u attains all values of modulus one in quite a sharp sense.

4. The Function Class H +

4.1 PROPOSITION: equivalent.

For h6L(m) with Reh~0 the subsequent properties are

92

i) e - t h 6 H 1 h-~6

ii)

for all H for all

I h--$~6 H for

iii)

In t h i s

t~0.

case

s6A:=

some

the o p e n

Re~(h)~0.

( 2) ~ ( e - t h ) = e -t~'h) V t~0

1 a n d ~(h-~-~)

3)

/(Reh)Vdm~Re~(h)

for all V6M.

4)

If R e ~ ( h ) = 0

h is a p u r e l y

Proof:

i)~ii)

For e a c h i h+s

since

for

0
r~ e-t £=i

Re s>0.

s6A.

I) h E H # w i t h

then

halfplane

sEA

imaginary

the p o i n t w i s e

oo - /e-t(h+S)dt o integral

i _ ~(h)+s Vs6A.

constant.

integral

is in H,

ie-t(h+S)dt o

~) (h+s) ( t ( ~ ) - t ( ~ - l ) ) 6 H

is the

with

limit

of R i e m a n n i a n

sums

0=t(0)
R u n d e r p o i n t w i s e and b o u n d e d c o n v e r g e n c e and h e n c e is 6H, and / e - t ( h + s ) c ~ + o co ÷ / e - t ( h + S ) d t e v e n in L ~ ( m ) - n o r m for R+ ~. F u r t h e r m o r e k0(e-th)=e -tc V t>0 o for

complex

some

complex-valued

c with function

RectO,

since

which

satisfies

t~(e

-th)

the

in

t>O

is

functional

a continuous

equation

of the

exponential

function and is of modulus ~1. The above approximations 1 1 t h a t <0(h--$-~)=c--$~ for all sEA. ii)~i) F o r t>0 we h a v e

show

e

-th

e

where

the convergence

iii)

ii)

that

The

set

it is open.

then

I i ~-~ - l i m - - - - - ~ , (i+ )

is p o i n t w i s e

and bounded,

ii)~iii)

is t r i v i a l .

{s6A:,}6 H} is c l o s e d in A so t h a t w e h a v e to p r o v e nts i___~ H. F o r I s - o l < R e o t h e n hs-~
and hence 1

I

h+s

(h+~)+(s-a)

Thus

i)

these above,

_

1

1

~

l+S-O h+o

h+g

~=0

ii) iii) h a v e b e e n p r o v e d

properties i) For

l +1e h E H

with

EH.

I

h+g

and retain

the

th+oj

to be e q u i v a l e n t ,

complex

c with

L e t us n o w

Re c ~ O

assume

as in i)~ii)

e>O we h a v e ~

1 + i for =<1 , ]-$~-~

£+0,

and T ~h- ~ =

1 (1- 1+--~)6H.

98

Therefore ~0(h)=c.

hEH #. Then

l=~0{(h+S)h+~)=(q0(h)+S)c+~

2) is then clear.

for sEA implies

that

3) For VEM and t>0 we have

e -tRec = ]e-tC l=I~0(e-th) ] =

< /e-tRe h Vdm,

I/e-thvdml

1 e-Rec__< (/(e-Reh) tVdm} t,

e-ReC<exp(/(-Reh)Vdml

for t+0 after

II.5.1.

Hence f(Reh)Vdm__0 then le-thI=1 and I~(e -th) I=Ie-tCI= I so that e - t h = c o n s t = c ( t ) from 3.1. It follows

that h = c o n s t = c .

The f u n c t i o n with

Reh>0

It is in close dered

class

which

H + is d e f i n e d

possess

relation

in the last

QED. to c o n s i s t

the e q u i v a l e n t to the class

of the f u n c t i o n s

properties

i) ii)

of functions

iii)

u6H w i t h

hEL(m) in 4.1.

lul~l

consi-

section.

4.2 P R O P O S I T I O N :

We have

1-u H w l t h lul~ 1 and u+ the c o n s t a n t - I } . H + = {i-~-~:u6 1-z m) The f u n c t i o n f:f(z)=1-T{V zED has Re f > 0 and is t h e r e f o r e I in HOI#(D). And f--~ for sEA is even in HOI~(D). Thus for uEH w i t h luI<1 Proof:

and

f(u)+s E H for s6A from

l%0(u)l<1 we have

The case have

u:

3.4 and hence

I -u~ E H + . f(u)=1-~-

I~(u) I= I w h e r e u is c o n s t a n t +-I is trivial, c) For hEH + we I -h 2 I -u l+h-l+h 16H w i t h luI<1 and u+-1, and one v e r i f i e s that h=1+ u.

QED. 4.3 RETURN (H~(D),~a)

to the UNIT

with

aED on

and the c o n n e c t i o n we have

with

incorporated

DISK:

Consider

(S,Baire,l). HoI#(D).

H+(D)

From

Now

the Hardy IV.3.15

1.4.8

and HoI+(D)

shows

into

algebra

we know

situations

that H#=H# (D)

that H+=H+(D).

the a b s t r a c t

Hardy

Thus algebra

theory. 4.4 REMARK: immediate

from 2.1.

for hEH # w i t h seen this

If h6H # w i t h Re h ~ 0

in S e c t i o n

even hEH # w i t h

Re h ~ 0

In p a r t i c u l a r

satisfies hEH w i t h

the r e s t r i c t i o n not e n f o r c e

lhl 6L #

Re h ~ 0

that

situation. Re~(h)~0

then h6H +. This

implies

elhlEL # cannot

1.4 in the unit disk

h~0 does

e

is

that hEH +. But

be dropped: we have And

3.5 shows

that

so that hEH + c a n n o t

94 be c o n c l u d e d .

4.5 REMARK:

Assume

that h n 6 H + and hn~h6L(m)

pointwise.

T h e n h 6 H + and

(hn) ÷ ~ (h) . 4.6 REMARK:

For h6L(m)

the s u b s e q u e n t

properties

are e q u i v a l e n t .

i) hEH + . ii)

There

iii)

exist

functions

There exist

h

functions

n

6H +

such

hn6H with

that

h ÷h. n

R e h n => 0 such that

lhnl~]h ] and

h ÷h. n Proof

of 4.5:

Obvious

can take the f u n c t i o n s

f r o m the d e f i n i t i o n . nh n2 h n = n - ~ = n-n--~6 H. QED.

The n e x t a i m is to t r a n s f e r theorem

a b o u t H +. To do this

the s u b s t i t u t i o n

it is n a t u r a l

the u n i t d i s k D o n t o the h a l f p l a n e h ( s ) = ~ s S V s6~ as in S e c t i o n

P r o o f of 4.6:i)~iii)

1.2.

theorem

We

3.4 into a

in v i e w of 4.2 to t r a n s f o r m

A v i a the f r a c t i o n a l - l i n e a r

Let us w r i t e

m a p h:

up the t r a n s i t i o n

in all

briefness.

~qe f u n c t i o n

h is e q u a l

to its inverse.

It m a p s

A÷D and i ~ ÷ S - { - 1 }

with h ( i x ) = e it

x = - t a n ~ t or t = - 2 a r e t a n x

I) T h e s u b s t i t u t i o n

Vx6~

formula

oo 1 f (eit) dt = ~I ff(h(ix)] dx V b o u n d e d 2z_z -~ l+x 2 shows

that the m e a s u r e

l:dl(eit)=2~dt

sure A : d A ~" ,=1 d_x_x o n i~. Thus if ~ix) Z1+x 2 for a B a i r e

set N c ~ w e have

2) L e t U 6 H + and u6H w i t h 4.2. this.

•

.

1 - ~ ( u )

For a B a i r e

fEHol(D)

set N c % w e

that m ( [ U 6 i N ] ) = 0

Lebesgue

into the m e a -

measure

~(N)=0~A(iN)=I(h(iN))=0. - u = h(u) and u~-1 such that U = I1+u so that R e ~ ( U ) > 0 ~ l ~ ( u ) I < 1 . have

[U6iN]=[u=h(U)6h(iN)]

a f t e r I) and 3.2.

after

i x 6 i ~ and h ( i x ) = e i t . ~ 6 S

f:S÷~

on S is t r a n s f o r m e d

9~ d e n o t e s

such that F = f o h : F ( s ) = f ( z ) V

F 6 H o l # ( ~ ) ~ f 6 H o l #(D)

measurable

lul<1

Thenko(U)=1+--~-~)=h(~(u))

implies

and Itl<~.

Section

the r a d i a l

3) A s s u m e

corresponding

after

L e t us a s s u m e so that

Z(N)=0

that F6HoI(A)

s6A and z6D.

1.4. 4) In c o r r e s p o n d i n g limits

on ~, t h e n

and

Then

boundary

points

95

limF(o+ix) 0+0 are n o t in o b v i o u s character

=: F(ix)

a n d lim f(R~)=: R+I

connection

f(T)

w i t h e a c h other.

of the m a p h it is c l e a r

B u t f r o m the c o n f o r m a l

that the e x i s t e n c e

of the a n g u l a r

limits lira F ( s ) = : F ( i x ) s÷ix s6~(ix,a)

on a(ix,e) : = { s 6 ~ : R e ( s - i x ) Z l s - i x l c o s e }

and

on ~(T,~) (as in 1.4.2)

lim f(z)=:f(T) z÷T

is e q u i v a l e n t

to e a c h other,

k n o w f r o m 1.4.2 i-almost 2)-4):

all

If F£HoI#(~)

and then of c o u r s e F ( i x ) = f ( T ) = f ( h ( i x ) ) .

for £ - a l m o s t

and R e ~ ( U ) > O

same s e n s e as f(u) w a s =f(u).

then F ( U ) 6 L ( m )

in the last s e c t i o n

Thus F ( U ) 6 H # a f t e r

3.4.

We can t h e r e f o r e

formulate

4.7 S U B S T I T U T I O N f(u)6L(m)

THEOREM:

f 6 ( H o l # ( ~ ) ) × then

Apply

Assume

Vs6~

theorem

the t h e o r e m

theorem

that u6H + w i t h

as follows. Re~(u)>O.

Let f£Hol#~). If

Re f ~ O

= loglf(q0(u)II V V £ M . t h e n f ( u ) £ H +- The same h o l d s

3.4.

to the m a i n b r a n c h f £Hol+(~)

In p a r t i c u l a r

the p o w e r of the s u b s t i t u t i o n

~

I

of the p o w e r for

the s p e c i a l

theorem.

w i t h s£A. QED.

ITI~I case

function

and h e n c e

T=-I

fT6

demonstrates

We s h a l l come b a c k to the case

in the n e x t section.

4.9 P R O P O S I T I O N : =

in the

(~(U))=F(~(U)).

to the f u n c t i o n s

for T6~. W e h a v e

Hol # (4) for all T6~.

ITI
is w e l l - d e f i n e d

and f ( u ) 6 H # w i t h ~ ( f ( u ) ) = f ( ~ ( u ) ) .

If in p a r t i c u l a r

Let us a p p l y the r e s u l t fT:fT(s)=sT

5) Let us c o m b i n e

f(u)6(H#) ×, and f u r t h e r m o r e

true for the s u b s t i t u t i o n

Proof:

We

for

Furthermore

/Iloglf(u) I IVdm<~ and / ( l o g l f ( u ) l ) V d m 4.8 C O R O L L A R Y :

limits exist

and F ( U ) = ( f o h ) (U)=f(h(U)) =

the s u b s t i t u t i o n

is w e l l - d e f i n e d

these

all x£~.

~(F(U))=~(f(u))=f(~(u))=f(h(~(U))=(foh)

Then

V 0<~<~

that in case of Hol ~ f u n c t i o n s

T6S and h e n c e

V0<~<~

L e t u6H + be +O. For all T 6 ~ t h e n u T £ H # and ~(u~) =

(~(u)) T. In p a r t i c u l a r

u T 6 H + if

I~I~I.

96

4.10

SPECIAL

In a d d i t i o n =logl~ main

CASE:

we

see

(u)! Y V £ M .

branch

p a r t ~0,

4.11

We

from

4.7

combine

IV.3.4

PROPOSITION:

this w i t h

the

u6H + be ~O.

and

/(loglul)Vdm=

the a p p l i c a t i o n

is in H o I # ( A )

to o b t a i n

Let

Slloglullv~<~

that

f:f(s)=log s Vs6A which

and with

IEH + . T h e n -u

L e t u 6 H + be ~0.

subsequent

Then

of

4.7 to the

since ~ ~if

has

real

result.

log u 6 H ~ and ~ ( l o g u) = l o g ~ ( u ) .

Furthermore

/[log u I Vdm<~

From class

and

/ ( l o g u) V d m = l o g ~ ( u )

4.10 we obtain

another

V VEM.

remarkable

closedness

property

of the

H+ .

4.12

REMARK:

Proof:

Let

u, v 6 H +

We can assume

with

Re(uv)~O.

t h a t v~O.

For

Then

t>O then

u v E H +.

u+~EH + from v

4.10

and

~O so t h a t 1 uv+t

t v

once more

1 H# t 6 and h e n c e u+-v

from

4.10.

To c o n c l u d e 3.6

into

the

a theorem

Thus

f r o m A to D d e s c r i b e d

4.13 THEOREM: and

I an o p e n

[u6iE])

m

[u6UN~])

Then m

transfer

the v a l u e

distribution

It is c l e a r

t h a t we can

EcIc~

E is a B a i r e

use

the

theorem transition

Assume

that

where

Let u£H + with

Re~(u)>O

set w i t h

Z(E)>O

fulfill

= O, = O for

some

open

Uc~ w i t h

iIcU.

[u6iI])=O.

The next function

section

class

for a l l V6M. an

we

H +.

above.

interval.

m

u v E H +. QED.

section about

6 H#NL~(m)=H,

example

will

be d e v o t e d

H +. F o r h 6 H + we k n o w

But

it is n o t

in the

Szeg~

always

situation

to a n o t h e r from

true

4.1

that

M={F}:

important

that

SlhlVdm<~.

If u 6 H

problem

on the

S(Reh)Vdm~Re~(h)<~ L e t us p r e s e n t

is a n o n c o n s t a n t

inner

97

function force

that h=const

However, from

1-U6H + w i t h t h e n h =I--+~ the

truth:

statement

certain

5. W e a k - L I P r o p e r t i e s

The which

section

a function

Thus

5.1

The

The

proof

is c a l l e d

that

the

functions

on d i s t r i b u t i o n

algebra

function

[O,~[

in

]O,~[.

situation.

implies

of

f is d e f i n e d

p-T P

/fTdm

type

t>=O.

with

W(O)=m(X)

and W(~)=O.

that

iff

tPw(t)÷O

l i m sup T+p In p a r t i c u l a r

5.3 LEMMA:

(m(X)

p-T P

]fTdm

of the a b o v e

t~tPw(t)

estimations

remark

is b o u n d e d

which

(~ S u p t P w ( t ) ] P t>O

are

the

function

for t+ ~. O u r

in the o p p o s i t e

for O < T < p ,

< lim s u p tPw(t) . '== t+~

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
< =

fix

to be

V t~O.

In v i e w

subsequent

PROPOSITION:

in

functions L e t us

tion.

5.2

en-

we h a v e

is i m m e d i a t e .

the

excursion

on

~ / fPdm [f~t]

of w e a k - L P ( m )

is to p r o v e

2.3 w o u l d /lhlFdm=~.

in H +

for a l l r e a l

left

For O
/ f P d m <~

be

a n d h 6 H + i s not far r~mote

be s h o w n

Hardy

distribution

the

tPw(t)

In p a r t i c u l a r

for all V £ M

it w i l l

a short

decreasing

from

REMARK:

with

= m([f~t])

W is m o n o t o n e

W is c o n t i n u o u s

section

w e r e <~ t h e n

So it m u s t

weak-L I properties.

to the a b s t r a c t

O~f6L(m).

W = Wf:W(t)

u=const.

of the F u n c t i o n s

starts

is u n r e l a t e d

If / l h l F d m

that

that h£L1(Vm)

in t h e n e x t

H + all p o s s e s s

Reh=O.

and hence

= - /F(t)dW(t). 0

then

/fTdm<~

a n d >O.

Then

for O
f

aim direc-

98 Proof [O,R].

of 5.3:

Let R>O and m:O=to
/ F(f)dm [O~f
the first

+ /F(t)dW(t) O

integral

and ~ M i n ( F l [ t £ _ 1 , t £ ] ) with

of

t~

R

I(R) := Here

be a d e c o m p o s i t i o n

Then

under

=

f

f F(f)dm Z~1<[ti_1~f
+ f F(t)dW(t)]. ti_ I

the sum is ~ M a x ( F l [ t z _ 1 , t l ] )

(W(tz_I)-W(t~))

and hence

(W(t~_1)-W(ti))

=F(TI) (W(tz_I)-W(tz))

t ~ _ 1 ~ T £ ~ t i. Thus

tf

r I(R)

=

[

Z=I where

6(T)=max(tl-to,...,tr-tr_

the f u n c t i o n

F on

tion

for R+~.

follows

Proof

of 5.2:

[O,R].

For

I) and ~ is the m o d u l u s

6(T)+O we o b t a i n

I(R)=O.

of c o n t i n u i t y Then

of

the asser-

QED.

Define

and m o n o t o n e d e c r e a s i n g we can a s s u m e

II(R) I ~[~(~)] (w(0)-w(R)),

and hence

(F(t)-F(T~)]dW(t)

t£_i

D : D(s)=Sup{tPw(t):t~s} in

that D ( s ) < ~

for s>=O. Then

D is <~

[O,~[ with

D ( s ) ÷ l i m s u p tPw(t) for s+ ~. Thus t+~ Now for O < T < p and O < s < R we have

in s~O.

R

R

D(s)

R

-f tTdw(t) = - R T w ( R ) + s T W ( s ) + y f W(t)tT-Idt < + T D(S)~ tT-P-ldt. s s = s p-T S It follows

for s>O that

+~ -/tTdW(t) < D(s) + TD(s) J7 tT-P-1 dt = p D(s) = s p-T p-T sp-~ ' s s -~oo

s

/f%dm = - / tTdW(t) < - / t~dW(t) O O

+

P D(s) p-T sp-~

from

5.3,

p-T ] (X)sT+ D(s) fTdm __


the last r e l a t i o n

lim sup Tip

p-T / fYdm < D(s) P =

On the other sp__~T D(O) re(x)

we d e d u c e

"

Vs>O and h e n c e < lim sup tPw(t) . = t+~

hand we can assume

It follows

p-¢p [fTdm=<

at first

that

that

m(X)+

Sp .

=

f#O and hence

D(O)>O

and then put

gg

We return to the reduced Hardy algebra situation

(H,~). A rather

simple estimation shows that the functions in H + are all of weak-L1(Vm) type VV6M. However, deeper result.

the subsequent precise limit relation is a much

It depends on the tauberian theorem 1.2.5°

5.4 PROPOSITION:

Let h6H + with $(h)=a+ib.

t(vm) ([ jh-ibI~t]) < 2a Thus h is of weak-L1(Vm)

For each V6M then

V t>O.

type.

5.5 THEOREM: Let h=P+iQ6H + with ~(h)=a+ib.

For each V6M then

-~t(Vm) ([lhJ>t]) + a - /PVdm for t+ ~. Proofs:

i) For t>O we have h _ I t h+t -~6

h H and ~0[h-~] -

~ (h) k0(h)+t"

For each V6M therefore h = ~(h)+t' ~(h) /~-~-{Vdm

with real part

/ lhj2+tp "-a2+b2+ta lhj2+2tP+t2vcum = a2+b2+2ta+t2

V t>O.

ii) We prove 5.4 and can assume that b=O. Then from i) we obtain ~ a--~t : /lhj2+tp Vdm> / lhJ2+tp Vdm lhI2+2tP+t 2 =[JhJ>t] lhl2+2tp+t 2 Z ~I [Jh|zt]~ Vdm=½(Vm)([ Jhlzt]) which is the assertion, t[

Vt>O,

iii) For t>O the difference

lh12 lhl2+tp ] = Pt 2 .... [hI2-t2 Jhj2+t 2 lhl2+2tP+t '2j (lhI2+t2) (lhI%2tp+t 2)

is of modulus ~P and tends ÷ -P pointwise for t+ ~. Since [ Jhl 2+tP

t ~lhl2+2~+t 2vdm÷a

for t+~

~00 after

i) and f P V d m < ~ we see t h a t

I hl 2

IPVdm

t i lh i 2 + t 2 V d m ÷ a -

for t+~.

N o w let W: W(x)=(Vm) ([ lhl>x]) Vx>O be the d i s t r i b u t i o n with respect

to the m e a s u r e

is m o n o t o n e origin.

From

increasing

V m and % : % ( x ) = W ( 1 )

and bounded

5.3 w e o b t a i n

Ibl2-vdm=-I0

t I lh l Z + t z ÷~ t(I) 2

x-+t-

says that ~

lhl Then

in the

w{xl: +t

1

÷~

(I) t2 + x 2 d ~ (x) =

We k n o w that this t e n d s ÷ a - / P V d m 1.2.5

[O,~[ and c o n t i n u o u s

=-S0 + x

2t~X2dw(xl

(x) = ~

of

for t>O

÷~

-~(1) 2+t2d~

in

function

V x > O and ~(0)=O.

~(x)+a-/PVdm

for

I

f~ d ~ O (~) +x

~0. Thus

(x)

the L o o m i s

for x+O w h i c h m e a n s

theorem

that ~ x W ( x ) ÷ a - / P V d m

for x+ ~. QED.

L e t us add a n o t h e r v e r s i o n functions deeper

t h e o r e m of t a u b e r i a n

5.6 P R O P O S I T I O N : ITI
of c o u r s e w i t h

Proof:

character

simple

of the

estimation

and a

nature.

Let h6H + be #0. For each V 6 M and each real

m with

prefer

(~(h)) T from the m a i n b r a n c h

c o u l d be d e d u c e d

It f o l l o w s

1+eh6H with

s=Isleit

ourselves

with

Itl~

m6 H

and ~((1+--~) T) =

6H +. Thus [ ~(h) < ~ j

F o r e a c h V 6 M w e thus have

cos/~]" ~

to the case O
theorem

and hence

TVdm < /Re(~)mV61m = R e , ~ ,

function.

a worse 4.7.

constant.

ii) For

for the m a i n e>O w e h a v e

f r o m 4.9 we o b t a i n IT

We

i) L e t s6~

sT=IsITe iTt

t h a t Re s T = I S l T C O S T t ~ I s I T c o s ~ .

real p a r t ~ O and h e n c e ( ~ )h

of the p o w e r

f r o m 5.4 a n d 5.2 w i t h

to b a s e the p r o o f on the s u b s t i t u t i o n

Re s >_O. T h u s

branch.

=< Re{~) T

In v i e w of 4.10 we can r e s t r i c t

The r e s u l t

h

a rather

then

cos 7~ I! h i ~ v ~

with

of the a l m o s t - L l ( v m )

in H +. As b e f o r e w e p r e s e n t

that

101

N o w let e+O and a p p l y

5.7 E X A M P L E : is n o n c o n s t a n t

the F a t o u

theorem

to o b t a i n

L e t u 6 H be a n o n c o n s t a n t with

inner

O
Thus we h a v e

5.8 T H E O R E M :

an e x a m p l e

5.9 LEMMA: of the p o w e r

as a n n o u n c e d

a -fPVdm

Let s = u + i v 6 ~ w i t h function

of 5.8:

for Tit.

Then

for the m a i n b r a n c h

that u>O and v+O.

T h e n we have

U ~ T / t T - I d t = u T. QED. 0

i) F r o m 5.2 and 5.5 we o b t a i n

lim sup T+I

cosV/lhl~Vdm

~< l i m s u p ~ t tim

(Vm)

= lim s u p ~ ( 1 - T ) / l h l T V d m T+I

[lh!~t])

F r o m 4.9 and f r o m 5.6 c o m b i n e d

(~(h)) T = ~ ( h T ) = / h T v d m

= a - fPVdm.

with

IV.3.4

we see

~/(Re(iQ)T+pT)Vdm

after

for O
that

and h e n c e

Re(~(h)) T = / ( R e h T ) V d m

T~ = cos~-/IQITVdm+/PTVdm~cosVflhlTVdm It f o l l o w s

2.

For e a c h V 6 M t h e n

u~O and O
u sT-(iv) T = T/(t+iv) T-Idt, 0 U IST--(iv) TI ~ / I t + i v I T - I d t 0

ii)

at the end of S e c t i o n

IST-(iv)TI~UT.

P r o o f of 5.9: We can a s s u m e

Proof

M={V} we know from

t h a t ~ l h l T V d m < ~ for all

Let h=P+iQ6H + with ~(h)=a+ib.

T~ COS ~ - f l h l T V d m +

QED.

1-U£H+ Then h =1--~u

function.

Re h=O. In the S z e g ~ s i t u a t i o n

2.3 t h a t f l h l V d m m u s t be =~. N o w 5.6 s h o w s

the result.

+

5.9

(/PVdm) T

that a = Re~(h)

~ liminf T+I

F r o m i) and ii) w e o b t a i n

The c o n n e c t i o n

imposed

f o r m of the r e l a t i o n it can h a p p e n

cos~flhlTVdm

the a s s e r t i o n .

+ fPVdm.

QED.

u p o n the f u n c t i o n s

O~P6L(m)

h : = P + i Q 6 H + is a r a t h e r w e a k one:

that P=O and Q#O w h e n e v e r

the e x i s t e n c e

and Q 6 R e L(m) in in p a r t i c u l a r of n o n c o n s t a n t

102

inner

functions

connection

has

6H has b e e n

been made

VI.2

it w i l l

be o b v i o u s

tion

is m u c h

weaker

Q 6 R e L(m) the

be its

latter

results

5.4

of

and

due

tend

to c o n j u g a t e

it is a s u r p r i s e

to see disk

6. V a l u e

The

Carrier

For

and L u m e r

section

of t h e

functions

which

are

a function

situation

definitions

that O~P6L(m)

(up to an a d d i t i v e

P a n d Q in the u n i t

as 5.6

and

5.8

these

valid

under

disk

the

1.4.

In S e c t i o n

that

the c o n n e c -

be c o n j u g a b l e real

algebra

assumptio~which

For

the above

classical

c a n be e x p e c t e d

Hardy

and

constant).

situation

are w e l l - k n o w n

theorems

in the a b s t r a c t

them

disk

theo-

to ex-

situation. are m u c h

But weaker

situation.

present

parations

function

functions

unit

respective

Of c o u r s e

even

In the

at the e n d of S e c t i o n

requirement

functions

to K o l m o g o r o v .

in the u n i t

the

the

5.5 as w e l l

rems

bution

from

than

conjugate

pairs

proved.

explicit

Spectrum

is to o b t a i n

further

in H a n d H #. T h e

unrelated

f6L(m)

results

first

to the a b s t r a c t

we define

the v a l u e

on the v a l u e

part

is d e v o t e d

Hardy

algebra

distri-

to p r e -

situation.

carrier

~(f) := {z6~:m([If-zl<6] ] > 0 V~>O},

~-~(f):=

We

list

some

simple

representative e(f)+~,

iv)

6.1LEMMA:

function

Then

the

then

m(h)nG=~.

ii)

Let

x ~ f(x) then

such

6>0}.

is c l o s e d ,

that

f(x)6m(f)

ii)

There

exists

for all x6X.

a

iii)

f=const=c.

and a6~

in the e x t e r i o r

of G. L e t

that

~h-z d m i O

same holds

i) L e t us

Choose

some

i) ~(f)

L e t G c ~ be a d o m a i n

flog

Proof:

for

properties,

If ~ ( f ) = { c }

h E L °(m) . A s s u m e

~(m).

{ZE¢:If-zl~6

true

V z6~G.

VzEG.

And

fix t 6 G s u c h

a representative

K and B denote

closure

if flog ~h-t[ _tldm=O

that

function

in some

point

h - t lId m > - ~ a n d h e n c e flog a_--:~ x ~h(x)

and b o u n d a r y

with

h(x)#t

of G r e l a t i v e

t6G

h-tl log a-~tl6

for all x6X. to the

Rie-

103

mann

sphere.

on G.

L e t AcC(K)

In v i e w

supnorm on A I B

isomorphism after

A÷AIB.

III.1.1

'

a~K.

consist

of the m a x i m u m

Let

For each

Thus we

86Prob(B) u6~

the

no~ that

be a J e n s e n function

u~/log

~u-z d8 (z) value

u ~

is a h a r m o n i c the m a x i m u m

From

log h ( x ) - t

respect

u-z

dg (z)

z~U-Z-1 _ u - a is in A since a-z z-a

extends

over

true.

To see this

function

on G w h i c h

on G. T h e r e f o r e

÷~ for u+t.

be > 0 in all p o i n t s

In v i e w

u+t

II+=-

in G as

V xEX,

~G.

We w a n t

theorem.

h(x)-t a-t

Ii°gll-h(x)-all+z-a

take

To see

to i n t e g r a t e that

h(x)-z + /(log[---~I

the

integral

for m - a l m o s t

be an m - n u l l

further

need

set.

on the

all x6Xo

Therefore

a simple

fact

=<

this

(~)

is c o r r e c t

+4^ ) a~(z),

ii)

~(h)DG=~

account

the

z6B,

assumption

< O.

left

Then

Ii°g(1+clh(x)-al)l+v

into

/log la-~tldm h-t h-z < f(/iogI~L-~Idm) d@(z)

If in p a r t i c u l a r

of

we h a v e

B or o v e r

~-log

tends

de(z)

the F u b i n i

holds

is h a r m o n i c

which

therefore

with c>O a suitable constant, and ,h - t , lOgla--/~tI6L1 (m). It f o l l o w s t h a t

We

is a

flB~f(t)

estimation

Ii°gI~

must

f~flB for

log u - t

=< /log ~

to m a n d u s e

the

an e q u a l i t y

measure

real-valued

and h e n c e

the a b o v e

we write Jkrll°g lh(x)-z ,--~l)-d@(z)

consider

are h o l o m o r p h i c

V ugh.

< relation

[u6G:u~t}

it m u s t

iii)

48(z)

is a c o n t i n u o u s

~0 on

principle

integration

strict

equation

/log

function

claimed,

(*)

u-z /log ~

<

for all u E G the

s a t i s f i e s the m e a n

we have

which

the r e s t r i c t i o n

have

We c l a i m t h a t

with

functions

principle

"

u-t log ~ - t

where

of t h o s e

modulus

is =O t h e n implies since

on c o n n e c t e d n e s s

(~) m u s t

that

have

{x£X:h(x)6G}

G is open.

in m e t r i c

QED.

spaces.

been

104

6.2 LEMMA: of d i s j o i n t

Let E be a m e t r i c (!) o p e n

subsets

s p a c e and TeE. A s s u m e

U , V c E we have:

that for e a c h p a i r

TcUuV~TcU

or TcV.

T h e n T is

connected.

Proof:

Assume

with T=PuQ

U:= where

t h a t T is not c o n n e c t e d .

and PNQNT=~.

U V(u,½6(u,Q)} u6P

6 is the m e t r i c

r a d i u s e>O. mains

Thus P N Q = P N Q : ~ .

and V:=

Then there exist nonvoid

U V(v,½6(v,P)), v6Q

of E and V(a,~)

is the open b a l l of c e n t e r

T h e n U , V c E are o p e n w i t h P c U a n d QcV,

to s h o w that UAV=~.

some u6P and

Indeed

6(v,x) < ½6(v,P)

so t h a t TcUuV.

for x 6 U A V w e h a v e

for some v6Q.

at a c o n t r a d i c t i o n .

6.3 P R O P O S I T I O N :

Let h6L°(m)

A+: = { a } U { z 6 ¢ : z + a is c o n n e c t e d .

It f o l l o w s

that

and a6~.

flog ~h-zdm~O}

and

open sets w i t h &+cUuV.

6.2 it suffices to p r o v e &+DV=~.

is d i s j o i n t

It f o l l o w s

that Vn&+=~.

6>0 such that the c l o s e d d i s k

To see this

~GAA+=~.

ii) A s s u m e

{z: I z - t I ~ } implies

T h e n 6.1 that t~A.

is d i s j o i n t that ~(h)NG=~

Assume

that

let G be a com-

in ~ and a in the e x t e r i o r

to V a n d to U and h e n c e

c a t i o n of 6.1 to G : = { z : I z - t l < 6 }

of G. N o w ~G

implies

that GDA+=~

Then t h e r e e x i s t s w i t h A. Thus a p p l i a n d in p a r t i c u l a r

QED.

After tion

T h e n the set

h-z and flog ~ : ~ dm>O}

p o n e n t of V. T h e n G is a d o m a i n

t~(h).

that UAV=¢.

QED.

i) Let U , V c ~ be d i s j o i n t

After

as well.

for

6(u,v)~6(u,x)+

uJ(h) c&.

Proof: a6U.

It re-

A n d the set

A := { a } U { z 6 ~ : z + a

satisfies

a6E and

$(u,x)<~6(u,Q)

+6 (v,x)<½6 (u,Q)+½6(v,p)<_½6 (u,v)+½6 (v~u)=6 (u,v) w h i c h p r o v e s We thus a r r i v e

p,QcT

Let us p u t

these preparations

(H,~).

For a function

sist of the c o m p l e x ~(h)6A(h)

numbers

so that A ( h ) ~ .

tions b e t w e e n

~(h)

above preparations.

we r e t u r n

to the r e d u c e d

h 6 H # we d e f i n e

Hardy algebra s p e c t r u m A(h)

z£~ such that h-z~(H#) x. T h e n

We w a n t

a n d A(h).

the L u m e r

Note

to p r o v e

several

results

situato c o n -

in p a r t i c u l a r on the r e l a -

that 6.4 a n d 6.5 do n o t d e p e n d on the

105

6.4 P R O P O S I T I O N :

Proof:

For each

i) We k n o w

us p u t ¢ - ~ ( h ) = P u Q

that

lh-sl~8>O. view

that

For

¢-~(h)={z6~:[h-zI~

some

6>O}

is open.

Let

a n d Q:={z{o(h):h1_-~{

Q is open.

z6~ w i t h

To see

Iz-s[ ~

that

then

H}.

P is o p e n

[h-z =2

let s6P w i t h

so t h a t

z~(h).

And

in

Iz-si
of h - s =2 w e o b t a i n

1

_

h-z

1

_

1

(h-s)-(z-s)

h-s

so that

z6P and h e n c e

we h a v e

s6&(h) ~h1-~s ~ H # ~ h 1 _ ~

is c l o s e d

and h e n c e

It f o l l o w s

that

in f a c t

Proof:

Assume

Assume

that

of p o i n t s

ICnl=1

such

that

Icl=1

c(h-a)6H + from

4.4

z-s

as well.

Therefore

in ~(h).

Z

(h_----~)6H,

ii)

is n o t

in

Qc&(h)

the c o n v e x

and hence

Qc(A(h)) °.

a n d e l h l 6 L #. T h e n

( c o n v ~ ( h ) ) °. T h e n

Vz 6 c o n v ~ ( h ) . and hence

which

hull.

tends + a. T a k e

Vz6conve(h)

for s6¢-~(h)

QED.

h 6 H # is n o n c o n s t a n t

denotes

Now

A(h)cQUm(h)=~-P

. We

can

Thus

1__h__ c(h-a) c ~+

there

complex assume

4.10.

c n + e.

that

Rec(h-a)~O. from

is a secn with

N o w we o b Thus

h-a6

(H#) ×

QED.

next

6.6

PROPOSITION:

on X. T h e n

result

is in the o p p o s i t e

Assume

for e a c h

We w a n t

of m.

is the

A(h)~.

a6~

and Rec(z-a)>O

The

Proof:

conv

~

Z 0

Furthermore

that

Re C n ( Z - a n) > O

tain

instead

H~s6Q.

oo

1

h-s

P is o p e n

a n ~ c o n y ~0(h) w h i c h

Then

or a~A(h).

_

is c o n t a i n e d

A(h) c (conv w(h)) °, w h e r e

quence

1

1_z-__{s h-s

A(h)cQU~(h).

8A(h)

6.5 P R O P O S I T I O N :

~(h)

SA(h)ce(h).

with

P:={z{~(h) : 16H}

It is o b v i o u s

hEH # we have

Let

to a p p l y

This

same

/lloglh-zll

6.3

is c o r r e c t

relative

z~A(h).

that

there

hEH ~ we have

Then

direction.

exists

to h and since

a:=~(h),

in v i e w

to Fm as r e l a t i v e h - z 6 (H#) x so t h a t

F d m < ~ and

an F 6 M J

which

is d o m i n a n t

~(h)cA(h) .

of

but

to the m e a s u r e

[F>O]=X

the v a l u e

to m. We c a n IV.3.5.i)

assume

implies

logl~(h)-z I = /(ioglh-zllFdm.

Fm

carrier that

that

106

Thus

we have

AcA(h).

h6L°(Fm).

Thus

6.7 P R O P O S I T I O N : connected

Proof:

In v i e w

z{A w h e r e

from

Assume

for e a c h

the m e a s u r e

And

~(h)cAcA(h)

6.3.

the

A is as in 6.3.

It f o l l o w s

that

QED.

Szeg~

situation

M={F}.

T h e n A(h)

is

h £ H #,

of

6.3 a p p l i e d

Fm i n s t e a d

to h 6 H # c L # = L ° ( F m )

of m we h a v e

to p r o v e

and

a:=~(h)

a n d to

that

(h) = {~(h) }U{z#~(h) : l o g [ ~ ( h ) - z [ < / ( l o g [ h - z [ ) F d m } .

But

this

The

is i m m e d i a t e

power

important

6.8

of the p r e s e n t

THEOREM:

6.9

the

Assume inner

of 6.8:

1.8

z~O

QED.

method

the

For

the

becomes

each

It f o l l o w s

Then

Szeg~

~(h)={q0(h)}

6.10 function 6.4.

complex h-z=u

that

z+~(h) h

z z

h~z-ZUz=hz

EXAMPLE: h6H.

that

and hence

and hence

as w e l l ) .

subsequent

M={F}.

L e t h 6 H # be s u c h

that

M={F}

and H~¢.

Then

there

with

we h a v e

and hence u =const z

ZUzfH

after

A(h)c{O,~(h)}.

h=const

~(h-z)~O

u 6H an i n n e r z

Now

(it is c l e a r

from

1.3. from

that

and h e n c e

function

and

the a s s u m p t i o n .

It f o l l o w s 6.7

6.6

that

a n d 6.6 w e o b alone

would

QED.

Let

us d e t e r m i n e

We h a v e

It f o l l o w s

But note

the

h=const.

situation

u -=-L~ 6 H so t h a t z u

h-z6(H#) x Vz~O,~(h)

suffice

with

in H.

z

tain

visible

situation

u6H.

functions

factorization

therefore

Szeg~

functions

COROLLARY:

hz6(H#)X. For

AssL~e inner

nonconstant

Proof from

1.7.

theorem.

h u 6 H # for all

exist

from

that

the

~(h)cS A(h)=D.

former

3.7

~(h)

and hence Hence

a n d A(h) A(h)cD

S=~A(h)c~(h)

is a s o m e w h a t

for a n o n c o n s t a n t

from

6.5

from

sharper

and

6.4

inner

~A(h)cS

so t h a t

from

w(h)=S.

statement.

Notes

Outer

functions

ture within

the

and

Szeg~

factorization situation

were

frame.

The

central

themes

introduction

in the of the

literaclass

H#

107

freed

the

theory

inclusions. zation tion

f r o m the

1.10 of H # in the

and result

The

It w a s

to it w a s ticular

Szeg~

theorem

the

LUMER

cases.

burden

form

of p e r m a n e n t

is in K O N I G

situation

labour with

[1967c].

is in K O N I G

The

Hp

characteri-

[1967a] (with d e f i n i -

reversed).

substitution

[1967a].

technical

1.3 in the p r e s e n t

reason

[1965] The

2.1 for

with

full

for the

the

theorem

the

Szeg~

situation

introduction

idea 2.1

of the

of s u b s t i t u t i o n and

the

is in K O N I G class

H #. A n t e r i o r

theorems

consequence

2.3

and p a r -

are

in K O N I G

[1967c].

The

class

3-5 w e r e volume

H+ was

announced

for the

theorem

[1976]. ration

with

HOFFMAN

The

Yabuta,

classical

The Lumer and

the

red

in K O N I G

sults

and

We

and

refer

of

was

6.5 w a s

in K O N I G

[1965].

independent

R. F r i t s c h .

For

the

proofs

6.7

elaborated

important

abstractions,

results

of S e c t i o n

presentation

announced

which

and

The

in c o o p e -

special

case

for e x a m p l e

in

[1967a].

introduced

[1967a][1967c] and

to L I f u n c t i o n s

contexts

distribution

[1973b].

[1974a][1974b][1975]

H + have been The

of S e c t i o n s

in the p r e s e n t

in Y A B U T A

is in Y A B U T A

class

the m a i n

to t h e i r

results

appear

is the v a l u e

result

[1973a][1973b].

in K O N I G

v e r s i o n s of

proofs

exception

research

of the

in v a r i o u s

spectrum

essential

lied with

of h i s

p.76

of the m a i n

The

The main

see Y A B U T A

before

[1925]o

and most

[1970a].

is the p r i n c i p a l

properties

[1962a]

stricted

time.

which

development

Several

4.10 appeared

GOROV

first

3.6=4.13

subsequent

introduced in K O N I G

(under

the n a m e

in L U M E R

in K O N I G

from our

of

[1965].

[1966b].

is u n e s s e n t i a l ) connectedness

5 are

in K A T Z N E L S O N

is one lemma

topological

in K O L M O -

[1968].

inner

spectrum)

Then

6.6

appea-

Theorem

6.8

(re-

of

the m a i n

6.2 we h a v e friends

T.tom

re-

been

supp-

Dieck

Chapter

The A b s t r a c t

In the

unit

tion which

disk

q 6 ReHarm(D)

such

the c o n j u g a t i o n define

is w h i c h

associates

some

how

For

with

each

functions

and

still the

this

tion

recall t£~o

e

were

it(G-Q)

one,

were

so t h a t

P6ReLI(I) such

each

P(s)dl(s)

V z6D.

such

6H#(D)

and hence

all h a v e

for all

to the a b s t r a c t

function

P6ReL(I)

ReLI(I) : In fact,

transit

t6~.

and

real

It is c l e a r

that

algebra

which

is c o n j u g a b l e

from et(P+iQ)6H#(D)Vt6~

classical

function

the

then

latter

the

of m o d u l u s Thus

for

is s u c h

definition

it f o l l o w s

Q6ReL(1)

corresponding

that

Furthermore

in the n e w

if

functions

to be the u n i q u e

this

condi-

In fact,

of H#(D)

the

To

et(p+iq)6Hol#(D)

G-Q=const.

that

situation.

the

function

constant:

elements

such

But

>O in the o r i g i n ,

Hardy

So let

D be a v o i d e d ?

t£~,

can be d e f i n e d t 6 ~ and

Then

also

t£~.

V t6~ and hence

values

of

non-

definition.

the c o n j u g a t e

for all

invertible

lots

from these

through

for all

that et(P+iG)£H#(D)

are

a boundary

to be

eP+iq6Hol#(D)

eit(G-Q)=const=c(t)

6HoI#(D)

and hence

up to an a d d i t i v e

a

f r o m D to S.

from

that

E of ReL(1)

is no o b v i o u s

p:=6ReHarm1(D).

et(P+iQ)6H#(D}

Q6ReL(1)

to e s c a p e

to reS: t h a t

it v i a P + i Q 6 H # ( D )

there

there

is o b t a i n e d

conjugate function # et(P+iQ)6H (D) for all

~olet(p+iQ)l>o extended

1.4.4

since

a not too n a r r o w

Q6ReL(1)

circle

subclass

And

to e x t e n d we h a v e

unit

to d e f i n e

definition

the a b o v e

the

that

functions

from

Therefore

determines

G6ReL(I)

can

situation

to fail

in o r d e r

preserve

we d e f i n e

But how

idea

function

In o r d e r

on the

in H#(D).

initial

q£ReHarm(D)

Of c o u r s e

of P 6 R e L I ( I ) .

for all

place

p+iq6Hol+(D)-Hol+(D)cHol#(D)

P+iQ6H#(D).

see

takes

immediate

(p+iq) (z) = S/ ~S+Z We h a v e

algebra

P6ReLI (i) we c a n f o r m

function

and q(O)=O.

P from a certain

The

is the o p e r a -

the u n i q u e

Hardy

E a n d H#(D)

to t r a n s p l a n t

conjugate

which each

conjugation

p 6 ReHarm(D)

of Q is b o u n d

real-valued

functions

seek

the c l a s s i c a l

each

abstract

Q6ReL(1).

to r e s t r i c t

Conjugation

p + i q 6 Hol(D)

normalization

constant us

that

to the

function

nonconstant idea

with

it as an o p e r a t i o n

unique plus

situation

associates

VI

that

c a n be we

sense

that

see

must

all

that

be in

these

109

functions tions

are

in

(H#(D)) x so t h a t

ht6(Hol#(D)) x have

ht(z)

= exp

It f o l l o w s

that

If S ~s+z Jz

P6ReLI(I).

that

etP=let(p+iQ)

unit

disk

After we

with

several is the

we

linear

of t h e m a i n

function

P£ReL(m)

boundedness the s a m e sense.

value

sharper

plications. have

for

Hereafter

those

we p r o v e

theorem:

E=ReLI(Fm)

and h e n c e

ReLY(m)

(the w e a k * D i r i c h l e t

Marcel

remainder Riesz

abstract

be d e d u c e d

derive

certain

The will

versions

relevant

be of i n t e r e s t

a separate

section.

not

then

The

too

has

from has

in the e x t e n d e d

which

is

immediate

so t h a t

the

here

a

but powerful

important

ReH

po-

is t h a t

fPVdm

situation

that

is t h e n full

far r e m o t e

integral

in E ~ in a s e n s e

has

the

im-

M={F}

we

approximation

is w e a k • d e n s e

a i m to e x t e n d

for c o n j u g a t e Kolmogorov

As to the M a r c e l results,

to i m p o s e conditions

in o t h e r

result

in t h e S z e g ~

estimations

particular

we have

result

result

situation.

from V.5.6.

additional

achievement the

E

~:

in

property).

of the c h a p t e r

sharp

The since

fundamental

algebra

or at l e a s t

is in E iff the

with that

functional

The principal

R e H c E ~ is d e n s e

out

It r e q u i r e s

a simple

and Kolmogorov

Hardy

course

hensive

the

IV.

chapter

combined

the

The main

of M.

As b e f o r e

The

It t u r n s

on w h i c h

linear.

E~=ReL~(m),

sharpens

conjugation

it e x i s t s

is c o n c e i v a b l e

In the

to do.

(X,I,m).

for w h i c h

then weak * density. This

on

for E ~ : = E n R e L ~ ( m )

that

V6M

so

to c o n c l u d e

P6ReLI (~).

EcReL(m).

the m e a n s

or o t h e ~

theorem

The

and

is b o u n d e d ,

sense all

abstract domain

of C h a p t e r

which

IV.4.1

we h a v e

(H,~)

of ReL(m)

of E w i t h

in s o m e

approximation much

of the

subspace

that

I=P t V t6~,

E=ReLI(I).

we k n o w w h a t

of its

theorems

to i n v o k e

situation

is f i n i t e - v a l u e d

the c h a r a c t e r i z a t i o n

func-

Pt6ReL1 (I).

with

V t6~ and hence have

algebra

the d e f i n i t i o n

largest

the c o r r e s p o n d i n g

tP=loglet(P+iQ)

faster

transcription

characterizations

ReL(m)÷[-~,~]

wer

be e v e n

thus

Hardy

V z6D

and hence

16L#=L°(~)

the a b o v e

fix a r e d u c e d

starts

Pt(s)dl(s))

It w o u l d

situation

i.4.7

the representations

loglhtl=

that

after

in o r d e r

rather

sharp

and their

contexts

as w e l l

to the

will

estimation

to o b t a i n

restrictions

mutual

classical

functions

estimation Riesz

the

more

be d e a l t

shall

compre-

upon

dependences

- will

of we

(H,~).

- which with

in

110 I. A R e p r e s e n t a t i o n

The subsequent theorem

Theorem

representation

of the n e x t section.

1.1 T H E O R E M :

Proof:

The u n i q u e n e s s proof

with

assertion

for t+O.

that h s h t =

Then t h e r e e x i s t s

is obvious.

differentiation

t~h t. We h a v e to face t e c h n i c a l

i) The

d i r e c t proof.

a unique

for all t6~.

is c a r e f u l

size of the f u n c t i o n

a simple

for the m a i n

lhtl =I for all t6~. A s s u m e

in L 1 ( m ) - n o r m

Q £ R e L(m) such that h t = e i t Q

existence

We p r e s e n t

Let ht6L~(m)

=hs+ t V s , t 6 ~ and ht÷1

t h e o r e m w i l l be n e e d e d

The b a s i c

idea of the

of the v e c t o r - v a l u e d

difficulties

function

due to the fact that the

Q £ R e L(m) to be c o n s t r u c t e d

is not r e s t r i c t e d

at all.

function

~ ÷ L I (m):t~h t is c o n t i n u o u s in L I (m)-norm. T h e r e f o r e t we c a n f o r m the e l e m e n t a r y integral H t : = f h u d U 6 L 1 (m) Vt£1~. F r o m t h e f u n d a o 1 mental theorem of calculus we h a v e _~ ( H s + t - H t ) + h t in L 1 (m)-norm for s÷O. In p a r t i c u l a r we o b t a i n numbers

1 H -I in L I (m)-norm s s

a sequence

of s u b s e t s

~ (n)+0 such that on each

for s÷O.

Thus

from the E g o r o v

E(r) 6 I w i t h E ( r ) + X fixed E(r)

theorem

and a s e q u e n c e

of

we have ~.~I~H£ (n~÷1, for n÷~

in L ~ (m)-norm. ii)

For f i x e d s 6 ~ the l i n e a r o p e r a t o r

It f o l l o w s

t hsH t = / h s h u d U o and hence iii)

L 1 ( m ) ÷ L 1 ( m ) :f~hs f is c o n t i n u o u s .

that

that

t t+s = /hs+udU = f hudU = Ht+s-Hs, o s

(hs-1)Ht=Ht+s-Hs-Ht=(ht-1)Hs

We c l a i m that t h e r e e x i s t s

for all s,t6~.

a unique

function

h t - 1 = i Q H t for all t6~.

In v i e w of i) the u n i q u e n e s s

To p r o v e

define

/He(n)

the e x i s t e n c e

for n s u f f i c i e n t l y

n. It is then i m m e d i a t e

large w h i c h

in v i e w of ii)

that t h e r e e x i s t s

such that Q I E ( r ) = Q r Vr~l. =(he(n)-1)Ht=iQHe(n)Ht

on e a c h E(r)

Thus h t - 1 = i Q H t on the w h o l e

such that

is i n d e p e n d e n t

a well- defined

F o r t 6 ~ it f o l l o w s

for n s u f f i c i e n t l y

Q6L(m)

a s s e r t i o n is obvious. the f u n c t i o n Q r : = [I( h e ( n ) - l ) /

on E(r)

that

large and h e n c e

of X. iv) The f u n c t i o n

Q6L(m)

function

from Q6L(m)

(ht-1)He(n) = that h t - 1 = i Q H t. obtained

in

111

iii) which

is real-valued. it follows

In fact, we have ht=h_t and hence Ht=-H_t , from

that ht-1=iQH t for all t6~. This implies

view of the uniqueness

assertion

in iii).

v) We claim that on each fixed E(r) we have ~(ht-1)÷iQ L1(m)-norm. I

In fact,

from

that Q=Q in

for t÷O in

i) ii) we see that for n fixed

I (ht-1) He (n) =t (He (n) +t-He (n)) -1Ht÷he (n) -I =iQHe(n)

in LI (m)-norm, sufficiently

and on E(r)

the function

for t+O

IH (n) I is ~ some 6>0 for n

large.

vi) We now prove that ht=e itQ for all t6~. For this purpose observe that the functions rem so that i)-v) Q~6L(m)

h~ :=hte-itQ Vt6~ fulfill

can be applied to them as well.

such that h[-1=iO~H[

~ Q ~ for t÷O in LI (m)-norm.

~(h~-1)

It follows

2. Definition

We return

Vt6~,

we have

~(h~-1~

But on each E(r) we have

that Q~=O.

of the abstract

For P 6 ReL(m)

for t÷O

Thus h~=1

or ht=e itQ for all t6%. QED.

Conjugation

to the reduced Hardy algebra

2.1 THEOREM:

of the theo-

Thus we obtain a unique

and on each fixed E~(r)

: ~(ht-1)e-itQ+~(e-itQ-1)÷0

in L I (m)-norm.

the assumptions

situation

the subsequent

(H,~).

properties

are equivalent.

i) a(etm)a(e-tP) = I Vt6~. ii) ~(P) 6 ~ iii)

and ~(tP) =t.e(P)

There exists

Vt6~.

a function Q 6 ReL(m)

In this case the function Q 6 ReL(m) constant.

Hence

there exists

~(et(p+iQ))=e te(P)

The function

such

that et(P+iQ)£H# V t6~.

is unique

a unique Q6ReL(m)

up to an additive

real

such that in addition

for all t£~.

class E is defined

to

consist

of the functions

P6ReL(m)

112 which possess the equivalent properties i)ii)iii) P 6 E are called conjugable.

in 2.1. The functions

For P6E the unique function Q6ReL(m)

such

that et(P+iQ)6H # and ~(et(P+iQ))=e ta(P) VtE~ is called the conjugate function of P and written Q=:P*. 2.2 LEMMA: Consider a sequence of functions UnEH# with

lUnl~ some

G6L # such that lim suplUni!1 and ~(Un) ~I. Then there exists a subsen÷~ quence Un(Z) which tends +I for £+~. Proof of 2.2: Take functions vz6H with vZ~I. we can assume that ~ ( v £ ) > 1 - ~ .

/lv~(Un-1)I2vdm

IviI~1, v£÷I and

Iv£1G~c Z

For VEM then

= ~1 {VzUn-1)+(1-v Z) ]2volta

2/]VzUn-11 2vdm + 2/ll-vzl2Vdm <

2/IV~Unt2Vdm + 2 -

4
Introduce now Gn:=Sup{lUsI:S~n}

n

) 9 4 - 4qo(vz)

and observe that

.

fun I
lim sup lUn]~1 for n ..... Then n÷~ lim sup ~Ivz(Un-1) ]2Vdm<8-8~(vi)< 8 n +~ = V

M Z > I. :

Thus there exists a sequence 1~n(1)<...
/Ivz(Un(~)-1)12Vdm<

78

V~,~I.

It follows that vi(Un(z)-1)÷O and hence that Un(1)÷1 Proof of 2.1:i) ~ii)

such that

for ~÷~. QED.

From V.I.2 we have O
VtE~ and hence

a(tP)6~ and a(tP)+e(-tP)=O Vt6~. Furthermore etP6L # V t6~. Now ~((s+t)P)<=~(sP)+~(tp)

and e(-(s+t)P)<~(-sP)+e(-tP) Vs,t6~,

SO that addition leads to a((s+t)P)=a(sP)+e(tP) show that the function t~e(tP)

Vs,t£~.

It remains to

is continuous on ~. But for tn÷t with

Itnl,ltl~c we have -clPl~ ~ tnP, ~tP with eclPI6L # and conclude from IV.3.12 that

113

e(tP) ~ l i m inf a(tnP) ~ l i m sup ~(tnP) n+~ n÷~ Hence e(tnP)÷~(tP)

=-lim inf ~(-tnP)~-~(-tP)=e(tP). n÷~

so that indeed t ~ ( t P )

is continuous

on ~• ii) ~ i )

is

obvious. iii) ~ i ) exists =e

is immediate

from V.I.1. i)&ii)=iii)

after V.I.3 a unique

-e(-tP)

=

ft6(H#) x with

eta(P) • It follows

follow if 1.1 can be applied have to prove

that ht÷1

Then /lht(n)-lldm÷e>O follows

that

ft(n(£))÷1

that fsft=fs+t

for a suitable

The assertion will

ht:=fte-tP

for t÷O. Assume sequence

Vt£~.

Thus we

that this is false.

t(n)÷O with O
Ift(n) I~elPI6L # so that from 2.2 we obtain a subsequence

for Z÷~. Thus ht(n(£))÷1

pointwise

so that we arrive at a contradiction. Q6ReL(m)

Vs,t£~.

to the functions

in L1(m)-norm

For each t6~ there

Iftl=e tP and ~(ft)=a(etP) =

and hence

in Ll(m)-norm

So we have obtained

a function

such that ft=et(P+iQ)6(H#) x and ~(ft)=~(et(P+iQ))=e te(P)

for

all t6~. The uniqueness

assertion

is seen as in the introduction:

is such that et(P+iG) 6H # Vt6~ then the functions hence invertible =c(t)

elements

of H # of modulus

V t6~ and hence G-Q=const.

We proceed

ii) EcReL(m)

so that eit(G-Q)=const =

QED.

i) For P6E we have e±P6L # and hence P£L #.

is a linear subspace.

iii) The conjugation iv) The restriction

E÷ReL(m) :P~P • is a linear operator. ~IE:E÷~

that TcReL(m)

is a positive

linear

is a linear subspace

such

and linear.

linear subspace

on which ~ is finite-valued

of ReL(m)

It remains

diate consequence

Then TeE.

functional.

tion ~IT is finite-valued

Proof:

are 6H # and

to collect a number of basic consequences.

2.3 PROPOSITION:

v) Assume

one,

e it(G-Q)

If G6ReL(m)

to show that ~IE is additive.

of ~ ( P + Q ) ~ ( P ) + ~ ( Q )

that the restric-

Thus E is the largest and linear.

But this is an imme-

VP,Q£E and ~ ( P ) + ~ ( - P ) = O V P6E. QED.

114

2.4

Let Pn6E(n=l,2,...)

PROPOSITION:

l e a s t o n e of the s u b s e q u e n t i) P n ÷ P p o i n t w i s e

ii)

and

conditions

and P 6 R e L ( m ) .

Assume

that at

be f u l f i l l e d .

IPnl ~ some G6ReL(m)

w i t h e G 6 L #.

~(IPn-Pl)÷0.

T h e n P6E and e ( P n ) + e ( P ) . Proof:

i) For t £ ~ we have - I t l G ~ ± t P n , ± t P .

that -~<e(±tP).

At f i r s t

IV.3.9

implies

Then from IV.3.12 we obtain

-~<~(tP) ~ l i m inf e(tPn) ~ l i m sup a(tP n) = - l i m i n f ~ ( - t P n) ~ - e ( - t P ) < ~ . n÷~ n÷~ n+~ It f o l l o w s 2.1

that ~ ( t P ) 6 ~ and ~ ( t P ) + e ( - t P ) = O

is f u l f i l l e d .

t=1.

Thus P6E.

ii) L e t t 6 ~ and

Furthermore

ItI~rE~.

V t 6 ~ so t h a t c o n d i t i o n

a(Pn)÷e(P)

f r o m the a b o v e

i) in for

Then

t P ~ t P n + It}IP-Pnl ! t P n + r l P - P n ! , e(tP) ~ e ( t P n ) Likewise

~(tPn) ~ ( t P )

n÷ ~ Vt6~.

follows.

is i m m e d i a t e

~(tP)6~

and ~ ( t P n ) ÷ ~ ( t P )

for

QED.

f r o m the s u b s t i t u t i o n

t h e o r e m V.2.1.

In p a r t i c u l a r

is a c e r t a i n

converse

to 2.5 w h i c h

w e can p r o v e

2.6 P R O P O S I T I O N :

the s u b s e q u e n t

Assume

is s o m e w h a t m o r e d e l i c a t e .

b u t we c a n n o t be sure that P ~ L # as well. assertions.

that P6E and P ~ 6 L ~ ~. T h e n P + i P ~ 6 H # a n d ~(P+iP~) =

=~(P) .

This with

is an i m m e d i a t e

IV.3.2

consequence

of the s u b s e q u e n t

and IV.3.3.

2.7 LEMMA:

ReHcE

for h = P + i Q 6 H .

For P6E w e h a v e P6L # from 2.4.i), However,

Thus

re(IP-Pnl)-

L e t h = P + i Q 6 H # such that e l h I 6 L #. T h e n P 6 E and Q = P ~ + I m ~ ( h ) .

and Q = P ~ + Imp(h) There

+ r~(]P-PnI).

The a s s e r t i o n

2.5 R E M A R K : This

+ ~(rIP-pnl ) ~ a ( t P n ) +

i) F o r z6~ w i t h

Re z ~ O

~fetZ-II < Izl t =

we h a v e

vt~o.

lemma combined

115

ii) For

z6~ we have

~letZ-ll ~ lle(T+E)z I + Proof: O~s~t.

Izl

vo
i) Apply the mean value theorem to the function

ii) We can assume

Now etX-l
Then P+iP~6H#and

Assume

follows.

that P6E is bounded below or bounded

that P~O. Then V.4.1

3.1 THEOREM:

Let P6ReL(m)

has the same value c6[-~,~]

Proof:

The asser-

sense.

final result.

with e±P6L #. Assume

that the integral

for all those V6M for which

it exists

/PVdm in the

Then P6E and e(P)=c6~.

We apply

IV.3.9

to tP with real t~O.

Thus c6~ and hence ~(tP)6~ Vt6~.

we obtain

can be applied.

of E with the means of M

In one direction we obtain an instant

~tc.

above.

QED.

3. Characterization

extended

QED°

~(P+iP~)=e(P).

We can assume

tion follows.

leitY-11.

and x<~(1+ex)
leity-II~21sint2~I~tly I . The assertion 2.8 PROPOSITION:

s~e sz in

that z=x+iy with x>=O. Then

letZ-11 = I ( e t X - 1 ) e i t y + ( e i t Y - 1 ) l ~ ( e t X - 1 ) +

Proof:

and ~>O.

~(tP)=tc

In the opposite

VtE~.

It follows

Then from ~(tP)~tc

that -~<~(tP)~ and ~(-tP)~-tc

QED.

direction

it would be most pleasant

to deduce

from

P6E that /IPIVdm<~

and

/PVdm

= e(P)

or at least for those V6M for which

for all V6M,

/PVdm exists

in the extended

But we cannot prove this unless we impose an additional dition upon PEE which appears tion e±P6L #.

sense.

boundedness

to be sharper than the implicated

con-

condi-

116

Let us start w i t h ties

as n o t e d

CONSEQUENCES:

3.2 J E N S E N

]IPIVdm<~

and e(P)=]PVdm,

[V>O]c[F>O]

such

Inf{/PVdm: Prior which

that

to the main

is in o b v i o u s

6>0.

Then

Assume

inequali-

i) For VEMJ we have exist

functions sense.

but i m p o r t a n t

such

that

VEM with

And

: all these

on the f u n c t i o n

o) Let P6E and V 6 M be

llPIVdm<~

V £ M}. assertion

class

H +.

/e-6Pvdm< ~ for

and /PVdm~e(P). below

then e ( P ) = S u p { / P V d m : V E M } .

then

a(P) = /PVdm

Let P6ReL~(m). for all VEM.

of 3.3:

a simple

to V.4.1

3.4 COROLLARY:

Then

o) The a s s u m p t i o n

for all VEM.

P 6 E ~ /PVdm has the same value

implies

that / P V d m < ~ .

From

for

IV.3.4

we

for o
so that

II.5.1

follows

from

above

that

-~(P)~/(-P)Vdm.

o) and 3.2.ii).

In the next quoted

implies

3.5 LEMMA:

a certain

The a s s e r t i o n

is then obvious. the s h a r p e n e d extension

follows,

I ~ [(tf_1)+]

and e+(cf)=ce+(f]

Inf{~((f-h) +] :h6ReL=(m)}<~+(f)

Vc>O.

i)

QED.

boundedness

of the basic

condition

theorem

Define + (f) := lim inf t+O

O<~+(f)=<~

ii)

lemma we i n t r o d u c e

and prove

]e-t(P+iP~)vdm,

I e-~(P) < (fe-tPvdm) ~ r

e-t~(P)~fe-tPvdm,

Thus

of the J e n s e n

in the e x t e n d e d

p o i n t we note relation

If P E E ~ : = E D R e L ~ ( m )

Proof

P6E.

VEM}~e(P) i S u p { / P V d m

all V 6 M ~ ( P ) = / P V d m

have

that

/PVdm exists

i) If P6E is b o u n d e d ii)

implications

ii) For F E M there

all these

3.3 P R O P O S I T I O N : some

the i m m e d i a t e

in IV.3.5.

V f E R e L (m) .

i) Then

< I n f { @ ((f-h) +] :h£ReL~(m) }.

IV.3.10.

117 ii) f bounded above ~e+(f) = O ~ e f £ L 8(f) = Sup ~I( t f ) t>O

#. iii) We have ¥ f6ReL(m) with ~+(-f)=O.

Proof: i) To prove the left estimation let On~((tf_1) + ) > n n_ ~ ( ( n + 1 ) (tf-1) +) ~[1-n~-~)~(~(tf-1) +)

Z(1-t) ~ ((f-~)+)z<1-t)~nf{~((f-h) +) :hCRe~ (m) }, so that the assertion follows. The second estimation is immediate from IV.3.9. ii) The first implication is obvious while the second follows from IV.3.13 and i). iii) Let f6ReL(m) with ~+(-f)=O.

~ @(tf)=@(f)

Then e-f6L # from ii)so t/nat -~<~(tf)

Vt>O from IV.3.9. Thus we have to prove that c:=Sup{~e(tf) :

t>O} is ~8(f). We can of course assume that c is finite. Note that e(tf)~ ~tc or e-tC~e-a(tf)=a(e -tf) Yt>O. I) Because of a+(-f)=O there exists a sequence O
that is lunI~1 and lUnI~e 1+t(n) f,

~(Un)=e-a((-t(n)f-1)+)>O,

and hence

I (~(Un))t(n)÷1.

From the second line t~n)log~(Un)÷O, so that the estimation t(--~log~(Un)~ ~ t -I~ ( ~ ( U n ) - 1 ) ~ O shows that t ( ~ ( ~ ( U n ) - 1 ) + O . Therefore Dn : = t ~n ) (~ (Un) e-t (n) c-I)

I (~ (Un) _I) e-t (n) c + e-tt(n) (n) c-I ÷ t(n) --C

And from the first line we see that I t(n) Unl (e-t (n) f-1 ) =< ef-.

This is obvious on [f>O] while on [f__
)=e(-f) =ef--

118

2) Let us fix a function V6M with /f-Vdm< ~. We have to prove that ffVdm~c.

Now for v6H with

Iv I= <e -t(n)f we obtain

• (Un) l~(v)l=I~(UnV)I ~ /lUnV[Vdm~ /lUnle-t(n) fvdm. It follows

that

~ ( U n ) e - t ( n ) c ~ ( u n) a [e -t(n)f] ~ /lUnle-t(n) fvdm, Dn~t(~ ~

f[lUnle-t(n)f-1]Vdm~t(--~

Thus the function Fn:=ef--t(~lUnl ~e]f-Vdm-D n. Furthermore elf-Vdm+~fVdm~eff-Vdm+c 3.6 THEOREM: with IP-Vdm<~}.

]lUnl [e-t(n)f-1]Vdm.

~-t(n)f-1]~O

Fn÷ef-+f~O. or IfVdm~c.

that

QED.

i) Let P6E with ~+(-P)=O.

Immediate

IFnVdm ~

Thus the Fatou theorem implies

Then ~(P)=8(P)=Sup{~PVdm:V£M

ii) Let P£E with e+(~P)=O. Then ~(p)=/PVdm ]PVdm exists in the extended sense. Proof:

has an integral

for all those V6M for which

from 3.5. QED.

3.7 COROLLARY: Let P6ReL(m) with a+(~P)=O. Then P 6 E ~ t h e integral fPVdm has the same value c6[-~,~] for all those V6M for which it exists in the extended sense. In this case ~(P)=c6~. 3.8 SPECIAL CASE: Let P£ReL(m) of functions

Pn6ReL~(m)

and assume that there exists a sequence

such that

e(IP-Pnl) : sup ]]P-PnlV~÷OV£M Then e+(~P)=0 and hence e~P6L #. Furthermore /IPIVdm~9(IPl)< ~ VV6M. Therefore P 6 E ~ /PVdm has the same value for all V 6 M ~ e ( P ) = / P V d m for all V6M. Proof: We have to prove that ~+(~P)=O. But for g=~l we have 8((eP-ePn)+~ ~8(IP-Pnl) so that 3.5.i) implies that ~+(~P)=O. QED. We conclude the section with a remarkable Szeg~ situation.

characterization

of the

119 3.9 THEOREM:

For F 6 M the s u b s e q u e n t

properties

are equivalent.

i) M = {F}, ii)

E = ReLI(Fm),

iii)

ReL~(m)cE.

Proof: hence

i) ~ i i )

P6E from

and hence implies

For PEReLI (Fm) we have

3.1.

On the other

PEReLI(Fm).

that

ii) ~ i i i )

~P(V-F)dm=O

such

is trivial,

and

that e~PEL#=L°(Fm)

iii) ~ i )

and hence

Let V6M.

Then

that V=F.

QED.

3.4

Theorem

THEOREM:

=P+iP*6H # and ~(h)=~(P).

e ~ P E L ° ( F m ) = L # from IV.4.1

PEE implies

for all PEReL~(m)

4. The basic A p p r o x i m a t i o n

4.1 A P P R O X I M A T I O N

hand

Assume

Then

for

that P6E

each

:=EDReL

¢>0 t h e r e

exists

(m), so that h:= a function

h 6H

that Ih

i)

t~lhl

and

IRehel ~ I R e h !

= IPI,

lh-hi £EMax(1,1hll+~),

ii)

iii)

~(h ) is real

Proof: hence

and ] ~ ( h ) - ~ ( h ) l ~ E .

I) For s = u + i v 6 ~

with

11-s21~1-u2+v2~1-u2>O.

<

lsl

lul<1 we have

It follows

1-s2=l-u2+v2-2iuv

and

that

Isi2,

<

= 1-u2+v 2 = 1-u

I

eS

1

1-s 2

=

eS
11-s212

<

lul

=

11_S21

<

= lu-ulsl21 Ii-s212

lul

lu!

=~ 1_U2+V 2 ~ 1_u 2"

2) Let us fix c~I w i t h

IPI~c.

we see that e - t ( c ~ h ) 6 H

Vt~O and hence

from V.4.1.

= luill-lsl21 I1-s212

Then

take

I

I - (~h)

(oc) 20

oc
c~hEH +. Thus

Let us put

fa:-- (I-(oc)2)

~>O with

2

I 1-~c+~(c~h)

1 = I_--~-~6H

120

Then

fd6H and

=

2d

It is i m m e d i a t e

I-~o (h)

from

(l+o~p h~"

i_ (oqg(h)) 2 "

I) that

Ifol =< 0-(~c) 2]

lhT 1 - (dP)

IRef(~I < (1-(qe) 2) =

Furthermore

,, IPl 1 -

q0(fd) is real

for o+O. But in order

lh; '

< IPI

(qP)

2

and ~(fo)+~(h)

to prove

2 =<

.....

"

for d+O. Also

ii) we need

a more

f0÷h p o i n t w i s e

elaborate

estimation

of Ifo-hl. 3) Let us fix 0
from

For

~>0 w i t h

oc
h(1-(oc)2)-h(1-(oh)

then

2) _ o2h(h2-c2)

1-(oh) 2

1-(oh) 2

I) we o b t a i n 2-c

Ifo-hl

2(~2 (Max.!c, lhl ) _<_ I- (oP) 2+ (op~) 2

_< 2oe(Max(c, lhl)

< 2°e(Max(c, lhl)] I+£ d2-e(c2+(p~)2)9 2 I- (dc) ~+ (oP ~) 2 I+~

(oc) 2-e+ (oP*) 2-s I-(OC)2+(jP~) 2

< 20~ (Max (c, lhl) 1- (oc) 2

1+e

+

(I- (dc) 2) c

I+E < 4d C(Max(c, lhl) 1- (dC) 2

For the third

estimation

2 £(Max(c, lhl))l+e

we used

4dCc1+e - - ~

the fifth

estimation

uv < t p -up - + =

which

we know

p

I

vq

tq

q

~ax(1, lhl 1+e) .

the c a l c u l u s

is based

from the p r o o f

I- (OC) 2+ (op~) 2

1-(oe)

(u+v) ~ __< M a x ( l , 2 e-l) (ue+v ~) while

(l_(oc) 2 ) e ( o p ~ ) 2 - e

V U,v~O

Vu,v>O

inequality and ~>O,

on the i n e q u a l i t y and t>O for c o n j u g a t e

of II.I.3,

for the c o n j u g a t e

1
2

121

a n d v2--~2~" _ It is n o w c l e a r t h a t we can take h s : = f o for ~>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<~

and / I v I 2 F d m < ~

in v i e w of the F a t o u

3) N O W f r o m h - / h F d m ± HF we c o n c l u d e

that

theorem.

123

f(h-fhFdmOn

the

wise

other

hand

and w i t h

of 2).

Un) (u n - iv n ) F d m

= -~(Un) (~(u n ) - i ~ ( v n)) ÷ O .

the i n t e g r a n d

the m a j o r a n t

It follows

converges

+(h-fhFdm-u)

(lhI+IfhFdmI+lul)

(u-iv)

(lul+Ivl)6L1(Fm)

pointin v i e w

that

0 = f (h-fhFdm-u) (u-iv)Fdrn = f ( ( P + i Q ) - ( P + i P * ) ) ( ( P + i P * ) - i ( O + i Q * ) ) F d m = fi(Q-P*)((P+Q*)-i(Q-p*))Fdm Thus we o b t a i n

Q = P*. T h e r e f o r e

= f(Q-P*)2Fdm

+ if(Q-P*)(p+Q*)Fdm.

h=fhFdm+P+iP*=fhFdm+u

and hence

h6H#N

gL~ (m) =H. QED. It is a p l e a s a n t characterization 4.7 COROLLARY: the s u b s e q u e n t

consequence

that

the functions

via m u l t i p l i c a t i v i t y Assume

under

that F 6 M is d o m i n a n t

properties

in H admit

integration

a simple

as follows.

over X. Then

for fEL~(m)

are equivalent.

i) f6H. ii)

f(f+u)2Vdm

iii)

/f2Vdm

=

=

(f(f+u)Vdm) 2 for all u6H and V6M.

(ffVdm) 2 for all V6M and f f u F d m =

ffFdm/uFdm

for all

u6H. Proof:

i) ~ i i )

cond a s s u m p t i o n that

and ii) ~ i i i ) shows

implies

The n e x t ponds

are obvious.

f ± H~F.

Thus

after

f I N. But for U , V 6 M and O ~ t ~ I we have (1-t)ff2Udm

This

that

at once result

+ tff2Vdm

that

(I-t)U+tV6M

((1-t)ffUdm

=

(ffUdm + tff(v-U)dm) 2.

ff(V-U)dm=O.

Hence

is the r e f o r m u l a t i o n

K = HF + N + iN L1(m)

t

The seto p r o v e

and hence

+ tffVdm) 2

fiN.

of t h e o r e m

We have

iii).

4.5 it r e m a i n s

=

to 4.4.

4.8 CONSEQUENCE:

So a s s u m e

QED. 4.5 w h i c h

corres-

124

whenever

F6M

Proof: tion and

is d o m i n a n t

The

h6L~(m) hIN.

inclusion which

Then

h annihilates L1 (m) the

m is o b v i o u s .

implies

the

that

follows.

with

still

In o r d e r

second

h6H

S i n c e tbe s e c o n d

assertion

We c o n c l u d e

X.

annihilates

4.5 K.

over

to p r o v e

member.

and ~ ( h ) = / h F d m = O .

member

is

= take

This means

a closed

a func-

that

h±HF

It f o l l o w s

that

linear

subspace

of

QED.

another

characterization

of the S z e g ~

situa-

tion.

4.9 T H E O R E M :

i) M = ii)

the

subsequent

properties

are

equivalent.

ReL=(m)

(the w e a k * D i r i c h l e t

property).

K n ReLI (m) = ~F.

Proof:

4.10 of the

F6M

[F}.

~-~weak~=

iii)

based

For

i) ~ i i )

REMARK: Szeg~

upon

is c l e a r

from

It is n a t u r a l

situation

4.5 and

4.8.

4.3,

i) ~ i i i )

to ask w h e t h e r

on the b a s i s The

and

answer

of 4.3 is no:

is c l e a r

the

and

above

4.4.

QED.

characterization

4.4 has

In the

from

a counterpart

Szeg~

situation

M={F}

we o b t a i n

H =

(H~F)

and these

two p r o p e r t i e s

from

one

them

cannot

and

are

~-~L 1 (m)

K =

seen

conclude

to be e q u i v a l e n t

that M = { F }

as the

for e a c h

subsequent

F6M.

But

example

will

show.

4.11

EXAMPLE:

as d i s c u s s e d

6 Pos(X,Baire). consider

H:=

We s t a r t

in IV.3.15. Let

the Hardy

from We

~:~(f)

the u s u a l some

(H~(D),~z)

O
us w r i t e f°:=fIS 6 L(1) algebra

{f6L=(m) : fQ6 H~(D)

= { (u,

(a)

fix

situation

and

f(a)

for

(H,~)

=(a)

= ff°d~

f6L(m).

defined

= / u P ( a , . ) d l ) :u6H ~(D) },

= ~ o ( f °) = (O)

for

zED on

and p u t X : = S U { a }

V f6H.

On

(S,Baire,l)

and m:=l+6a6

(X,Baire,m)

to be

= /fCP(a,')dl}

we

125 We have

the

subsequent

I) It is clear we k n o w

that

=(u,
(a)) /fGdm

Note

properties.

that

G>O

Xs6M.

Furthermore

from S e c t i o n

I.I

I -a I -a G:=(I-I~P(a,'),I-- ~ )

initial

remark

In fact,

where

for f=

6 H we have

= /ull-

1-a 1~aP(a, • )Idl +
(a)1-1-a ~ = /udl = /f°dl

= ~(f).

that G6M is dominant.

2) M = {(I-t)Xs

+ tG:O
6L I (m) and Yu6H~(D)

= f uV~dl Hence V ' + V ( a ) P ( a , . ) = 1 . =V ( a ) ~ < l = -

In order

to see c let V c M so that O<=V6

we have

%Oo(U) = ~ { ( u , < u l > ( a ) ) }

= /(u,
(a))Vdm

+
(a)V(a) It follows

= fu[V°+V(a)P(a,.)Idl.

that V ( a ) P ( a , - ) < 1

and hence O<=t:=

" Then V° = I - V (a) P (a, • ) = (l-t) +t (1-1-ap l+a (a,')) =((I-t)Xs+tG)"

hence V = ( I - t ) X s + t G

and %0(f)=~o(f° ) Yf6H #. This

4) One c o m p u t e s

H#={f6L(m):f'£H # (D) and

is immediate.

that (I-Z) 2 (a-Z) (1-aZ)"

G° = a Hence I/G°6H#(D)

and

from the d e f i n i t i o n s .

3) L#={f6L(m):f°6L# (Ha(D))=L°(I) }. F u r t h e r m o r e f(a)=kOa(f~)}

that

i).

6M,

and q0a(I/G°)=0

in view of V.4.4

and V.4.10.

Thus

3) shows

(I/Go,O)6H #.

5) For f6K we have have

f/G6H # and ~0(f/G)=/fdm.

~o(U) ffdm = ~ [ ( u , < u l > ( 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, fn÷1

and fnf/GEH.

It fol-

lows that f fEHG and f f÷f in L I (m)-norm so that fELI (m)-norm c l o s u r e n n (HG). T h u s w e h a v e f u l f i l l e d the p r o m i s e of 4.10. 7) M o r e o v e r real-valued but also

the p r e s e n t

situation

h6H # and d o m i n a n t

for c e r t a i n

since otherwise a f t e r V.2.3.

p>1,

furnishes

examples

namely

for 1
This

lh[6L1 (Gm)~e lh [ £ L ° ( G m ) c L # w o u l d

One e x a m p l e

is h = ( I / G ~ , O ) 6 H # a f t e r

/lhlPGdm

of n o n c o n s t a n t

G 6 M such that / l h l P G d m < ~

P-ld

=

not o n l y for p=1

implies enforce

that L ° ( G m ) ~ L # that h=const

4) s i n c e

aP-1 2p-2 dt < ~

for I
2za-P-l_~ [ [l _ e i t I) Another

example

•

,I+Z.2

is n=t~_7)

which

is 6H" in v i e w of V . 4 . 4

and V.4.10

as above.

5. T h e M a r c e l

We r e t u r n

Riesz

and K o l m o g o r o v

to the c o n j u g a t i o n

s e n t s e c t i o n we s t a r t to e x t e n d gorov estimations. additional

We s t a r t w i t h from V.5.6

operator

that for h£H + we have

:P~P*.

Marcel

w i l l be c o n t i n u e d

on the H a r d y

the K o l m o g o r o v

E÷ReL(m)

the c l a s s i c a l

The discussion

assumptions

Estimations

algebra

estimation.

Riesz

In the p r e and K o l m o -

in S e c t i o n

situation

7 under

(H,~).

Let us fix O
Recall

127

T'fr

COS-~-- / lhlYVdm < Re(~(h)) T T~ cos-~@

In p a r t i c u l a r =~(P)=6(P)

if O < P 6 E

Y (lhl)

< Re(~0(h)) T

then P + i P ~ 6 H + f r o m the d e f i n i t i o n s

f r o m 2.8 a n d 3.3.i).

It f o l l o w s

cos TTZ e (Ip+iP* 1~) <_ (~ (P) ] ~ Our a i m is to r e m o v e

VV6M,

the r e s t r i c t i o n

(~ (p)] ~

=

a n d q0(P+iP~) =

that

P > O at l e a s t

V O
functions

P6E m.

5.1 T H E O R E M :

F o r O
cos~e Proof:

we h a v e

(Ip+ip*l ~] < 21-~(0(IpI)) ~

We m a k e e s s e n t i a l

VpeE%

use of the f u n d a m e n t a l

fact IV.2.5.

Let

f 6 R e H w i t h f >IPI. T h e n f t P 6 E ~ a n d > O , and hence ~(f)±a(P)=a(f±P)>O. + h-:=(f_+P)+i(f+_P)~=(f+if ~)± (P+iP ~) w e k n o w that

cos-~-e (lh-+l ~) a

Now P+iP~=½(h+-h-).

(P)

With repeated

•

use of the c a l c u l u s

inequality

f r o m the p r o o f of 4.1 we thus o b t a i n

Ip+iP*l ~ =< ~(lh+l+l h-l]~< =

cos T

0 (IP+iP*IT)

I

< 7

l(lh+l~+lh-l~ ], 2T

COS T• ~

(e(lh+l<)+O(lh-l~))

I

<2-7 ! But

from

IV.2.5

and

IV.3.10

Inf{~(f) : f6 ReH w i t h

f~

w e see that

IPI} = Inf{Re<0(u) : u 6 H

with

= ~°(IPl) = 0(IPb). The a s s e r t i o n

follows.

QED.

For

Reu>IP

I}

(~)

128

Let us turn to the Marcel Riesz estimation.

For 1 ~ < = and V£M the

estimation in question assumes the subsequent forms which for each O~R<__~ will be seen to be equivalent. oo

(-)

IIP*IILP(Vm) ~ R IIPII LP (Vm)

V P6E ,

(o)

IIImhlILP(Vm~< RIIRe hIILP(Vm)

V h 6 H with Imp(h) = O,

(#)

IIImhI~P(vm ~ RIIRehIILP(Vm)

V h £ H # with I m ~ h )

The implications

(=) ~ (0) and

= 0 and/lhlPVdm <~.

(#) = (0) are obvious. The implication

(O) ~ (~) is an immediate consequence from the basic approximation theorem 4.1 combined with the Fatou theorem, and the implication

(O) ~ (#)

is clear after the definition of H #. We define O~R(p,V)~ ~ to be the smallest constant O ~ R ~

for which the above estimations

(~) (0) (#) hold

true. 5.2 REMARK:

i) For each V6M we have either R(p,V)=O VI<__p<~ or

R(p,V)~I V I ~ < ~ .

In fact, if ul [V>O]=const=~(u)

Vu6H then R(p,V)=O

V1<__p<~. Otherwise take some u6H with ~(u)=O and u![V>O]~O and apply (0) to u and iu. ii) For each V6M we have R(2,V)~I

from 4.6 and hence

R(2,V)=O or=1. 5.3 REMARK: i) If I ~ < ~ and VEM is dominant such that there exists some nonconstant real-valued h6H # with /lhlPVdm< ~ then R(p,V)==. fact, if R(p,V)< ~ and h6H # is real-valued with /lhlPvdm< ~ then applied to i(h-Re~(h))

enforces that h=Re~(h)=const, 4.11 we thus have R(p,G)== V I ~ < ~ .3 5.4 PROPOSITION:

ii) In the example

For each V6M we have R(np,V)~6nR(p,V)

n£~. In particular R(2n,V)~6n<~ Vn£~

In

(#)

V I ~ < ~ and

(this estimation will be improved

in 5.14). Proof: We can assume that I~R(p,V)<~.

i) We fix h=P+iQ6H with ~(h)=O.

Then I e l n ~ lhl n : li]-nhnl ~ IRe(i]-nh n)

[IQ In[ILp (vm)


+ IILp (Vm)

+ IIm(i]-nhn) I, Im(il-nhn)I [ < L p (vm) =

=< 2R(p'V)IIRe(il-nhn) [ P(Vm)

.

129

NOW we have i 1-nhn =

~ x.1-£rn, ( ~ ) p i ^U n - i, Z=O

Re(i 1-nhn) =

n )p2k+lQn-1-2k ~ (-I)k [2k+I k>O 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__
We claim that B<3nR(p,V)A. For the proof we can assume that B>A>=O and =~ n-1 I n>2. Let us put O<x: <1 and O<S: 2n R(p,V-------~
x f(1+t)n-ldt -x

x <_R(p,V)n fe(n-1)tdt = R(p,V)n--~1(e(n-1)X-e-(n-1)x), -x [e(n-1)x)2-1>2se(n-1)x (n-1)x>log(S+ I + ~ S2) = ~ -

or e ( n - 1 ) X > s + !

0 1~+t ~

SO that 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

1
p=2n(n=l,2...)

_

<ePS(Max(1, lhl s)). The assertion

5.6 THEOREM:

results

[h -hlP<

= p

<~P(Max(1, lhl)) (1+~)P<~P(Max(1, lhl))s=~PMax(1, lhl s) and hence

We turn to the central

and is

ii) Fix 1<_p<s< ~. For 0<~ < s - I then

follows

=

8(lhe-hlP)<

from i).QED.

of the section.

Assume that H contains

nonconstant

inner functions.

For each V6M then R(p,V) In particular

For each V6MJ we have

R(p,V) We combine

V l<__p<~.

R(I,V)=~.

5.7 THEOREM:

subsequent

~ Max(tan~p,COtan~p)

< Max(tan~,cotan~)

these results with V.6.9

particular

5.8 THEOREM:

and Section

IV.4 to obtain the

theorem.

Assume

R(p,F)

V I ~ < ~.

the Szeg~ situation

= Max(tan~p,

cotanTp)

M={F}

and H#~. Then

v I ~ < ~.

Proof of 5.6: We fix an inner function u6H with ~(u)=O which exists after V.3.5.i).

Then h : = ~1-u 6

have

VV6M

flhlVdm=~

H + with Re h = O

and ~(h)=1

since flh[Vdm < ~ implies

=~hVdm and hence Re~(h)=O which contradicts T T the main branch h p is 6 H + with ~(hP)=1

after V.4.2.

after IV.3.4

~(h)=1.

after V.4.9.

We

that ~(h) =

For I ~ < ~ and O
131

h =

h lhle l~sgn(Y)

~ h~ =

and hence

! ~sgn(~) lh[ p e T

A n d the e s t i m a t i o n

V.5.6

the F a t o u

lemma we have T

apply

to hp to obtain

(#)

shows

that

flhlTVdm÷~

/lhPlPvdm = /lhlTVdm< for T+I.

and hence

apply

(#)

I

p < R(p,V)(cos

sin

-~--~R(p,V) cos-~ T to i(hP-1) to obtain

for

~)(flhlYVdm)

T+I.

ii)

p

VV6M,

For the

cotan

T II (cos ~ ) l h l

and hence

p - 11 P(Vm)

damental the Green

C,(~),

L,(~),

formula For

with

P(vm)"

lemmata.

We define

support.

ca,(~), .... We start with formula

which

V V6M,

QED.

compact

f6C2(~)

C~(~)

to con-

In the same

sense

the s u b s e q u e n t

has a s t a n d a r d

proof

fun-

based

on

we have

= 2~ /loglz-tIAf(t)dL(t)

G£ReC(~)

is d e f i n e d

is iff the d i s t r i b u t i o n a l

on ~. In the case G£ReC2(~)

¥z6¢.

to be s u b h a r m o n i c

/G (z)Af (z)dL(z) > 0 that

~)II lhlPl

A.3.2.

f(z) The f u n c t i o n

several

6C~(~)

representation

5.9 LEMMA:

=< R ( p , V ) ( s i n

sln~-c_ zp for Y+1.

of 5.7 r e q u i r e s

sist of the functions we d e f i n e

T

cos z~-z
The p r o o f

estimation

I 1 = (]lhlTVdm) p < I + R(p,V) (sin ~p) (/lhlYVdm) p

TIT

(COS ~ )

From

i) For the tan e s t i m a t i o n

I (sin ~ ) ( / l h l T V d m )

~VV6M.

iff

V O~f6C, (~) ,

derivative

this m e a n s

AG is a Radon m e a s u r e

that the usual

> O

derivative

AG is

~ O on ¢ . 5.10 LEMMA:

Assume

that G6ReC(~)

G(~0Cu)) < ] G ( u ) V d m

is subharmonic.

Vu6H.

For each V 6 M J

then

132

Proof of 5.10: i) We first consider a subharmonic We fix R>O and F6C~(~) with F(z)=1

VIzieR.

function G6ReC2(~).

From 5.9 applied to GF6C~(~)

we obtain / loglz_tlA(GF)(t)dL(t), I { R) ioglz_tiAG(t)dL(t ) + I G(z) = 2--~V 2 ~ - V (R) with V(R) :={z6~: IzI
VzEV(R),

a harmonic function on Let now u£H with luI~

~c
I lloglz-tl]~G(t)dL(t)

llog]z-t[IdL(t)

< const

] V(R)

=< const

/ llogltIIdL(t) V(2R)

V(R)

Vz6V(R),

so that we can apply the Fubini theorem to obtain for V6MJ the desired estimation

/G(u)Vdm = ~ ]

+ Re/h(u)Vdm

( / loglu-tl~G(t)dL(t))Vdm V(R)

I

~R (/l°glu-tlVdm]AG(t)dL(t)

= 2-~v )

I /R l°glq°(u)-tIAG(t)dL(t) > 2--~V )

+ Re%0(h(u))

+ Reh(q0(u))=

S(q0(u)).

ii) Consider now a subharmonic function GEReC(~). We fix some O<__F6C,(~) with fF(z)dL(z)=1 and put F :Fe(z)=s-2F(Z) Vz6~. We form the convolution G e = G*F :Ge(z) = /G(z-t)F£(t)dL(t) Then GeEC~(~)

and is subharmonic AGe(z)

= /G(t)F

(z-t)dL(t)

Vz6~.

since

= fG(t)AFe(z-t)dL(t)

Moreover GE+G uniformly on each compact set

~ O

V z6~.

c ~ for e+O. Thus for V£MJ

and u6H we have Ge(<0(u))
ftan ~

for 1
[cotan ~p for 2 ~ < ~ ]

] > 1.

133 In v i e w of 5.10 the p r o o f blished

the s u b s e q u e n t

5.11LEMMA: G6ReC(~)

of 5.7 w i l l be c o m p l e t e

o n c e we h a v e e s t a -

lemma.

For e a c h

1
a subharmonic

function

such that

(,)

G(z)

< ApP l x l P - l y l p

=

V ~=x+iyC¢,

and G ( z ) > O V z 6 ~ . P r o o f of 5.11 ~ 5.7: We c a n a s s u m e Im~(u)=O we obtain

from 5.11

(APlIRe u llLp P

)P - (llImu

(Vm)

F o r V E M J and u6H w i t h

)P

> /G(u)Vdm

T h e p r o o f of 5.11 d e p e n d s For

1
IILP(vm)

= /(APlReulP-IImulP)Vdm

lities.

that

and 5.10

on c e r t a i n

> G(q)(u)) > 0. QED. sophisticated

calculus

1 < p < = we d e f i n e (sin2~) P-I for

I
cos Bp:

> O.

= (cOS~p) p- I for 2 ~ < ~ sin

The inequalities 5.12 LEMMA: B

ii)

p

in q u e s t i o n

are as follows.

i) In the case

I
cospt

< AP(cost) p - (sint) p =

¥ O
In the case 2_<_p<~ w e h a v e

-BpCOSp(t-)

This

~p

implies

=< A P-( c ops t ) p

(sint)P

V 2~- p =~ < t < = 5~"

(sint) p

V O
that B p =< A

p

(cost) p -

inequa-

134

s i n c e the e s t i m a t i o n notone decreasing

is m o -

in O ~ t ~ 2 .

P r o o f of 5.12: assertion

is true for t = ~ - ~ and the s e c o n d m e m b e r

For p=2 we h a v e A_=Ip and Bp=1 so that b o t h t i m e s the 2 2 -(sint) V O <= t =< and h e n c e is o b v i o u s .

r e a d s cos 2t (
T h u s we can a s s u m e

that p~2. M o r e o v e r

lities

in the r e s p e c t i v e

in q u e s t i o n

i) In the case

1
it s u f f i c e s

the i n e q u a -

the f u n c t i o n

(sin t) P + B p C O S pt F:F(t)

to p r o v e

open intervals.

T V O
= (cos t) p

It is C ~ w i t h F(2~) = (tan~p) p= A pp" The a s s e r t i o n One c o m p u t e s

that

F' (t) = p (sint)P-1 (cos t) p+1 f'(t)

=

(p-l)

(1-Bpf(t))

sin(2-p)t (sint) p

We see that f' (t)>O in O0 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~-~<

t < ~. One c o m p u t e s

1-Bpf and h e n c e F' are increases

in ]0,2~]

~p )P=A pP.

~ :T ---< t < 2 p 2"

The a s s e r t i o n

is F ( t ) < F ( 2 -

(s in t) p- I

cos (p-1) (t-2) (1-Bpf(t))

p

~)

that

= F' (t)

and

z

V

w i t h F ( 3~- ~ )~ =(cotan

1-Bpf

the f u n c t i o n

(cost) p It is C

and h e n c e

follows.

(sin t) P - B ~ c ° s P (t-2) F:F(t)

= sin(p-1)t (sint) p-I '

V O
Thus

]0,2 [ . N o w

and <0 in ] ~ , ~ [ ,

strictly d e c r e a s e s ii)

is F ( t ) <~=F ( ~zp -z ) V O < t < ~2"

with

f:f(t)=

(cost)P+ I sin(p-2) (t-2)

f' (t) = -(p-l)

, (sint)P_ I

~_~ < t < ~ V ~ P 2"

(sint) p We see that f' (t)>O in ~ - ~ 1-Bpf strictly d e c r e a s e s

< t < 2" Thus

in ] 2 - p '~ 2 ~ [" N o w

f strictly

increases

I- Bp f (~2 - ~ )~

and h e n c e

=0 " Thus

1-Bpf

135

and hence F' are

>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 I
G(z) : =

and

Itl~2 then

BplzlPcospt = BpRe(z p)

(the main branch),

and if Re z ~ O then G(z) :=G(-z). Note that if Re z = O then z=-~ so that G(z) is well-defined. tinuous.

We obtain the subsequent properties.

I) G is con-

2) G is harmonic in the open halfplanes ~:Re z >O and -4: Re z
3) For all z=x+iy 6 ~ we have G(x+iy) = G(-x+iy) = G(x-iy).

4) A standard

application of the Green formula A.3.2 combined with I) - 3) leads to /G(z)~f(z)dL(z)

= 2pBp sin 2P~ /ItlP-lf(it)dt

Thus G is subharmonic.

5) In order to prove

v fEC~(~).

(~) we can in view of 3)

restrict ourselves to z=x+iY=Izle it with x,y~O and hence O ~ t ~ .

But

then 5.12 shows that G(z) = B p l z l P c o s p t <

AP(Izlcost) p - (Izlsint) p = ApPlxlP-lyl p.

6) For z 6 ~ we have G(z)=G(Izl )=Bplzlp>O" ii) In the case 2 ~ < ~

we define G:$÷~ for z=Izleit£$ as follows: if

Imz=Re z > O and O
{

- B p l z ! P c o s p ( t - ~ ) = - B p R e ( ~ ) P ( t h e main branch)

G(z) : =

BplZl p

for It-21 ~ I for

'

It-21 ~ j

and if Im z< O then G(z) :=G(z). Note that G(z) is well-defined in all cases. We obtain the subsequent properties.

I) G is continuous. 2) G is and is C ~ P

harmonic in the open double cone U:z=Izleit+o with IItl-~l <

with ~G(z)=p2BplZl p-2 in the open double cone V:z=Izleit+o with <~ 3) For all z=x+iy 6 ~ we have G(x+iy)=G(x-iy)=G(-x+iy). Z
136

leads to

fG(z)~f(~)dL(~)=p2Bp~IZlP-2f(z)dL(~) v f~c~{¢) Thus G is subharmonic.

5) In order to prove

restrict ourselves to z = x +

iy =

(*) we can in view of 3) zle it with x,y > O and hence O ~< t ~<2Z"

But then 5.12 shows that

2

~<

<~

p=t=~:

G(z

=-Bp[zlPcosp(t-~)

<

< AP(Izlcost) p- (Izlsint)P=APlxlP-lyl p,

O~t~-~:

G(z

= BplZl p p- (Izlsin t) p=A~Ixlp-lyl p. =< AP(Izlcost) p

6) For z65 we have G(z)=BplzlP~o.

QED.

At this point the proof of 5.7 is complete. The subsequent results are important by-products of the above discussion. 5.13 THEOREM: Assume that V6M.

i) In the case I < p ~ 2

llImh ± A p ~ (h)llLP{Vm) l]Im hlILP(Vm)

V h6H + with Imp(h)

we have

Ap[IRe hllLP(Vm) and hence AplIRe h[ILP(Vm)

= O and /lhlPVdm < ~.

ii) In the case 2 ~ p <~ we have

IIRe h IILp (Vm) =< llApImh +- ~°(h)IILP(Vm) Vh6H + with Im ~(h) = 0 and In 5.13.i)

/lhlPVdm< ~.

the restriction to H + instead of H # is essential as is

evident from 5.3.ii). We shall come back to this question in Section 7. Proof:

i) In the case I < p < = 2 we know from 5.12.i) BpCOSpt

< A (cost) p - Isintl p

that

V Itl < ~ .

137 For h£H + with /lhlPVdm< ~ we have h p E H # with ~(hP)=(~(h)) p after V.4.9 and ~(h p) = /hPvdm after IV.3.4. that

Thus BpRe(h p) ~A~(Reh) p-

]IImh IpVdm ~ A~ /IRehlPVdm

- BpRe(~(h))P

We apply this to h°:=h + ieAp~(h) We see that ~(h °) = ~(h) I1+iCApl

with c=±1 under the assumption

-

~(h)~ COS t

I~(hO)) p = I~(h)------~)Pexpl i ~ l cos t so that the result that

follows,

-BpCOSp(t-3) -BpCOSpt

~ A

IImhl p implies

Im~(h)=O.

exp ll~l ,

and hence ReI~(h°)l p = O,

ii) In the case 2 ~ p <~ we know from 5.12.ii)

costl p - (sint) p

< APlsintl p - (cost) p

V O~t~,

vltl <=3"

For hEH + with flhlPVdm <~ we obtain as above /IRehlPVdm

<

A p /IImhlPVdm

+ BpRe<%0(h)l p.

We apply this to h ° : = h + i c ~ ( h ) ~ with e=±1 under the assumption Im~(h)=O. P As above we see that Re(~(h°))P=o, so that the result follows. QED. 5.14 THEOREM:

Assume that V£M and p = 2n(n=1,2,...).

IIImh ± Ap~(h) llLP(Vm)

Then

AplIRe hlILP(Vm) and hence

I!ImhIILP(Vm) =< ApliRehI~ P(Vm) V h6 H # with Imp(h) = O

and /lhlPVdm < ~.

Thus R(p,V) ~ A p = cotan2~. Proof: we know from 5.12.ii) -(-1)nB p c o s p t

that

p - (sint) p =< AP(cost) p

V ItI<~. =

138

Thus

for h 6 H # w i t h

/lhlPVdm<~

we o b t a i n

/(Imh) p V d m =< AP/p ( R e h ) P V d m

We a p p l y We see

this

to h ° : = h +

+

is Ap~(h)

(~(ho))p

so t h a t

the

6. S p e c i a l

In the

=

6.1

(_1)n

result

~=±I

under

the

assumption

Im~(h)=~

ice(h) sin)~

e x p (-i~p) r

(~(h)._~)Pexp(_i~)

and hence

sln-~--~

follows.

R e ( ~ ( h ° ) ) p = O,

QED.

Situations

first

an i n d i v i d u a l

part

of the

function

REMARK:

For

section

we d i s c u s s

F 6 M as to t h e i r

F6M

i) M c F ( R e L ~ ( m ) ) ,

the

that

subsequent

mutual

several

properties

of

dependence.

properties

is for e a c h V 6 M t h e r e

are

equivalent.

exists

a constant

c > O

t h a t V < cF.

i')

There

ii)

F is an i n t e r n a l

each

exists

V 6 M there

a constant

exists

exists

point an

c > 0 such of the

e>O such

ii')

There

iii)

N = {c(V-F) :V 6 M a n d c > O}.

The

functions

in 6.1

are

an e>O such

F 6 M which

called

internal

d i m N <~ i n t e r n a l f u n c t i o n s not

with

that ~ ( h 0) = ~ ( h ) ( 1 + i g A p )

such

( - 1 ) n B p R e ( ~ ( h ) ) p.

use

the reducedness

Proof: exist

i) ~ i ' )

functions

= ~ 1V n=1 2 n

6M

and hence

that

V~cF

e(V-F) 6 M V V

for all

is to

ii)

Note

we

that

6M.

properties see

that

i) - iii)

in the c a s e

the p r o o f

the assertion

that V n~ n2nF for

some

is n o t

is false

of 6.1 w i l l

c > O.

true.

sufficiently

large

Then

(n=I,2...)°

It f o l l o w s

n

c2nF ~ n2nF

that

F - e(V-F) £ M .

From

exist.

M.

(H,~).

that

such

F-

VV6

set M c R e L 1 ( m ) ,

the e q u i v a l e n t

functions. always

of

Assume V n £M

convex

that

possess

that V~cF

n. So we o b t a i n

that V

there Now V:= < 2nv n =

a contradiction.

139

i') ~ i i ' ) F

For V 6 M and e>O the r e l a t i o n

e(V-F) ~ 0

trivial,

or V ~

(I+~)F.

ii) ~ i i i )

f =c(U-V)

F- e(V-F)6M

Thus the i m p l i c a t i o n

We have to p r o v e

is e q u i v a l e n t

is clear,

the i n c l u s i o n

to is

ii') ~ i i )

c. Let f £ N, that is

w i t h U , V 6 M and c>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 O
F>O 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~s<~ t h e r e e x i s t s

< c(s) Ilfll v f6T. = LS(m)

around

is u n r e l a t e d

that F>O 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 ~p<~

closure.

QED.

The s e c o n d p a r t of the s e c t i o n We s t a r t w i t h a r e m a r k a b l e dy a l g e b r a

that RI (Fm) N ReLY(m) = E ~ iff

and N have the same L 1 ( m ) - n o r m

is o b v i o u s

c(s)>O

such that IiflI ~ < L (m)= for p<s<~. For I < s < p we have =

llfIlP = SlflPdm__< IIfIIP~s ] I f l S d m < cP-SllfIlP~ s llfIILs(m ) , L ~(m) L (m) L ~(m) P-1 IIf IILP (m) < cs

and h e n c e

Iifl~LS(m)

the a s s e r t i o n

¥ f 6T,

as well.

iii)

N o w we d e d u c e

< cIIfll v f6T t h a t T m u s t be f i n i t e - d i m e n s i o n a l . = L2(m) and o r t h o n o r m a l f l , . . . , f n 6 T. For t l , . . . , t n 6 ~ then I n n n , [ t£fi, ~ cl, ~ tifi,IL 2 = c[ ~ Iti, 2] £=I ~=I (m) Z 1 If r e p r e s e n t a t i v e equality

functions

is true o u t s i d e

with rational

components

take tz:=fi(x) (~=1,...,n)

for f l , . . . , f n

a fixed m-null and h e n c e to o b t a i n

are c h o s e n

set

N 6 Z

f r o m IIfll ~ L (m) T a k e an n < d i m T =

t h e n the a b o v e

in-

for all t l , . . . , t n £ ~

for all t l , . . . , t n 6 ¢. N o w for x 6 X - N

143

n

2 Ifz(x) 12 ~ c .

£=I After

integration

it follows

6.6 PROPOSITION:

that n ~ c 2 m ( X ) .

Thus d i m T < ~. QED.

Let F 6 M. Then the subsequent

properties

are equi-

valent. i) N ReLI (m) c F ( R e L = ( m ) ) . ii) N QF(ReL=(m)) conditions iii)

is L1(m)-norm

closed and F fulfills

the equivalent

i) - vi) in 6.4.

dim N <~ and F is internal.

Proof:

i) ~ i i i )

The subsequent 6.7 THEOREM:

follows results

from 6.5 and iii) ~ i i ) ~ i) are obvious. do not depend on lemma 6.5.

Let 0 < F 6 ReL(m)

be such that N ReL1 ( m ~ F ( R e L ~ ( m ) )

do not assume that F be in LI (m) and hence be finite-dimensional).

QED.

cannot conclude

(we

that N must

Then

1~ReL1(m) i) E N ~ m = {0}, I ~ReL1(m) ii) Re H + ~ Proof:

i) Assume

f ± N implies

in ReLY(m).

1~ReL1(m) that f 6 E A ~ N . Then f is bounded

that f i N ReL1(m) . In particular

that f F f 2 d m = O ReH

is weak,dense

and hence

f =0.

ii) Assume

I~ReLI (m) E~ and ~N . Then f ± and hence

bipolar

theorem.

that / ~ f 2 d m = O

and hence

6.8 COROLLARY: NcF(ReL~(m)).

Hence

I f ± ~f which f =0.

f.L Ff 6 N ReLl(m)

that f 6 ReL1(m) f 6

~ReL1(m)

is a bounded

so that It follows

annihilates

from 4.3 and the

function.

It follows

QED.

Assume that dim N <~. Let O < F 6 ReL (m) be such that

Then

1 i) E A ~ N = { O } , ii) E ~ e

~IN =

ReLY(m).

6.9 PROPOSITION:

Assume

that d i m N < ~ a n d

that F 6 M is internal.

Then

144 i) RP(Fm) = E N ReLP(Fm) ii)

RP(Fm)

iii)

K=

Proof:

for

ReLP(Fm)

(H#NL I ( F m ) ) F + N +

the r e l a t i o n

i). iii)

for I ~ p <

iN.

s i n c e RI(Fm) c E

(O} from 6.8.i)

I ~ p < ~,

we see that RP(Fm) + ~ N =

F r o m 6.8.ii)

c E N ReLP(Fm) N~N=

I ~ ~N=

from 6.4.

to d e d u c e

Combine

the d i r e c t n e s s

ReLP(Fm). And RP(Fm) c this w i t h

(ENReLP(Fm)) N

of the sum in ii) and

F r o m 4.8 we h a v e

K = ~ L1 (m) + N + iN = HLI (Fm)F + N + iN,

so t h a t H L I ( F m ) = the s u b s e q u e n t

H # N LI (Fm) is to be shown.

But this

r e m a r k w h i c h w i l l be s e p a r a t e d

is c o n t a i n e d

in

for f u t u r e r e f e r e n c e .

QED. 6 . 1 0 REMARK:

Let F 6 M

be e n v e l o p e d

we see that H# N LP(Fm) i s norm closure

F r o m LP(Fm) c L ° ( F m ) c L #

LP(Fm)-norm c l o s e d and hence i s t h e LP(Fm) -

of H. L i k e w i s e

is L P ( F m ) - n o r m

and I ~ p < ~ .

for H # : = { u 6 H # : ~ ( u ) = O }

closed and hence

w e see that H # N LP(Fm)

is the L P ( F m ) - n o r m

closure

of H

(see

4.5).

7. R e t u r n

to the M a r c e l

We s h a l l a s s u m e obtained

and K o l m o g o r o v

5. We s t a r t w i t h

Assume

that d i m N

the K o l m o g o r o v

<=. For O<~
Tz 8 ( 1P+iP* l ~) £ 21-T ( e ( I P ! ) ) cos --~Proof:

Estimations

that d i m N < ~ and use 6.9 to e x t e n d

in S e c t i o n

7.1 T H E O R E M :

Riesz

Let P 6 E w i t h

e(IPl)<~.

s t a n t c > O such that V ~ c F there exists

a sequence

f £ ReLI(Fm).

From

VV£M.

T

the e s t i m a t i o n s

estimation.

then

v P6E.

We fix an i n t e r n a l

F 6 M and a con-

T h e n P 6 E N ReL1(Fm) = RI(Fm)

of f u n c t i o n s

Pn 6 E = w i t h P n + P a n d

so that

Ipnl ~ some

5.1 we have

C O S V S ( I P p - P q l :~ T ) _<_21-'[ (O(iPp-Pql)) "~ =< 21 -"[c'[i iPp-Pqlr L I (Fm)

145

Ilence t h e r e e x i s t s that P~n ( Z ) ÷ /gYVdm

a subsequence

some Q 6 ReL(m)

I < n(1)<...
for ~÷~ and

<~ for all V 6 M. O n c e m o r e

cos~/[pn(z

such

iP (Z)I ~ some g 6 ReL(m)

with

from 5.1 w e o b t a i n

) + ip~(z) [TVdm ~ 2 1 - T ( 8 ( i p n ( 1 ) i) T

=< 2 I-~ (e(Im-Pn(~)l)+0(Iml)) < £ 21-T[clIP-Pn(z)IIL1(Fm)+s(IPl)] ~ COS~IIP+iQq ~ v ~ ! 2 1 and h e n c e

cos~8(IP+iQl

~(e(!Pl))~

VVCM,

T) ~ 2 1 - T ( 8 ( I p I ) ) T. It r e m a i n s

Q = P~. B u t this is a r o u t i n e

application

to s h o w t h a t

of I V . 3 . 2 - 3 . 3

and 2.4.i)

in

v i e w of e Itlf 6 L°(Fm) m E # V t 6 $. QED.

Let us turn to the M a r c e l above proof

Riesz

estimation.

that it can be e x t e n d e d

7.2 T H E O R E M :

Assume

It is c l e a r

a f t e r the

as follows.

that d i m N < ~

and t h a t F 6 M is internal.

For

I ~ p < ~ then

Jlp~

_< R(p,F)IIPII IILP (Fm)

V P~E. L p (Fm)

The l a s t a i m is the s u b s e q u e n t 7.3 T H E O R E M : R(p,F) < ~

Assume

theorem.

that d i m N < ~

In v i e w of 5.4 we can r e s t r i c t we h a v e

to r e m o v e

the p o s i t i v i t y

we n e e d the s u b s e q u e n t 7.4 LEMMA: linear

and that F 6 M is internal.

Let

subspace

ad-hoc

ourselves

Inf{IIfI~ P(m)

to the case

restriction

in 5.13.i).

I ~ p <~ and ReLP(m) = T + R, w h e r e with

I
Thus

To do this

lemma.

1 6 T and R c ReLY(m)

s u b s p a c e w i t h T N R = {O}. T h e n t h e r e e x i s t s :IPi < f 6 T N R e L ~ ( m ) } =

TcReLP(m)

is a c l o s e d

is a f i n i t e - d i m e n s i o n a l a constant

< cIIPIl = LP(m)

P r o o f of 7.4: F r o m the c l o s e d g r a p h such t h a t

Then

V 1
linear

c > O such that

v P6ReL~(m).

t h e o r e m we o b t a i n

a constant

a>O

146

IIfll , IIull < a IIf+ull v f 6T L p (m) L p(m) = L p(m)

and u 6 R .

Furthermore there exists a constant b > 0 with

IIull ~

a b llUlLpL

L (m) Let now P6ReL~(m). and hence

v u c R. (m)

Then

IPI = f + u with f 6 T and u 6 R. Thus f6TNReL~(m)

]P[ ~ f+IlUllL~(m)6T N ReLY(m).

It follows that the infimum in

question is I

I


from 6.9.ii)

and take into account 6.9.i). It follows that

there exists a constant c > 0 such that

Inf {I[f[ P(Fm) :IPI =< f £ E ~} =< cI[PIILP(Fm) V P 6 ReL=(m). ii) We assume that 1
IPI~fcE

=~(f±P) ~ O

~

we have O~f±P 6

E~

. Thus

h ±

:=(f±P)+i(f±P)*6H + with ~(h±) =

and Slh±IPFdm < ~ after 5.5.i). From 5.13.i) we obtain IEf*±P*II < (tan~ L p (Fm) = IIP*IlLP(Fm) : I ~ ~(tan~)

, L p (Fro)

l!½(f*+P*) - ~(f*-p*)llLP(Fm)

[llf+PllLP(Fm)

+llf_p 1 ]<2(tan2~)ii f , ILP(Fm) = IILP(Fm)

so that from i) the assertion follows with R(p,F) ~ 2 c ( t a n ~ ) .

QED.

Notes

For the classical conjugate function theory we refer to ZYGMUND [1968] Chapter VII, KATZNELSON

[1968] Chapter III and DUREN [1970] Chapter 4.

147

Substantial

portions

let a l g e b r a

situation

in L U M E R

[1965].

transition

was

LUMER

[1968]

duced

in p a r t

did not text

came

rSle

had

question are

dealt

Szeg~

and LUMER

up.

its The

full

situation

to b e c o m e

and

of the

[1967c]

more

with

(exept

emphasize

of

the

the

definition.

relation

if O ~ P 6 E

then

The

in G A M E L I N -

theory

is r e p r o theory

of the p r e s e n t

since, The

but

their

definition

presented

in

in S e c t i o n s

3.6 w h i c h

to the

the

for c o n j u g a b l e

ever

f o r m of

that

conjugation

exp(t(P+iP*))

literature

situation

step.

out

the d e f i n i t i o n

the p r e s e n t

V.4-5:

serious

abstract

fundamentals

the c l o s e

in S e c t i o n s

the

until

to be c o n v e r t e d

to the D i r i c h -

clear

and worked

version

But

in the

3.9 m a k e s

is a v e r y

functions

been

extended

a n d to the S z e g ~

[1965]

Their

coherence

composite

been

theorem

in L U M E R

[1969].

the v e r s i o n

have [1966]

our

[1968].

in G A M E L I N

and t 6 ~ have

in K O N I G

us o n c e

the

initiated

attain

P 6 ReL(m)

the t h e o r y

in D E V I N A T Z

In r e t r o s p e c t

beyond

transition

of

function

I-3

is new). class

h:=P+iP*6H + from

Let

H+

the de-

finitions.

In S e c t i o n estimation. unit via

disk

situation,

the u s u a l

4:9.ii). algebra 1
also

YABUTA

I
cotan and

Moreover

KONIG

[1965]

from YABUTA Bochner.

The

to

the

sharp

YABUTA

leads

to the

results

approximation

theorem

in S e a t t l e

after

result.

The

the m a i n

an o r a l

an e x a m p l e theorem

The

theorem

in K O N I G

of a p a r t i a l IV.4.5.

papers

4.1

is

result

in the

4.9

the

in 5.7,

results

In 5.4

method

to from

another

of p r o o f

of

and

due

4.11

of the F u n c t i o n

Cole

due

idea to

of 4.8)

is q u i t e

literature

[1965].

to

weaker o n e of

different.

we quote

achievement

and K O N I G

Algebra

are d u e

of a s o m e w h a t

the proof

is the m a i n

[1965]

Hardy

the e s s e n c e

example

(in the v e r s i o n

where

ROSSI

are n e w

2
property

a n d the c a s e

v e r s i o n a n d its p r o o f

[1967c],

Theorem

HOFFMAN-

for

in the

[1978].

from Brian

4.5

and

2
5.3.ii).

is a r e s u l t

present

communication

annihilator

results

multaneous

1970.

and also

that

a classical

see K O N I G

in 5.6

the c a s e

5.14

discovered

counter-example

further

Seminar

5.8 as b e f o r e ,

with

method

Riesz

5.8

to the a b s t r a c t

tan e s t i m a t i o n

estimation

combines

estimation

on the w e a k * D i r i c h l e t

in 5.6 a n d

[1977]

on the M a r c e l

sharp

the p r e s e n t

based

the

estimation

[1977]

For

results

the

[1977] t r a n s f e r r e d the m e t h o d

combine

The

brand-new obtained

argument

in 5.7 w h i c h

5.13.ii)

[1969]

for

duality

some

[1972]

to o b t a i n

K~nig.

K~nig

are

situation

5.13.i).

As

5 there

PICHORIDES

GAMELIN

of the

si-

148

The

considerations

the respective AHERN-SARASON there

in S e c t i o n

sections [1967a],

GLICKSBERG

are some new results.

f e r to G R O T H E N D I E C K

6 on

in G A M E L I N

[1954].

For

special

[1969] [1968]

the

situations

which

are close

in t u r n a r e b a s e d

and GAMELIN-LUMER

finite-dimension

[1968].

l e m m a 6.5 w e

to upon But re-

Chapter

Analytic

The

first

proved the

that

famous

Hereafter

Disks

section a Hardy

and

then

d e r of the

develops

to be

in some

or other.

all

sense

true

theorems theorem

are due

properties sesses sense

that

are

independent.

either

We a l s o purely (H,~)

prove

possesses

isomorphic

I. T h e

Then

the w e a k e s t

above

two

and

that to

structure

kind

an

(H,~)

are

theorems

equivalent.

the a n a l y t i c the

two

an

(H,~)

(H~(D),~O)

spaces.

of

possharp

from a

Here M={F}. that

is a

implies

that

that

in the

disk

sets

which

of i s o m o r p h i s m ,

and hence

The

in the

comes

H a n d H~(D),

(H~(D),~O)

remain-

which

that

which

measure

of p r o p e r t i e s

situation

set are

of

(H~(D),~o)

equivalence

shows

it is p r o v e d

possible

such

form

situation.

The

situation

to M u h l y

of the a l g e b r a s

sets

disk

whether

either

the b a s i c

Szeg~

of p r o p e r t i e s

is i s o m o r p h i c

between

isomorphism the

due

of t h e e n t i r e

to the u n i t

described

(H,~)

sharp

is

sense

above.

Invariant

Let(H,~) near

that

algebraic

within

of e x a m p l e s

s e t of p r o p e r t i e s

map(X,Z,Fm)÷(S,Baire,l)

sets

It is

a certain

is a s s u m e d .

disk

and prove

theorem

A class

of an i s o m o r p h i s m

two

situation

the m a x i m a l i t y

(H,~)

Situation

sequel:

satisfies

the q u e s t i o n

the p r o p e r t i e s

to W e r m e r .

for t h e

Disk

iff it is a r e d u c e d

to the u n i t

We c o n s i d e r

disk

tool

situation

around

the U n i t

(H,~)

theorem

isomorphic

in the u n i t

to the e f f e c t

the b a s i c

Szeg~

centers

can be p r o v e d

with

situation

subspace

a reduced

chapter

Isomorphisms

algebra

invariant

VII

Subspace

be a H a r d y

subspace

TcL~(m)

Theorem

algebra which

situation.

Consider

is i n v a r i a n t

in the

a weak~

sense

that

closed

li-

H T c T.

We

define T

where

H

notation variant that

:={u£H:~(u)=O}

weak~

closed

with

linear

{f T +T t h e n

are obvious

Lin

H ~weak~,

as i n t r o d u c e d

is in a c c o r d a n c e

T =T.

There

:=

the

in V I . 4 . 5 former

subspace

T is d e f i n e d

examples

(in the

one).

as well,

Then

and T c T .

to be s i m p l y

of s i m p l y

invariant

case

T=H

T c-L~(m)

the

is an in-

It can h a p p e n

invariant.

subspaces:

If P 6 L ~ )

150

is of m o d u l u s s p a c e of L Consequence P~T

IPI=1 t h e n T : = P H

(m)

(the w e a k *

IV.3.14)

and T =PH

. In the o p p o s i t e

space

is an i n v a r i a n t

closedness

weak*

is o b v i o u s

closed

, so that T is s i m p l y

direction

linear

sub-

f r o m the K r e i n - S m u l i a n invariant

we c l a i m the c e l e b r a t e d

since

invariant

sub-

theorem.

1.1

INVARIANT

SUBSPACE

The s u b s e q u e n t

THEOREM:

assertions

are e q u i -

valent. i)

(H,M)

ii)

is a r e d u c e d

Each simply

of the form T = P H In this case IPI=I

Szeg~

invariant

situation.

weak*

closed

for some P6T of m o d u l u s for each s i m p l y

invariant

such that T = P H is of c o u r s e

unique

linear

subspace

TcL~(m)

is

IpI=1. TcL~(m)

the

function

up to a c o n s t a n t

P6T w i t h

f a c t o r of m o -

d u l u s one.

Proof:

i) ~ i i )

an H i l b e r t

space

The i n c l u s i o n there exist

We have M = { F } argument

we h a v e

and

uif÷f.

functions

Thus

if f6L~(m)

3) T a k e a f u n c t i o n

that hence tion

and

Ifnl ~ some

luil~1 , uz÷1

It f o l l o w s

of L 2 ( F m ) - n o r m

space

sense.

Ip[2F=F or

and

lu~iG~c%.

u£f6T.

luzfl~Ifl~const<~ and h e n c e

Thus

f6T L2(Fm) . In p a r t i c u l a r

IpI2F6M a n d h e n c e PHcT.

as well.

P6T L2(Fm)

in the H i l b e r t

f£T and h e n c e

uz6H w i t h

then

is via

to p r o v e c let f6T L2(Fm) . T h e n

Iulfni~iuziG~c Z so that for n÷~ we o b t a i n

W e h a v e T--L2(Fm) N L ~ ( m ) = T

T--L2(Fm)

In o r d e r

proof

I) We c l a i m that TL2(Fm) AL~(m)=T.

fn6T such that fn÷f p o i n t w i s e

G 6 L 2 ( F m ) c L ° ( F m ) = L #. Take Then Uzfn6T

in L2(Fm).

m is trivial.

functions

w i t h F>O on X. The e a s i e s t

f6T.

2)

t h a t T--L2(Fm)#T L2(Fm)

one w h i c h

/Puf F d m = O

/]pI2uFdm=O

IPI=I.

For Z + ~

is o r t h o g o n a l

to

for all u E H M and

V u 6 H M. It f o l l o w s

In p a r t i c u l a r

P6T from

I) and

4) To see the c o n v e r s e

/Puf F d m =O V u £ H

P f 6 H or f6PH.

means

Thus T c P H and h e n c e T=PH.

ii) ~ i ) : I) We p r o v e that Xy(x)6H. s p a c e of L~(m)

o b s e r v e for f6T t h a t the a b o v e e q u a l that P f 6 ( H F) so that V I . 4 . 5 i m p l i e s that

that (H,~) m u s t be r e d u c e d .

Hence T:=Xy(x)H (the w e a k .

is an i n v a r i a n t

closedness

is c l e a r

weak*

From

IV.1.10 we know

closed

linear

f r o m IV.3.14).

Also TM=

= X y ( x ) H ~ a n d h e n c e T ~T s i n c e X y ( x ) ¢ T M in v i e w of M ( X y ( x ) ) = I . are no f u n c t i o n s P6T of m o d u l u s

IPI=1

except when m(Y(X))=1.

sub-

But there

Therefore

151

(H,~) m u s t be reduced.

2) L e t f6L~(m)

IV.3.14

that T : = f H

shows

subspace T=PH

as a b o v e

of L~(m)

and that T =fH

for some P6T w i t h

element

[P[=I.

and t 6 ~ to o b t a i n

from VI.2.1

so t h a t

1.2 C O R O L L A R Y :

(H,M)

Let

1_
Then T=P(H#ALP(Fm)) variant

subspace

P6T w i t h

as well,

B=PH.

IPI=I

iff T is s i m p l y

is an i n v a r i a n t

the o p p o s i t e

and m r e s u l t s

T P A L ~ ( m ) L P ( F m ) = T p. It f o l l o w s

of L~(m).

implies

and in

in-

Furthermore

s i n c e TPAL~(m)~ a function

closed

for some

T h u s T~+T s i n c e direction.

3)We

from LP(Fm) c

that T ~ N L ~ ( m ) ~

is an i n v a r i a n t

H BcH ToT p and h e n c e is w e a k ,

closed

P6B of m o d u l u s

IP[=I

weak* H Bc

as well. such that

that T = B L P ( F m ) = p H L P ( F m ) = p ( H # N L P ( F m ) ) .

QED.

Theorem

For the r e m a i n d e r (H,~) w i t h M={F}.

with M={F}

is i n v a r i a n t

Tp = P ( H ~ N L P ( F m ) ) .

that T P + T and p r o v e

1.1 we o b t a i n

2. The M a x i m a l i t y

tion.

shows t h a t

so that B c T ~ A L ~ ( m )

Then VI.6.10

But this

which

2) If T = P ( H # N L P ( F m ) )

Here c is o b v i o u s

subspace

Thus B +B. F r o m

that R e L ~ ( m ) c E

QED.

situation

4) F r o m I V . 3 . 1 4 w e see that B : = T N L ~ ( m )

linear

cT~AL~(m)

and t h a t TPcT.

VI.6.10

c L ° ( F m ) = L #. L i k e w i s e

is an i n v e r t i b l e

we take f=e th for

T LP(Fm)

I) It is c l e a r t h a t TPcLP(Fm)

IPl =1 t h e n

now

that T ~ T .

have TDL~(m)LP(Fm)=T.

closed

Szeg~

linear

. F r o m ii)

It f o l l o w s

subspace

6>0. T h e n

closed

and d e f i n e

SO let us a s s u m e

~TDL~(m).

linear

f{T

that ~f=Pf

a(eth)a(e-th)=1.

(H,~) be a r e d u c e d

weak*

since

a f t e r v.I.3,

for some P6T of m o d u l u s

in the s e n s e

P r o o f of 1.2:

P*T~.

It f o l l o w s

be a c l o s e d

that HTcT,

IfI~ some

is S z e g ~ a f t e r V I . 3 . 9 .

TP: = L i n H

linear

. Thus T M + T

of H so t h a t a ( I f l ) a ( I ~ I ) = 1

h6ReL~(m)

such that

is an i n v a r i a n t

of the c h a p t e r we fix a r e d u c e d

We c o u l d of c o u r s e

is not a d v i s a b l e

assume

in p a r t i c u l a r

that F=I

Szeg~

situation

as it is o f t e n done.

for the b e n e f i t

of the n e x t sec-

152

In the p r e s e n t is a s u r p r i s e

section we want

because

2.1MAXIMALITY

THEOREM:

i) H is a m a x i m a l ii)

If TcL~(m)

iv)

For each nonzero H contains

Fix a n o n z e r o :fa6H}

lows

interest,

and h e n c e

from ii)

T=L~(m).

the s c h e m e

iii)~iv)

weak~

XATCT.

the w e a k ~

closed

linear

of L~(m)

closed

linear

It is not h a r d to p r o v e 2.2 LEMMA: T(T)=L~(m).

Proof: ties.

linear

for some V6Z.

for the step i)~ii).

it fol-

a=O.

The s u b s e q u e n t situation.

we i n t r o d u c e

the t r a n s -

is a w e a ~ c l o s e d

the c o n s t a n t s .

T:=XvL~(m)

then be

and h e n c e

Hardy algebra

TcL~(m)

contains

be a w e a k ~

com-

In p a r t i c u l a r

w i t h V£Z we h a v e

for

T(T)=L~(m).

closed

linear

subspace

such that

for some V6I.

A:={D6Z:XD£T}

then UUV6A.

3) If D n 6 ~ for n~l

t i o n s are b e t w e e n

We h a v e

ii)=iii).

It is s e e n t h a t ~(T)

subspaces

Let TcL~(m)

In fact, w e h a v e

of L~(m).

laI2=const

which proves

subspace

T h e c l a s s of s u b s e t s

v i e w of I).

is simple:

Then T:={f6L~(m) :

the c o n v e r s e .

Then T=xi~(m)

2) If U , V 6 &

A n d ii)~iii)

com-

deser-

But s i n c e HoT it m u s t

[a126H so t h a t

which

The m o s t

XA~H in v i e w of the a s s u m p t i o n

I) If D6A and U6E is c o n t a i n e d

Xu=XuXD6T.

such

for some V£E.

a b i t of w o r k w h i c h

is o b v i o u s .

to the a b s t r a c t

Y ( T ) : = { f E L ~ ( m ) :fTcT}.

plex subalgebra

subspace

i)=ii)~iii)=iv)~i).

involves

subspace

a6T or

is u n r e l a t e d

For a w e a k ~

linear

closed

L e t us turn to the p r e p a r a t i o n

porter

closed

of L~(m).

has m ( A ) > O .

at a c o n t r a d i c t i o n

consideration

subalgebra

are e q u i v a l e n t .

that A : = [ a = o ]

Since

that T=XvL~(m)

In p a r t i c u l a r

So we a r r i v e

properties

is ~H then T=XvL~(m)

and i)=ii)

a6H and a s s u m e

is an i n v a r i a n t

XAL~(m)cT

weak~

It

no zero d i v i s o r s .

p a r t is iv)~i),

ves i n d e p e n d e n t

which

theorem.

look q u i t e d i f f e r e n t .

u6H w e have m ( [ u = O ] ) = O .

The p r o o f w i l l be a f t e r plicated

closed

is an i n v a r i a n t

the s u b s e q u e n t

involved

The s u b s e q u e n t

proper weak~

t h a t fTcT for some f£L~(m) iii)

to p r o v e

the p r o p e r t i e s

has the s u b s e q u e n t

in D then U6A.

This

follows

0 and I and +X[f+O]

proper-

follows

from

from XUuv=Xu+Xv-XuNv

and D n + D then D£&.

Ifl2/(In-+IfI2)=(f/(ln-~IfI2))f6T

This

in

4) If f6T t h e n [ f + O ] 6 ~

for n~l,

and t h e s e

for n÷ ~. N O W we can p r o v e

func-

the a s s e r -

153

tion:

F r o m 2) and 3) we o b t a i n

Then DcV

(modulo an m - n u l l

[ f ~ O ] c V or fEXvL~(m) T=XvL~m)

P r o o f of 2.1.i) ~ i i ) : follows

It f o l l o w s

that TCXvL~(m)

and h e n c e

QED.

to be ~ H. H e n c e

contains

Y(T)=L~(m)

H since T

after

i). T h e n

from 2.2. QED.

The p r o o f of iv) ~ i ) sequent

is m a x i m u m .

F r o m 4) in p a r t i c u l a r

The t r a n s p o r t e r T(T)cL~(m)

and is a s s u m e d

the a s s e r t i o n

for all D6A.

for all fET.

s i n c e = is trivial.

is i n v a r i a n t

a set V E A such that m(V)

set)

requires

several

steps.

Let us i s o l a t e

the sub-

lemma.

2.3 LEMMA:

Let Bc-L~(m) be a w e a k ~

Define A:={DEE:XDEB} contains

the m - n u l l

closed

subalgebra

so t h a t A is a o - a l g e b r a sets.

Then

mE which

for e a c h n o n z e r o

fEL(m)

such t h a t H~B. in p a r t i c u l a r there exists

a

set DEA such that f X D ~ O and fXX_D~O. Proof: function

I) D e f i n e and h e n c e

Let L(mlA)

consist

f6L(m)

of the A - m e a s u r a b l e

f6L(m).

m(D)>O

and is m i n i m a l or m ( D - U ) = O . numbers

in A and h e n c e

in the s e n s e

T h e n each

carrier

~(fID).

measurable.

To see this

of f w h i c h

is r e a l - v a l u e d

of the f u n c t i o n s

h:~÷~

the p o l y n o m i a l s ,

ii)

continuous ~R~].

b o t h on

functions

so that

4) T h e r e

exist

and b o u n d e d

:=e

tf

Reft,

e It

for

function

fEB is A-

be a f i x e d r e p r e s e n t a t i v e IF(x) I~R vx6X.

Define

in B. T h e n

function

~ to c o n s i s t

i) ~ c o n t a i n s

and h n E ~ are such that h n ÷ h and

lhnl ~ some

then hE~.

Therefore

iii)

and hence

the B a i r e

functions

h:~÷~ bounded

set A c ~ w e h a v e X A ( F ) = X [ F E A ]

~ contains

and h e n c e

the

on

XA(F)modm=

[fEA]EA as claimed. sets U£A w i t h O < m ( U ) < m ( X ) .

failure

Otherwise

2) a n d 3) all r e a l - v a l u e d

Fix n o w f=P+iQ6B.

Thus

so a and b cannot both

[-R,R]

(P~iP~) eit (Q-P~) EBX"

ImftEB.

in D f u l f i l l s

on D. In fact,

m-measure,

such that h ( F ) m o d m i s

the s e n s e of 2) so that a f t e r were constant.

is c o n s t a n t

3) Each r e a l - v a l u e d

let F:x~F(x)

If h : ~ + ~

N o w for a B a i r e

=X[fEA]EB

that DEA has

that each U 6 A c o n t a i n e d

fEL(miA)

cannot both have positive

constant

of f is A - m e a s u r a b l e . 2) A s s u m e

a+b the sets D N [ i f - a I < ½ [ a - b I ] and D N [ i f - b ] < ½ i a - b I] are

be in the v a l u e

common

iff some r e p r e s e n t a t i v e

function

m(U)=O complex

to be A - m e a s u r a b l e

each r e p r e s e n t a t i v e

For t6~ consider

We have

IftI:1

of the a s s e r t i o n

X were minimal functions

et(P+iP~)£HX

so t h a t f t = l / f t E B would

in

f6B

a n d ft:= and h e n c e

i m p l y t h a t ft=const=c(t)

154

and h e n c e BcH

that

etf=c(t)et(P+iP~)6H

in c o n t r a d i c t i o n

5) To e a c h <m(D), 2).

that

D6A w i t h m ( D ) > 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)<

fn6H × with

that

the

X6A(f).

that m(D-U)=m(D)

sense

H~B~L~(m).

visors that

from our

in the

Proof that

2)

in D a n d m ( U ) < m ( D )

fXu=O

such

D were minimal.

It f o l l o w s

If U , V 6 & ( f )

nimal

Thus

But

in the

so t h a t

t h a t b =I n

a n d D +D t h e n D 6 A ( f ) . N o w we a s s u m e the n d e d u c e a c o n t r a d i c t i o n as f o l l o w s . F r o m

contained

a subset U6A

of m o d u l u s

A(f):={D6&:f×D=f}.

A(f)~

iv)

sense

f6H.

is a c o n t r a d i c t i o n .

and define

i) & ( f ) c A .

m(D)>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~<

IlU![

= iv)

I~5~ ~ then ~6HoIP(D) (see S e c t i o n

~

~=I

I.I).

L ~ (Fm) A

F o r u , v £ H # and cE¢ we h a v e

Furthermore

and

A+A

A

(u+v) :u v,

(cu) = ca

A

and

A A

(uv) = u v .

and ~=Z.

v) For a6D we have {u6H#:~(a)=O}

Proof:

a n d h e n c e uz~f6H.

Then

assertion

Inun=~(Un)+In+lun+

Then u£f6H

such that u = u o and

I Vn~O

so that we o b t a i n

The o t h e r

assertions

3) If u , v 6 H # and uv have the T a y l o r

coeffi-

a n , b n a n d c n then n cn = ~oa~bn_£

follows

upon m u l t i p l i c a t i o n

lor c o e f f i c i e n t s . fine a f u n c t i o n converges nest

and u£fEH.

Un6H#(n=O,1,2,...)

in i) w i t h a n : = ~ ( U n ) .

in i) are then i m m e d i a t e .

This

m is trivial. To see c let

[uz I~I, uZ÷I

that If6H # or f6IH #. 2) For u6H # w e o b t a i n

of f u n c t i o n s

Un=~(Un)+IUn+ 1Vn~O. the e x i s t e n c e

uz6H with

It follows

I) a s e q u e n c e

cients

I-a H #. I -EI

I) we h a v e H # = I H #. The i n c l u s i o n

f £ H ~ a n d take f u n c t i o n s

from

=

V n~O. of the r e l a t i o n s

u6H # to be h o n e s t

iff its p o w e r

for all zED and thus d e f i n e s

functi6ns

which

For u+v and cu the c o r r e s p o n d e n c e

EH # f o r m a s u b a l g e b r a

series

a function

which

define

4) De-

expansion

~6Hol(D).

contains

the T a y -

is obvious.

in ii)

T h e n the ho-

the f u n c t i o n s

I and

157

I and on w h i c h

the relations

5) For u6H#nLl(Fm) tion.

And

claimed

we have

in iv) are fulfilled. func-

so that u is an h o n e s t

]anl~/lulFdm

for zED we have 00

/ui___IzFdm=Su

00

1 F d m = / u ( [ (~z)n)Fdm = [ an zn= ~(z), I -~z n=O n=O

/u I Z _ F d m = / u ( ~ (Iz)n)Fdm = ~ (/uInFdm) zn=o, 1 -Iz n=1 n=l fu~Fdm=

/U(~z+

IZ_)Fdm=~(z), 1-Iz

II-zl

as c l a i m e d

in ii).

of the r e l a t i o n s

F r o m this one o b t a i n s

/~Fdm=1

II-zl -

Vz6D

as in I.I.2 with

from the above

the aid

for the c o n s t a n t

func-

~

tion =I and /P(z,s)dl(s)=1 S 6) We next p r o v e

iii)

v)

Vz6D

from

for H#NL 1 (Fm)

1.I.1. instead

of H #. Here h:= I-a

is an

I -az A

inner

function,

and from

(1-aI)h=I-a

we see that

Z-a

A

(1-aZ)h=Z-a

or h=

I -~z A SO that h(a)=O.

Thus we have D. In order

to prove c let u E H # D L 1 (Fm) w i t h

(a) =0. For all v6H then 0=0 (a)~ (a)= (vu) ^ (a)=/V~#aFdm. fiu_--IaFdm=O. Thus From

from V I . 6 . 9 . i i i ) w e

I) we o b t a i n

6) we d e d u c e

see that

Hence iu_~IaF6K with

IU#a6H#AL I (Fro)with

~0(iu_--Ia)=O.

I--~a6H#ALI (Fro) and hence ~=(1-aI)I_--ua6H#AL1 (Fro). 7) From

that

a function

fies UA(z)~O for all

u 6 H # N L I (Fm) w h i c h

is i n v e r t i b l e

in H # satis-

zED.

8) We can now prove

that all

functions

fEH # are honest.

To see this

write

f uu w i t h u,v6H and v6(H#) x after V.I.10. Let u,v and f have the v T a y l o r c o e f f i c i e n t s an,b n and c n. From 7) we know that ~ ( z ) ~ O Vz6D. Thus A A

u/v is an h o l o m o r p h i c

function

expansion

in the o r i g i n

an h o n e s t

function.

in D, and

3) tells

has the c o e f f i c i e n t s

us that

its power

se~es

c n. It follows

that

f6H # is

take

uz6H w i t h

A

lu£l~l,

uz÷1

9) To see that fEHol#(D)

and uzf6H.

Then

~i6Hol~(D)

^ 1_i~I~1_z 2 and we see that u~(z)=#u£~--u:~Fdm+1_~_,~

with

VzED.

functions

I~iI~I

Thus

and ~ = ( u z f ) A 6 H o ~ ) ,

A fEHoI#(D)

as claimed.

II-zl 10)

It r e m a i n s

to prove

v). The i n c l u s i o n

m is obvious.

To see c let

158 A f(a)=O

f6H ~ w i t h ~(a)=(fv}

A

and w r i t e

A (a)=f(a)O(a)=O

f=~ w i t h v

and hence

u,v6H u=

I-a

a n d v6(H#) x as above. w with w6H

after

6).

Then

It f o l l o w s

I-~I that

f= I-a

For

~6

I - a H #. QED.

the m o m e n t

we o b t a i n

the

we

3.2 C O N S E Q U E N C E : function

I6H

such

A i) ~ z : U ~ U ( Z ) on H.

points

z6D p r o d u c e

with

M

The

that

H =IH. m

In p a r t i c u l a r

Hardy

Z ilI_zl2

iii)

Assume that

to l o o k at t h e

individual

points

z6D.

Then

consequence.

is a n o n z e r o

tional

ii)

prefer

subsequent

is s i m p l y each

weak*

~e=~.

different

algebra

H For

invariant

continuous

Furthermore

functionals

situation

and

fix an i n n e r

z6D t h e n

multiplicative ~z(I)=z

linear

func-

so t h a t d i f f e r e n t

~z"

(H,~ z)

is a r e d u c e d

Szeg~

situation

J

The extension

of ~z

to H # in the s e n s e

of

IV.3.3

continues

to

be ~ z : U + ~ ( z ) . iv)

We h a v e

H

= I-z H.

~z 1-1I Proof: that

Re H

cause

Most

V.I.10

IV.3.3

of the

is w e a k , tells

come

ii)

weak, Assume Then

obtained iii) linear kest

that

the

result

DISK THEOREM:

that

this

after

extension

multiplicative

continuous

the

i) H

is true

and

For

ii)

iii)

observe

is t r u e b e -

to H # in the

sense

of

QED.

linear

invariant

are

iff H p o s s e s s e s

functionals

fix an i n n e r

in q u e s t i o n

3.1. And

section.

is s i m p l y

multiplicative

functionals

of ~z

extension.

of the

from

VI.4.9.

function

the ~z

other I6H

t h a n ~.

such

for the p o i n t s

that

z6D as

above. On the

set Z*(H)

functionals

topology

tinuous. Proof: cative

in Re L~(m)

to the m a i n

3.3 A N A L Y T I C nonzero

H =IH.

us

is the u n i q u e

Now we

a s s e r t i o n s are i m m e d i a t e

dense

Then

in w h i c h the m a p

I) A s s u m e

linear

of all n o n z e r o

on H i n t r o d u c e for e a c h

u6H

D÷Z*(H):Z~z

that

functional

~:H÷~ +~.

weak*

the G e l f a n d the

topology,

function

multiplicative that

~,(H)÷~:~(u)

is the w e a is c o n -

is a h o m e o m o r p h i s m .

is a n o n z e r o

Then

continuous

weak,

continuous

H~:={u£H:~(u)=O}cH

multipli-

is an i n v a r i a n t

159

weak~

closed

variant.

linear

In fact,

~6H~ but u ~ H obtain stant =/QFdm

of L~(m).

take a function Q6H~ with

function

since

is 6D. We introduce

IQ[=I

1-1bl~IQ-bl~1+Ib

that

and

~(u)~O.

( H g ) ~ H ~. F r o m

inThen

1.1 we

Then Q is a n o n c o n -

we had H~=H or ~=O.

Thus b:=~(Q) =

(1_~Q_b~+ibl2).

1-1bl 2

< G < ~
9(u)=/uGFdm

IGFdm QG6H with

H @ is simply

I we see that

0 < ~I- b

We claim

. Thus

such that H~=QH.

I

1-1bl 2

that

function

G:: ~

From

(H~) cH

otherwise the

We c l a i m

u6H such that ~ ( u ) = O

so that u is not in

a function inner

subspace

for all u6H.

To see this note

that

I (1-Sb-bZ+ b l2) 1_ibl 2 l = I,

~(QG)

= IQGFdm

= 1

(b_~b2_b+ibl2b)=O.

i-Ibi 2 NOW for u6H we have

u=9(u)+Qv

=9(u)+~(v)~(QG)=~(u) ation

(H,~)

with

as claimed.

is a r e d u c e d

Szeg5

v£H and hence

It follows situation

/uGFdm=](~(u)+Qv)GFdm=

that

with

the Hardy

algebra

representative

situ-

function

GF£ReLI(m). 2) W i t h ~=~.

@:H+~

It follows

H =IH.

as b e f o r e that

there

of c o u r s e

be =F.

proved

functions

of iii)

(H,@)

and to the functional

a function

IEH

I6H~.

a:=~(1)

Then

with

I!-aI2 I_IaI2GF 6 Re L1(m).

that

G = lqlal2 ii_al2 w h i c h

that

Now the functions TcvID.

let us i d e n t i f y

T is then

Zm~z~z(U)=~(z),

Yu6H w h e n c e

I) to

III=1

such

that

is E D and ~ : H ÷ ~ This

means

function that ~=~a"

must Thus

i) and ii).

topology

to prove

continuous

function

It follows

3) For the proof The G e l f a n d have

exists

Let us fix such a f u n c t i o n

has the r e p r e s e n t a t i v e

we have

we apply

the w e a k e s t

via the map

~ is the a b s o l u t e

~ are h o l o m o r p h i c On the other

hand

that ~IDcT.

value

on D and hence the i d e n t i t y

Thus

T=~]D.

Vu6H. We

topology

continuous

on ~.

in ~ID

A I=Z: (D,T)+(~,~)

QED.

z ~ z.

on D such that the

that is the functions ~ are c o n t i n u o u s

T=vID w h e r e

which means

DmE*(H) topology

is

160

4. T h e

Isomorphism

In the p r e s e n t and

the

unit disk

Szeg~

situations

plain

what

contain

theorem

±nvariant 3.3.i)

be t h a t

under

must these

(H~(D),~o)

fixed

inner

is n o t

be

so t h a t

too w e a k

satisfied.

in a s e n s e w h i c h I6H~ w i t h

map H#÷HoI#(D):u~

obtain

isomorphism

(X,X,Fm)+(S,Baire,l) is n o t m of

a measurable

s u c h maps.

But

between

then

as the

The

result

there

such

nevertheless

L(m)+L(1):foI~f.

striction

H # ÷ H # ( D ) :foI~f

we

It w i l l to be

inverse

of

that

H~ be

the a n a l y t i c

disk

is an i s o m o r p h i s m

between

as o n e

an

could

wish.

For

isomorphism

comes

from

in 3.1.

However,

map which

measure

comes

spaces.

out

to the a b o v e

(H,~)

the to

from a map above

class

a well-defined

to be b i j e c t i v e

will

a

we w a n t

N o w the

b u t an e q u i v a l e n c e

can o b t a i n

turn

i)-iv)

if the n o t i o n

comprehend of

H can-

section

substitution the b a s i c

Then

conditions

ex-

compre-

of the p r e s e n t

is as s h a r p H~=IH

H~(D).

(H,~)

reduced

we must

it s h o u l d

Furthermore

conditions

(X,X)÷(S,Baire)

tion map

H and

it s h o u l d

those

Of c o u r s e

that

the e q u i v a l e n t

established

between map

c a n be done.

the e q u i v a l e n t

between

to c h a r a c t e r i z e

be s a t i s f i e d .

two assumptions

function

and

It is c l e a r

isomorphism

evaluation the

this

is to be.

2.1 m u s t

so t h a t

to c o n s t r u c t i s o m o r p h i s m s

(H~(D),~o)

for w h i c h

zero d i v i s o r s

theorem

and

(H,~)

isomorphism

simply

we w a n t

situation

an a l g e b r a i c

the m a x i m a l i t y of an

section

an i s o m o r p h i s m

h e n d at least not

Theorem

I

modulo

substitu-

and

evalution

its map

reH #÷

H o l # (D) : u ~ .

We

start with

substitution

4.1 i)

definition

PROPOSITION:

Assume

I ( B ) = ( F m ) ([I6B])

ii)

For

sentative

each

jugation. iii)

for all

the b a s i c

there

fn,f6L(1)

We h a v e

I£H

is an

Baire se~

is a u n i q u e Baire

J:x~J(x)ES

Furthermore

For

for all

homomorphism

pointwise

iv)

f6L(1)

that

representative

functions

an a l g e b r a i c

and

properties

of the

desired

map.

d u l o m for all

plies

the

of

/fdl=f(foI)Fdm S O~f6ReL(1).

Then

foI6L(m)

I. T h e m a p

such

R:S÷~

of

that

foI=RoJ

f a n d all

L ( 1 ) + L ( m ) :f~foI

with

respect

mo-

repre-

is of course

to c o m p l e x

con-

injective.

pointwise

convergence

function.

BcS.

functions

and homomorphic

it is

inner

convergence

f oI÷foI n for all

fn÷f

in the L(m) f6L~(1).

in the L(1)

sense

im-

sense.

Hence

the

same

is t r u e

161

v) For

!IfIILp(l)= Ilf°lllLP

lip! ~ we have

(of c o u r s e w i t h

(Fm) for all f6L(l)

the v a l u e ~ i n c l u d e d ) .

For

f6L(l)

therefore

f£LP(I)~

foI6LP(Fm). vi)

F o r f6L(%)

Proof:

I) For a B a i r e

sentative

functions

equivalent c l a i m that functions isclosed

modulo

f 6 H # ( D ) ~ f o l 6 H #. function

J:x~J(x)6S

R:S÷~

the s u b s t i t u t i o n

of I leads

to f u n c t i o n s

m so that R o I : = R o J m o d m 6 L ( m )

under bounded Hence

pointwise

4) For R6L(B)

it f o l l o w s

~ iff RoI=O.

is w e l l - d e f i n e d

convergence

it m u s t be = B ( S , B a i r e ) .

for all O ~ R 6 R e L(S).

tion modulo

RoJ w h i c h

is w e l l - d e f i n e d .

and c o n t a i n s

It f o l l o w s

are 2) We

from 2) a p p l i e d

to

the s u b s t i t u t i o n

as c l a i m e d

in ii),

]R I that R is a null

and we h a v e

from i). Thus we h a v e

i)-v).

6) For

f=Z n w i t h n ~ O w e h a v e

foI=In6H.

foI=O

It r e m a i n s Hence

H~(D)

with

Then uioI6H with

(u£oI) ( f o I ) = ( u i f ) o I £ H .

It f o l l o w s

T h e n of c o u r s e

see this take an h6L1(1)

which annihilates

a n d the

take f u n c t i o n s

that f£L(1)

We c l a i m that f6H~(D). H~(D).

u£6

luloI]~1,uZoI÷1

t h a t f o I 6 H #. 8) A s s u m e

f6L~(%).

5) i i i ) i s

f r o m 1.3.3.iii)

7) F o r f6H#(D)

is such that foI6H.

func-

foI:=RoI6L(m)

iff f=O.

iii) we see that f 6 H ~ ( D ) ~ f o I 6 H . and u l f 6 H ~ ( D ) .

is i).

to p r o v e vi).

above

luzl~1,ui÷1

set we o b t a i n

Thus

for f£L(%)

the f u n c t i o n s

t h a t the r e l a t i o n

3) For R = X B w i t h BcS a B a i r e

immediate

and

of the r e p r e -

/ R d l = ] ( R o I ) F d m for all R 6 B ( S , B a i r e ) . In fact, the set of all S R 6 B ( S , B a i r e ) for w h i c h this is true is a l i n e a r s u b s p a c e w h i c h

R=Z n Vn6$. true

we h a v e

From VI.6.9.iii)

To then

h6H#NLI (1) w i t h

/hd%=O. H e n c e h o I 6 H # n L l ( F m ) a f t e r 7) w i t h ~(hoI) = S ] ( h o I ) F d m = /hd%=O. It f o l l o w s that / h f d l = / ( ( h f ) o I ) F d m = / ( h o I ) (foI)Fdm = S S = ~ ( h o I ) ~ ( f o I ) = O . H e n c e f6H~(D). 9) A s s u m e that f6L(l) is such that f o ~ H #.

Then

foI£~=L°(Fm)

and h e n c e

f6L°(1)

in v i e w of iv).

S i n c e L°(1)

is=L#(D): =

the c l a s s L # for the u n i t d i s k s i t u a t i o n

t h e r e are f u n c t i o n s

with

(uif)ol=(u£oI) ( f o I ) £ H # n L ~ ( m ) = H

luz]~1,

and hence

uZ+I

and u l f 6 L ~ ( ~ ) .

uzf6H~(D)

N o w let I6H

from 8).

be an i n n e r

tution map H#(D)÷H#:f~foI HOI#(D): u~

It f o l l o w s

function

the c o m p o s i t i o n

Assume

t h a t f6H#(D).

such t h a t H =IH0

can be c o m p o s e d

f r o m 3.1. The r e s u l t

4.2 P R O P O S I T I O N : Then

We h a v e

with

uz6H~(D)

QED.

T h e n the s u b s t i -

the e v a l u a t i o n

m a p H ~+ ~

is as follows.

that I6H

is an i n n e r

function

such t h a t H =IH.

162

H # (D) + H # + Hol # (D) :f ~ f o I ~

is the i d e n t i t y of

m a p in the s e n s e

(foI)^£Hol#(D).

that f6H#(D)

An equivalent

~z(f) = ~z(fOI)

(foI)A

formulation

Vf£H#(D)

is the b o u n d a r y is

and z6D,

where

the f i r s t ~z is r e l a t i v e

while

the s e c o n d ~z is in the s e n s e of 3.2.i).

Proof:i)

Let U6H~(D)

to the u n i t disk s i t u a t i o n

and u = < U I > 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)
and <0z(V)=~0z(VOI)+O

from i) it f o l l o w s

as w e l l

that Mz(f)=Mz(fOI)

QED.

F o r the s u b s e q u e n t 1_
I~

~ this

consequence follows

4.3 C O N S E Q U E N C E :

Assume

T h e n the e v a l u a t i o n

map u~

map H#nLP(Fm)÷HolP(D) Proof:

Immediate

observe

t h a t I6H

combined

is an i n n e r

is s u r j e c t i v e

for all

that H # ( D ) A L P ( 1 ) = H P ( D )

from I V . 6 . 1 0

function

in p a r t i c u l a r

combined with

for all

1.3.3. s u c h t h a t H =IH.

as a m a p H # ÷ H o I # ( D )

1<~p!~ , h e n c e

from 4.2 and 4.1.v)

with

and as a

as a m a p H÷HoI~(D).

the a b o v e remark.

QED. Let us k e e p (H~(D),~o)

we f o r m u l a t e equivalent

in m i n d

that we w a n t

via s u b s t i t u t i o n the d e c i s i v e

an i s o m o r p h i s m

m a p and e v a l u a t i o n

bijectivity

property

between

map.

(H,~)

and

In the n e x t r e m a r k

of t h e s e m a p s

in s e v e r a l

versions.

4.4 REMARK:

Assume

T h e n the s u b s e q u e n t The s u b s t i t u t i o n

that I6H

is an i n n e r

nine p r o p e r t i e s m a p f~foI

function

such that H =IH.

are e q u i v a l e n t .

is s u r j e c t i v e

and h e n c e b i j e c t i v e

as a m a p

163 1#)

L(1)+L(m),

I)

LP(1)÷LP(Fm)

I~)

L~(1)÷L~(m)

The

evaluation

V I ~ < ~, ,

map

u~

2#)

H # (D)÷H #

2)

HP (D) =H# (D) NLP (l) + H # A L P (Fm) V I~<_~,

2~)

H ~ (D) +H.

is i n j e c t i v e

and hence

bijective

as a m a p

3#) H#+HoI#(D),

The proof

will

3)

H#ALP(Fm)÷HolP(D)

3~)

H÷HoI~(D).

be a f t e r

the

I#)

V l ~ ! ~,

scheme

; 2#) <-

"~ 3#)

iI/i, if

1~)

The

implications

and t h e The

I#) ~ I) ~ I~)

implications

implications

Furthermore

is s e e n

alike.

Hence

of

there

lows

that

Proof

of

It r e m a i n s

exists

uoI=P

we

ISOMORPHISM

are

i) T h e An

are

and

I=) ~ 2~)

are

immediate

And

3~) ~ 3#)

are o b v i o u s .

to p r o v e

consequence

2#)=I~)

The

such

functions

at the m a i n

THEOREM:

from

4.1.vi).

and

after

2~) ~ 3~)

1~) = I#).

4.1.v).

VI.3.9

and VI.2.8.

such

that

([I6[v=O]])=(Fm)

It fol-

The a s s e r t i o n follows. U P = ~ w i t h U , V g L (m)

c a n be w r i t t e n

follows.

after

4.1.v),

is i m m e d i a t e

foI=uoI+i(voI)=P+iP ~.

that

u,v6L~(1)

l([v=O])=(Fm) assertion

and

after

of 4.2,

P+iP~6H # after

u 6 R e L~(1) P6L(m)

immediate

uoI=U

and voI=V.

([V=O])=O.

Hence

v£

QED.

result.

Assume

that

H~.

Then

the

subsequent

proper-

equivalent. invariant

invariant

some

2#) ~ 2) ~ 2~)

function

that

are n o w a r r i v e d

4.5 ties

and

an f = u + i v 6 H # ( D )

We h a v e

I 1 (L(1)) × a n d 5 o I = ~ .

-'~ 3 ~ )

is a d i r e c t

a n d of c o u r s e

see

--~

L e t P 6 R e L~(m). T h e n

I~)=I#) : E a c h

4.1.i)

We

2#) ~ 3#)

2#)~I~):

and m ( [ V = O ] ) = O . From

I#) ~ 2#)

3#) = 3) ~ 3~)

V.I.10.

Proof

:- 2 ~ )

weak~

P 6 T of m o d u l u s

subspace closed IPI=1

theorem

linear

is true

subspace

or of the

in its

TcL~(m)

form T=XvL~(m)

classical

is of the for

form: form T=PH

some V6Z.

for

164

H

ii) i) -iv)

is s i m p l y

invariant,

in the m a x i m a l i t y

iii)

H is s i m p l y

function

I6H

and hence

such

In this

for all

invariant, that

bijective

case

~z(f)=~z(fOI)

the

conditions

H = I H the

for

some

substitution

substitution

isomorphism

f6H#(D) a n d

(and h e n c e map

for each)

f~fol

is s u r j e c t i v e

H#(D)÷H#:f~foI

z6D and h e n c e

inner

satisfies

in p a r t i c u l a r

~o(f~p(foI)

f6H#(D).

Proof:i)=ii)

I) If H

H =XvL~(m)

Thus

H =O or H = ~ w h i c h

ly i n v a r i a n t .

for

H. T h e n

P6B w i t h

]PI=I

so t h a t

B=H.

B=L~(m).

ii)~iii)

has b e e n that

Vn~O.

Thus

closed

linear

condition

Let

in L~(m). P I ~ £ L (Fm)

I6H

linear

It f o l l o w s

is a w e a k .

subspace

16HOB

that

some

of the

of L~(m).

V£E.

But

3~)

then

in 4.4

be an i n n e r

combinations

of

such

u6H w i t h

that

of the

Xv=O.

H

is s i m p -

subalgebra

either

XV=I

that

Taylor

InH Vn~1 Also

ITcT

Xv£TCIH

B=PH

which for s o m e

and h e n c e

that

H =IH.Then

T:=

coefficients

and h e n c e so that

enforces

an=O

an i n v a r i a n t

ii)

that

implies Xv=O

so

is s a t i s f i e d .

function the

such

that

functions

To see

this

take

so t h a t

from

I) in 4.4 we o b t a i n

It f o l l o w s

that

closed

so t h a t

implies

function

functions

intersection

t h a t T=O.

the

case

subspace

for

iii)~i)

it w e r e

some V6E. In the first case p26B=PH or P6H

be an i n n e r

that T=XvL~(m) Thus

then

is f u l f i l l e d .

IEH

T is the

invariant

excluded.

for

second

2.1.i)

Let

simply

BcL~(m)

or B = X v L ~ ( m )

c o n s i s t s of the

weak*

not

B is an i n v a r i a n t

In the

Thus

{u£H:~=O}

were

some V6~. T h e n Xv6H ~ w h i c h e n f o r c e s

2) A s s u m e

contains

a function

H =IH.

In w i t h

PEL] (m) w i t h

n6~

I) We c l a i m are weak*dense

/Inpdm=O

a function

Vn6$.

u6L] (~) w i t h

Then P uoI=~.

/znudl=/(ZoI)n(uoI)Fdm=/Inpdm=OVn6$. Thus u=O and hence S 2) L e t TcL~(m) be an i n v a r i a n t w e a k * c l o s e d linear subspace. Then

P=O.

H T=IT the

and

of 4.4.

for all

the e q u i v a l e n t

2.1.

in the s e n s e

form

that

and H fulfills

theorem

that

so t h a t

invariant

hence =L~(m).

[T=T

so t h a t

Then

2.2

In t h e n e x t tion H~(D).

T =IT. subspace

theorem

1.1.

section

that

we

T=XvL~(m)

shall

prove

of a p u r e l y

an i s o m o r p h i s m

for

Assume

I,I6T(T) (see S e c t i o n

shows

is the e x i s t e n c e If s u c h

If T #T t h e n T = P H

some now

2).

for

that

Thus

some

then

IPI=I Then

I) i m p l i e s

after IT=T

that

and

T(T) =

QED.

V6E.

another

algebraic

exists

P6T with

t h a t T =T.

equivalent

isomorphism

H contains

no

condi-

between zero

H and

divisors,

165

which be

is c o n d i t i o n

simply

5. C o m p l e m e n t s

be

on

The

first

5.1

THEOREM:

Assume

Invariance

linear

OPERATOR

algebra

on V w h i c h

nonzero

vector

simply

5.2:

space

contains

the

We w a n t each

identity

ty I and

fix an e i g e n v a l u e

In fact,

Px6U.

invariant

successive

an o p e r a t o r

c6~

It r e m a i n s

is n o t

of R. T h e n to s h o w

for x 6 U we h a v e

that

dimV>1

N o w we are

a multiple

The

done

of the

which

under

means

if

identi-

U c V of R-cI

U is i n v a r i a n t

(R-cI)Px=P(R-cI)x=0

a

of V w h i c h

subspace.

the n u l l s p a c e

that

exists

in c a s e

linear

application.

R£A which

there

P6A.

subspace

to p r o v e

finite-

operator

is

each

that

QED.

5.3

REMARK:

ant weak~

Proof (T)~

Then

linear

upon

the p r o o f

folklore.

be a c o m m u t a t i v e

operator.

It s u f f i c e s

into

L e t V be a n o n t r i v i a l

a one-dimensional P6A.

So t a k e

and +V.

introduced

to e a c h

dimA=1.

P6A.

Let S,TcL~(m) T SeT and dim~< .

that

is an e i g e n v e c t o r

a nontrivial proper common follows

be

LEMMA:

there

nontrivial

invariant. such

a n d AcL(V)

under

then

must

H

result.

to be m a t h e m a t i c a l

ALGEBRA

vector

a6V which

will

seems

is i n v a r i a n t

assertion

is t h a t

of H

subspaces

hypothesis

lemma which

complex

exists

to be p r o v e d

illustrative

t h a t H is n o t

closed

5.2 C O M M U T A T I V E

of

fact

at all o b v i o u s .

subsequent

finite-dimension

Proof

the

.

an a l g e b r a i c

dimensional

Thus

is n o t

simple

is the

weak~

S = T or S=T

The via

This

the

aim

invariant

Then

2.1.iv).

invariant.

closed

of 5.3:

implies

Proof

Assume

of

S=T,

and

that

dim(T/T)

will

be p r o v e d

linear

by

is not

subspace

H Tc(T

i)

in the c a s e =I

H

The assumption

that

5.1:

that

T %T the

is that

induction

For each

H =LinH~H~weak~ that

to p r o v e

invariant

indeed

invariant.

invari-

TcL~(m)then(T~)~=T~.

)~ a n d h e n c e

It s u f f i c e s

so t h a t

simply

T mS.

.

dim(T/S)=:n.

Thus

H H TcH T~

) . QED.

In the

subspace

S = T or S = T

after

T c(T

c a s e T =T t h e n

theorem

ii) The

The

1.1

implies

assertion

case n=O

T mS

is t r i v i a l .

166

So let us t u r n to c o n s t r u c t that

ScUcT

with

hypothesis which

subspace

It is o b v i o u s a n d c6~, quired ~:H÷~

U we

such

e(I)=I.

iii)

as =

that

and

~ is w e a k ~

3.3.i)

theorem

uf-~(u)f6S

The lemma

other which

tells

main

then

Proof:

i) T h e

the L ~ ( m ) - n o r m s . theorem

interest

be a w e a k ~

linear

map

proof

that

consequence

such

that

From

i) w e o b t a i n

Alaoglu Then

f6T

T÷PT:f~Pf

a constant

sequence

since

e=~ a f t e r

the a n a -

invariant. Then

that

Thus

we

it is c l e a r

that

that

S c U c T a n d dim(U/S)=1.

HUcS.

QED.

a functional

linear

analytic

subspace

and P6L=(m),

If PT is L ~ ( m ) - n o r m

is s u r j e c t i v e

C>O

closed

closed such

and continuous

then

that

in

the o p e n m a p p i n g

to e a c h

g6PT

there

~cHgilL~ (m)

(m)

PT is w e a k ~

IV.3.14.

Thus

closed

functions

exists

a weak~

will

consider

Ign =
there

that

and

L The

with {u6H:

such

as well.

if PT is L ~ ( m ) - n o r m

us w i t h

g=Pf

lian

It is c l e a r

nullspace

requires

closed

subspace

IIfll ~ ii)

u6H.

functional

closed.

Thus

provides

is an f6T w i t h

section

is as re-

its

simply

and

Vf6T.

for u , v 6 H

in itself.

is a l i n e a r

PT is w e a k *

for all

of L~(m)

T/S.

f~S a n d a f u n c t i o n

linear

U:=S+~f.

U is i n v a r i a n t

of the

L e t TcL~(m)

PTcL~m)

is n o t

subspace

us that

result

is of

5.4 LEMMA: so t h a t

linear

H

=

It f o l l o w s

Now define

space

:[f]~[uf]

an f£T,

5.3,

intermediate

quotient

since

such

induction

A:={:u6H}cL(T/S)

continuous

since

the

the

via

and

exists

closed.

for all u6H.

closed

above

Thus

it s u f f i c e s

UcL~(m)

) cU cS a f t e r

to c o n s t r u c t

6L(T/S)

there

From

T =(T

is a m u l t i p l i c a t i v e

lytic

the

U cS.

the n - d i m e n s i o n a l

identity.

is w e a k *

And

Then

T cU so t h a t

< u > [ f ] = £ ( u ) [f] or u f - ~ ( u ) f 6 S

Furthermore

U is a w e a k ~

linear

=+,<cu>=c

is u n i q u e

disk

subspace

operator

g(u)=0}={u6H:uf6S}

have

To do this

In o r d e r

consider

It f o l l o w s

that

step O~n-1~n. closed HUcS.

that

a linear

as w e l l

g:H÷~

and

follow

that

in 5.2.

weak~

dim(U/S)=1

it w i l l

u6H defines

that

induction

is the a s s e r t i o n ,

linear Each

to the

an i n v a r i a n t

pointwise.

fn6T with

gn=Pfn

adherence

T is w e a k , closed.

be b a s e d

a sequence

For

l~n(1)<...
point

on the K r e i n - S m u of

functions

The claim and

Ifnl =< C b < ~ ° A f t e r

f6L~(m)

each

fixed

such

that

gn6PT

is t h a t g£PT.

of the

u6L1(m)

Banach-

fn for n÷ =.

there

is a sub-

/UPfn(z)dm÷/uPfdm

for£+~.

167

But /uPfn(z)dm=/Ugn(z)dm÷]ugdm. and hence

that g = P f 6 P T .

5.5 T H E O R E M : situation o:H÷H.

that

(H,~)

5.6 C O N S E Q U E N C E :

converse

invariant

Assume

H and H~(D).

the e q u i v a l e n t

(H,~) on

(X,~,m)

and that there

If H~ is s i m p l y

between

that / u g d m = / u P f d m

Vu6L1(m)

QED.

Assume

besides

It f o l l o w s

conditions

then H

is s i m p l y

i)-iii)

reduced

an a l g e b r a i c

is s i m p l y

that t h e r e e x i s t s

Then H

is a n o t h e r

exists

invariant

an a l g e b r a i c

invariant.

Hence

in the i s o m o r p h i s m

Szeg6

isomorphism as well.

isomorphism (H,~)

satisfies

theorem

4.5(the

is trivial).

P r o o f of 5.5:i)

W e use the n o t a t i o n

that o:H÷H preserves

L~-norms.

H iff f° is i n v e r t i b l e

o : f ~ f O for f6H. W e s h o w f i r s t

In fact,

in H. H e n c e

a function

the s p e c t r u m

cides with

the s p e c t r u m

= spectral

r a d i u s we o b t a i n I ] f l ~ ( m ) = lif~Ii L~(m)

I 6 H ~ be an i n n e r

of f° r e l a t i v e

function

to H. S i n c e

such that H~=IH,

f6H is i n v e r t i b l e

of f r e l a t i v e

in

to H coin-

in b o t h c a s e s n o r m =

Vf6H as claimed,

ii) Let

and let I6H such that I°=Io

Then o:H

)

U

U

IH U

>iN = N~ U

I2H

where

in e a c h

line we have an i s o m o r p h i s m

i). T h e l i n e a r III=1.

Hence

spaces

in the s e c o n d c o l u m n

the l i n e a r

and t h e r e f o r e

1~2~,

weak~

u n d e r H. F u r t h e r m o r e

spaces

closed

the c o d i m e n s i o n s are all=1

in the f i r s t column.

force b o t h IH and I2H to be =H It f o l l o w s

If n o w H

that H

L~-norms closed

are L ~ ( m ) - n o r m

5.4. A n d t h e y are of c o u r s e

hence

6. A C l a s s

are L ~ ( m ) - n o r m

in the f i r s t c o l u m n

after

would

tradiction.

which preserves

closed

invariant

in the s e c o n d c o l u m n

w e r e not s i m p l y

invariant

so that we w o u l d a r r i v e

m u s t be s i m p l y

after

since

invariant.

and

then 5.1

at a con-

QED.

of E x a m p l e s

We construct of e q u i v a l e n t

a c l a s s of e x a m p l e s

properties

which

in o r d e r

dominate

to show that the two sets

the p r e s e n t

c h a p t e r are independent.

168

The measure space is nate functions ~Z I

Plz2P2

(X,E,m) :=(SxS,Baire,lxl).

We have the coordi-

Zl,Z2:Zz(s)=s ~ for s=(sl,s2)6X and the monomials

for p=(pl,P2)6$×$.

zP: =

It is an immediate S t o n e - W e i e r s t r a S

conse-

quence that the linear c o m b i n a t i o n s of all these m o n o m i a l s are weak~ dense in L~(m).

For each subset FcSx$ with 06F and F+FcF we define -weak~ H(F)

Thus H(F)

:= Lin { zP:p£F}

cL~(m).

is a weak~ closed complex subalgebra of L~(m)

the constants. H(F)+~

We need two immediate properties,

is weak~ dense in L~(m),

R e L Y ( m ) . ii) If rD(-F)={O} tive on H(F). Therefore:

which contains

i) If FU(-F)=$x$

then

that is Re H(F) is weak~ dense in

then the functional ~ : u ~ / u d m is m u l t i p l i c a -

If Fu(-F)=$x$

a reduced Szegb situation with M={I}.

and FN(-F)={O}

then

(H(F),~)

is

In the sequel we fix a subset

FcSx$ w i t h all these properties. 6.1 PROPOSITION: i)-iii)

(H(r),~)

in the i s o m o r p h i s m

same time possess

does not fulfill the equivalent conditions theorem 4.5. That is

the equivalent properties

theorem 2.1 and have a simply invariant

(H(F),~)

i)-iv)

(H(F))

cannot at the

in the m a x i m a l i t y

.

Proof: Assume that this is not true. Then there is an inner function I£(H(F))~ such that H(F):f~foI

(H(F)) =IH(F)

is bijective.

zP=fpOI with fp6H~(D).

and that the s u b s t i t u t i o n m a p H~(D) +

Also ~o(f)=~(foI)

Then fo=1 while

for all f6H~(D). For p6Ft/lus

for nonzero p6F we have • o (fp) =

=~(ZP)=O so that after 1.3.6 there is a unique N(p)6~ such that fp =zN(P)h P with hp6H~(D)

and ~o(hp):~O. Of course we put N(O)=O.

implies that m(p+q)=W(p)+N(q).

For p,q6F then fp+q=fpfq

For p6$X$ not in I' we have p%O and -p6F

so that we can define N(p) :=-N(-p)
The function N:$x$÷$

fies N(p+q)=N(p)+N(q)

as a look at the different possi-

bilities

shows,

for all p,q6~X$

then satis-

and we have N(p)%O w h e n e v e r p~O. But this is impossible:

we have N ( p ) = a l P 1 + a 2 P 2 Yp6$X$ of nonzero p6$x$ with N(p)=O.

for some a1,a26 $, so that there are lots This c o n t r a d i c t i o n proves the assertion.

QED.

After the above result it remains to show that the subset FcSx$ can be chosen such that equivalent

(H(F))~ is simply invariant,

conditions

i)-iv)

in the m a x i m a l i t y

and also such that the

theorem 2.1 are satisfied.

169

6.2 P R O P O S I T I O N : a]inonzero

Proof: prove

p6F.

Assume

Then

We h a v e to p r o v e

z-af6H(F)

the i n c l u s i o n

/zPdm

6.3 EXAMPLE:

Let

Thus

(H(F))

are = O a s

tions

f6L~(m)

a consequence

F consist

and ii)

s£S d e f i n e s

a member

The o t h e r

mality

that z - a f £ ( H ( F ) ) # D for p=O,

task

p=(pl,P2)6$x$

QED.

with P1>O

that is of the p > O in the l e x i c o g r a of 6.2 w i t h a = ( O , 1 ) .

It is n o t h a r d to see t h a t H(F)consists the f u n c t i o n

all s6S the s e c t i o n

S~o(f(-,s))=/f(-,s)dl

of the f u n c f(-,s)6L~(1)

for h - a l m o s t

all

of H~(D).

is

Gelfand

and for n o n -

of the a s s u m p t i o n s .

of the p o i n t s

such t h a t i) for l - a l m o s t

is in H~(D)

commutative

implies

is o b v i o u s

of $x$. T h e n we are in the s i t u a t i o n

=Z2H(F).

. In o r d e r to

V p6F.

equation

a n d of t h o s e w i t h P 1 = O and P2>O, phic ordering

c. Fix an f6(H(F))

so t h a t V I . 6 . 9 . i i i )

B u t the d e s i r e d

zero p£F b o t h s i d e s

for

to s h o w that

= /z-afdm

For t h e n z - a f 6 K A L ~ ( m )

a £ F is s u c h that p - a 6 F

=ZaH(F).

it s u f f i c e s

/z-afZPdm

NL~(m)=H(F).

t h a t the n o n z e r o

(H(F))

somewhat theory

more

involved.

combined with

We n e e d a b a s i c

the s u b s e q u e n t

fact from

restricted

maxi-

theorem.

6.4 R E M A R K :

Let

(H,<0) be a r e d u c e d

BcL~(m) is an L ~ ( m ) - n o r m that ~ can be e x t e n d e d

closed

complex

Szeg6

situation.

subalgebra

to a m u l t i p l i c a t i v e

linear

which

Assume

that

contains

functional

H such

on B. T h e n

B=H. Proof:

Let ~ : B ÷ ~ be a m u l t i p l i c a t i v e

T h e n ~ is L ~ ( m ) - n o r m ded l i n e a r e x t e n s i o n implies A(f)

u6H w i t h and hence M={F}. remains

A(f)>O

for all f 6 R e L ~ ( m ) .

Re u > f = R e ~ ( u )

~°(f)~A(f)

It f o l l o w s

Vf6L=(m).

functional

of ~ w i t h IIAII= I. A f t e r A . I . 5

t h a t A is p o s i t i v e :

is r e a l

linear

which

continuous w i t h II~II =1. A n d let A : L ~ ( m ) ÷ ¢

Thus

that A ( f ) ~ / f F d m

multiplicative

IIAII = A ( 1 ) = and h e n c e

I

that

for f £ R e L ~ ( m ) w e o b t a i n

= Rei (u) = A(Re u) ~A(f) ,

f r o m IV.2.5.

In p a r t i c u l a r

then

for all O ~ f 6 R e L ~ ( m ) ,

e x t e n d s ~.

be a b o u n -

~(f)=/fFdm on B. Thus

But ~ ° ( f ) = / f F d m Vf6ReL~(m)

from I V . 3 . 1 0 w h e r e

and h e n c e

that A(f)=ffFdm

Vf6B so t h a t the i n t e g r a l f6B i m p l i e s

f~/fFdm

that fF6K and h e n c e

f6H#N

170

DL~(m)=H

after VI.6.9.iii).

N o w for Fc~x~ that

as a b o v e d e f i n e

for each n o n z e r o

F+FcF

so t h a t H(~)

expect

It f o l l o w s

subalgebra

Proof:

It is c l e a r

If BcL~(m)

Then

that ~ U ( - ~ ) = $ x $

if ~ = $ ~

~:B÷{:

z-P6H(F)cB

is i n v e r t i b l e

there exists an n 6 ~ s u c h

such Fc~ and

but we must

is a m a x i m a l

6.6 EXAMPLE:

with H(F)~B proper weak~

This

~(zq)=o.

It f o l l o w s

[=$x$.

Thus

for

to prove linear

theory

that

So let us a s s u m e

then a nonzero

is i m p o s s i b l e

that

q6F and c h o o s e (~(zq)) n=

that 9 ( z q ) = ~ ( Z q) in v i e w of 6.4.

for all Hence

it

QED.

For ~6~ an i r r a t i o n a l

p1+~P2~O}={O}U{p:p1+~P2>O}.

Gelfand

from z n q = z p + n q z -p we o b t a i n

But this

as claimed.

that zP6B.

Take

is c l e a r

In o r d e r

multiplicative

from c o m m u t a t i v e

~(z-P)=o.

Then

and h e n c e

and h e n c e ~ I H ( F ) = ~ . be ~ ( z - P ) + o

for e a c h n o n z e r o

in B and h e n c e

that p+nq6F.

~(Zp+nq)~(z-P)=o

the d e s i r e d

t h e n H(F)

subalgebra

t h a t zP6B for e a c h p6[.

then it f o l l o w s

such a ~ w i t h

tions a n d t h a t

closed

t h a t p6~ a n d p{F so t h a t p ~ O and -p6F.

functional

must

of the p 6 $ x $

of L~(m).

W e h a v e to p r o v e

So a s s u m e

is a w e a k ~

t h a t zP6B w e s h o w that ~ ( z - P ) ~ o

q6F

to c o n s i s t

that ~N(-~)#{O}.

t h e n H(~)c_B. In p a r t i c u l a r

p6F.

~c~x$

QED.

qEF we can achieve p+nq6F for some n6~.

is d e f i n e d .

6.5 P R O P O S I T I O N :

closed

that B=H.

number

It is o b v i o u s 6.5 tells

define

F:={p=(p1,P2)6$x$:

that F f u l f i l l s

us that

(H(F),~)

our a s s u m p -

is an e x a m p l e

of

kind.

Notes

The e v o l u t i o n [1949]

of the i n v a r i a n t

who discovered

idea of the a b s t r a c t

the L 2 - v e r s i o n proof which

is d u e to H E L S O N - L O W D E N S L A G E R the u s u a l v e r s i o n

[1958].

condition.

f r o m the fact t h a t it m i s s e d

theorem

started with BEURLING

in the u n i t disk s i t u a t i o n .

carried

of the i n v a r i a n t

to the w e a k ~ D i r i c h l e t let. A l s o

subspace

as far as to the S z e g ~

SRINIVASAN-WANG

subspace

theorem

[1966]

t h e i r p r o o f of the e q u i v a l e n c e

implication theorem

o b s e r v e d that

is in fact e q u i v a l e n t

But t h e i r e q u i v a l e n c e

the d e c i s i v e

The basic situation

theorem

suffered

Szeg6~ weak~Dirich-

s e e m s to us to beircomplete

171

(in the d i a g r a m sive

because

MERRILL-LAL

on p.

[1969]

invariant

is in A H E R N - S A R A S O N

MAN

width

[1962a],

and depth

is f r o m S A L I N A S

[1976].

in c a s e

termined

in N A K A Z I

of m a x i m a l

dorff

space.

c a n be

The

the

It f u r n i s h e s

of a n o n - m a x i m a l [1976].

quite

to t h e w e a k ~

The maximality subalgebras

See W E R M E R

[1961]

where

also

some

of H O F F -

The present

information

of C(X)

algebras

of t h e s e

is o f t e n

B are de-

dealt

with

proof

o n the o - a l -

intermediate

classes

the

situation

is of an im-

treatises

[1972].

theme

closed

clas-

[1973].

closed

H. C e r t a i n

Szeg6

theme

in the

conclu-

situation

to c e r t a i n

the

subspace

to M U H L Y

is n o t

Szeg~

theorem

RADJAVI-ROSENTHAL

is due

(vii) the

A step beyond

invariant

proper

analytic

spectra

is of w i d e

we r e f e r

The

essence

as in 4.1 [1976].

theorem

with

X a compact

respective

in Haus-

version

of 6.6

appeared

for the

the

new result was m

first

to W E R M E R

(in the

Dirich-

analytic

interest

and was

in fact

of the m a i n

theory.

[1969]

For

the

Chapter

VI.

isomorphism was

first

earlier in L U M E R

theorem

time.

Section

results

theorem

4.5

in M ~ R M A N N [1966b].

5.5 w i t h

one

5.4

6 is f r o m

beyond

Theorem

SALINAS

certain

direct

no p r i o r

explicit

example

of a r e d u c e d

invariant

the

The 5.1

is due

replaces

simply

structures

situation

[1968].

is due

[1976]

verifications.

(and n o n t r i v i a l ) .

in for

The

substitution

to K ~ n i g

Szeg~

stimuli

Szeg~

is in M E R R I L L

[1967].

6.1

not

[1960]

to d i s c o v e r

The p e r m a n e n c e

here

is d u e

problem

of the

i)~iii)

3.3

The

of the

to G A M E L I N

equivalence

H

disk

situation).

the d e v e l o p m e n t

with

2.1

to

Whithin

found.

let a l g e b r a

there

The

and

correspond

H~B~L~(m)

terms

leads

as it is d e m o n s t r a t e d

[1964]

theorem

~cZ which

which

H~:=H2nL~).

subspaces.

[1967a].

HELSON

The maximality

gebras

the a r r o w

a n d N A K A Z I [1975] e x t e n d e d

ses of n o n - s i m p l y

pressive

239

of the d e f i n i t i o n

to

and

except

appears that

It seems

situation

map

SALINAS

that (H,~)

Chapter

Weak

The tion

present

(H,~)

sentative This

chapter

on

functions

assumption

VI.6.3.

the

was

result

of i n d e p e n d e n t

will

parts,

linear

gence

for s e q u e n c e s topics

will

finite

I. T h e

there

exists that

T(f)

= T(fXE)

nal

the

of

with

one

subspace

1.1 T H E O R E M :

Each

of

repre-

and

compact

in-

two

auxiliary

means

decomposi-

into w e a k . c o n t i n u o u s notion

to o u r p u r p o s e s I and

and

of c o n v e r -

in L~(m).

2 where

we

fix

a non-

(X,E,m).

of H e w i t t - Y o s i d a

m(E)<~

iff

to be

such

that

collection

is c l o s e d

Furthermore

requires

is a p a r t i c u l a r

space

under

there

singular T(f)

iff

= T(fXE)

of s u b s e t s

intersection.

exists

for e a c h

~>0

Vf6L~(m).

E6E Hence

such

that

a functio-

a decreasing

sequence

and

r(f)

= T ( f X E ( n ))

Vf6L~(m)

it is o b v i o u s

that

the

functionals

that m(E(n))+O

a linear

in I V . 4 . 5

is the H e w i t t - Y o s i d a

adapted

T 6 ( L ~ ( m ) ) " the

such

situa-

O(ReLI ( m ) , R e L ~ ( m ) ) .

t h a t M is w e a k l y

in S e c t i o n s

measure

Vf£L~(m)

algebra

set M of the

topology

on L~(m)

functions,

a set E6~ w i t h

E(n)6E

(n=I,2,...). form

one

T6(L~(m)) " is d e f i n e d

for any

Hardy

the

consideration

case

The proof

second

T 6 ( L ~ ( m ) ) " is s i n g u l a r

subsets

short the

first

Theorem

that

N is f i n i t e - d i m e n s i o n a l .

3.3.

be d e a l t

Decomposition

Note

that

The

positive

A functional

under

be

case

in the w e a k

functionals

while

These

trivial

case

of M

to the r e d u c e d

special

in IV.4

interest.

of b o u n d e d

singular

already noted

important

Our main

tion

in the

is c o m p a c t

As w e h a v e

cludes

Compactness

is d e v o t e d

(X,E,m)

Vlll

singular

of

(L~(m)) ".

functional

16(L~(m)) " a d m i t s

a unique

decomposi-

tion

I = Gm + T

where

G6LI(m)

(that

is l ( f ) ~ O

: l(f)

= ffGdm

+ T(f)

a n d T 6 ( L ~ ( m ) ) " is s i n g u l a r . whenever

f~O)

iff G ~ O

Vf6L~(m)

In p a r t i c u l a r

a n d T is p o s i t i v e .

,

I is p o s i t i v e

173

We remark that the above decomposition The Gelfand

transformation

tive B'algebra

L~(m)

the functionals measures

with C(K)

isomorphism

on the structure

16(L~(m)) " are in one-to-one

~6ca(K)

=

positive measures ~6Pos(K).

can be interpreted

is an isometric

(C(K))',

and positive

~. In particular

as follows.

of the commuta-

space K of L~(m).

correspondence

functionals

i correspond

respect to ~

to the Lebesgue

(see BARBEY

to

m itself has a representative

Now it is not hard to see that the above decomposition

16(L~(m)) " corresponds

Hence

with the

[1975] p.524

decomposition

of ~6ca(K)

for more details).

of with

This remark

will not be used in the sequel. The proof of 1.1 requires quent

the implication

iii) = i) in the subse-

lemma.

1.2 LEMMA:

For a positive

T6(L~(m)) " the subsequent

properties

are

equivalent. i)

T is singular.

ii)

Inf{fhdm + ~(1-h)

iii)

If O~F6LI (m) with ~hFdm~T(h)

Proof of 1.2: i)~iii)

: h6L~(m)

ii)~i)

Shndm < 2-2ns

Define A(n) :=[hn~2-n]6E

T(1-hn)

union of the A(n). that T(f)

= T(fXA)

f ~ some positive

o:o(f)

Define

implies VfEL~(m).

< 2-nT(1)

= ~(fXA).

we have

From this the asser-

functions

hnEL~(m)

with

(n=I,2,...).

~ h n ~ 2 -n + XA(n ) . From the

< 2-ns while the second one implies ~ T(1)

- 2-n+1T(1).

Let now A6Z be the

And in view of ~(1) ~ Y(XA) ~ T(XA(n ) )

that T(1)

= T(XA).

We can of course

c and hence O ~ T(f-fxA)

so that indeed T(f) iii)~ii)

- 2-nT(1)

Then m(A)~s.

the above inequality

then F=O.

fFdm = IFXEdm.

so that 2-nxA(n)

first relation we obtain m(A(n)) that T(XA(n )) ~ T(hn)

VO~hEL~(m)

Fix s>O and choose

and

= O.

For E6Z with [(f)=T(fXE ) Vf6L~(m)

IF(1-kE)dm ~ T(I-X E) = 0 and hence tion is clear, O~hn~1 and

with O~h~1}

From this it follows assume that f~O. Then

= %(f(I-XA))

Thus T is singular

~ T(c(I-XA))= O

as claimed.

the functional

= Inf{Shdm + T(f-h):hEL~(m)

with O
for O
174

We c l a i m

the

subsequent

I)

0 < o(f)

2)

~(tf)

3)

o(f+g)

Here

< ffdm

= t~(f)

with

for e x a m p l e

extend

o to a p o s i t i v e

la(f) I ~ f l f l d m

extension

course

F must

of

O~h~f+g

linear

be ~ O. N o w Thus

1.1:

in v i e w

of

which

2) W e p r o v e

note

that F,G6T

Sf S u p ( F , G ) d m I(fX[F~G]) Put

implies

+ I(fX[F
Gn:=

Sup(FI,...,Fn)

Beppo

Levi

theorem.

linear

I as claimed. trary that

then

F6L~(m)

1) the

obtain

assumption

assertion

assertion

the

of m o d u l u s

is c l e a r

that we via

represen-

IFI~I.

iii)

ii).

and

Now

Icl=l

. Hence

Of

implies

QED.

f r o m the

defini-

of the d e c o m p o s i t i o n

for

%(f)

Sup(F,G)6T

= 1(f)

:= 1(f)

Gn+G6T

we h a v e

Rel

and

f£ReL~(m).

Fn6T

such

that

with

fGdm

= s after

- ffGdm

VfEL~(m)

of the

above

a decomposition

the e x i s t e n c e

if

of

for O~fEL~(m).

< ~ and choose

Thus

consider

in v i e w

+ SfX[F
~£(L~(m)) " . In v i e w

1.2.i).

is r e a l - v a l u e d

for all O ~ f £ L ~ ( m ) } .

of the

Iml

define

and

= Sup{1(h) :h6L~(m)

with

O~h~f}

the

a posi-

it s a t i s f i e s ~ = Gin + T of

decomposition

separately

Then

S F n d m ÷ s.

defines

thus

for can

the p o s i t i v e

1 + of I to be

1+(f)

O~u~f

clear.

c with

~ flf[dm

6(LI (m)) " w e

is the

6T and h e n c e

3) To p r o v e

16(L~(m)) " we l(f)

Now

functional

and hence

are

complex

the e x i s t e n c e

that

= ffX[F~G]Fdm

s: = S u p { f F d m : F 6 T }

1.2.iii)

h = u+v where

16(L~(m)) " . D e f i n e

Then

tive

To p r o v e

o 6 ( L ~ ( m ) ) ". It f o l l o w s

cf) ~ ~([f[)

for some

c(I)=0

~ is o b v i o u s .

details

for some

functional

I) T h e u n i q u e n e s s

functional

The

functional

T = {O~F6LI(m) : f f F d m ~ 1(f)

First

inequality

can be w r i t t e n

= o(Re

Vf6L~(m)

t i o n of s i n g u l a r i t y . a positive

3) the

u = Inf(h,f).

= ~(cf)

= ffFdm

F = O or o=0.

Proof

In

of o to a l i n e a r

o(f)

f6L~(m).

V f 6 L ~ ( m ) : in fact,

Io(f) I = c~(f)

tation

that

with

f£L~(m).

for 0 __< f , g 6 L ~ ( m ) .

are clear.

t h a t h6L~(m)

for 0 <

for t > O a n d O <

O=
have

a n d < T(f)

= o(f)+o(g)

I) a n d 2)

note

properties.

for O ~ f 6 L ~ ( m ) .

arbiassume part

175

As in the p r o o f of 1.2.iii)=ii) tional

l+6(L~(m)) " w h i c h

functional

as well.

The

it e x t e n d s

is such that result

to a p o s i t i v e

linear

func-

I-:= l + - 1 6 ( L ~ ( m ) ) " is a p o s i t i v e

follows

upon a p p l i c a t i o n

of 2) to I+

and I-. QED.

2. S t r i c t C o n v e r g e n c e

The c r u c i a l change

step in the p r o o f of the m a i n r e s u l t

of two l i m i t p r o c e s s e s .

is to use a s o p h i s t i c a t e d w h i c h w e call s t r i c t those properties

3.3 is the i n t e r -

A n e l e g a n t w a y to m a s t e r

n o t i o n of c o n v e r g e n c e

convergence.

of the s t r i c t

The p r e s e n t

convergence

this d i f f i c u l t y

for s e q u e n c e s

section

which

in L~(m)

is to d e v e l o p

are n e e d e d

in the

sequel. Assume f stricty

that

fn,f6L

(m)

We say that the fn c o n v e r g e

(n=I,2,...).

iff fn÷f p o i n t w i s e

to

and in a d d i t i o n

co Ifll

+

~ Ifn-fn_11 n=2

W e note two i m m e d i a t e

Ifnl ~

Ifll +

properties,

~ Ifl-fl_ll I=2

In p a r t i c u l a r

it f o l l o w s

in L ~ ( m ) - n o r m

then t h e r e e x i s t

fn(1)

÷ f" In fact,

Then ~

I

Assume

i) If fn÷f s t r i c t l y

=< c o n s t < ~

that f f n G d m + f f G d m

choose

1=2 2.1 REMARK:

< const < ~

strictly

then

(n=1,2 .... ).

v G 6 L I (m). ii)

convergent

If fn ~ f

subsequences

1
such that

IIfn (1)-fn (i-i)II <

t h a t h£L(m)

with

Re h > O and let O
+ I strictly.

Proof:

It is s u f f i c i e n t

to s h o w t h a t

for 1 = t ( O ) > t ( 1 ) > . . . > t ( n ) > t ( n + l ) >

> . . . > 0 we have

I_~

+

~ 1=I

i I )z 1+t(1)-----~- 1+t{i-1

~

V Z6~ w i t h Re z > O.

176

To see this fix z = u + i v w i t h = I/(1+tz) VO<__t<1. T h e n

u>=O and R:= 1 zI>O and d e f i n e b:b(t)

z

R

b" (t)

Ib" (t) i (1+tz)2

R

< 1+2tu+t2R 2 = 1+t2R 2

'

=

vO!t!1 -- _ •

Thus

Ib(t(1))-b(t(1-1)) 1 1=I

+

<

+

1+t(1) z

1+t (i-I

Ib" (t) Idt <

in R~O.

f6L~(m)

< - -

+

~+jds=:B(R),

shows

o

t h a t B(R)

is m o n o t o n e

in-

QED.

For a l i n e a r s u b s p a c e

Sc-L~(m) w e d e f i n e

such t h a t t h e r e e x i s t s

strictly.

~

(I/jiq+u)2+v2

is ~ ~ s i n c e d i f f e r e n t i a t i o n

creasing

1=I

+

+

o which

z

Of c o u r s e

ScL~(m)

a sequence

is a l i n e a r

S to c o n s i s t of f u n c t i o n s

subspace

of the

functions

fn6S w i t h ^fn+f

w i t h SoS,

a n d S=S w h e n -

e v e r S is w e a k , c l o s e d .

L e t Sc.L~(m) be a l i n e a r defined

to be s t r i c t l y

that l ( f n ) + l ( f ) . ¥f6S

of the s e q u e n c e 2.2 LEMMA: are l i n e a r

property

lemma.

space

11

(I=1,2,...)

series

continuous

F i x fn£S w i t h

for n~2 t h e n

(see K O T H E

f =

fn+f6S

convergence

[1966]

implies = ~fGdm

If I is s t r i c t -

is f o r m u l a t e d

sequential

in the

completeness

p.283).

subspace.

Assume

then I is s t r i c t l y

strictly.

with

ii)

such t h a t ll(f)+l(f)

that I I , I : S + ~ Vf6S

We h a v e

for i÷~.

continuous

If w e p u t g1:=fl

[ gn and Ignl ~ const. n=l n=1

X(g n) is c o n v e r g e n t

i) If l(f)

continuous,

upon the w e a k

L e t Sc-L~(m) be a l i n e a r

functionals

If all Ii are s t r i c t l y

Proof:

rests

I : S ÷ ~ is

continuous.

of the s t r i c t

The proof

functional

properties,

then i is s t r i c t l y

then 1IS is L ~ ( m ) - n o r m

The decisive subsequent

A linear

iff fn6S and fn÷f s t r i c t l y

We n o t e two i m m e d i a t e

for some G6LI(m)

ly c o n t i n u o u s

subspace.

continuous

as well.

and gn:=fn-fn_1

to p r o v e t h a t the

s u m = X(f) . F o r ~ =

(~n)n 6 1 ~ w e de-

n=1 fine

f~:=

~1~ngn6S. n

For each

1~I then the s e r i e s

~ l ~ n l l ( g n ) is conn-

177

vergent each

with

s u m = ll(fe).

e 6 1 ~ we h a v e

<e,Tl>

From

=

[ ~nT~ n=l

the w e a k

such

Now

first

in p a r t i c u l a r

=

that

TI: =

(I 1 (gn) ) n 6 1 I. T h u s

for

convergence

[ ~ n l l ( g n ) = l l ( f a) ÷ l(f ~) n=1

sequential

T=(Tn)n611 choose

the

In p a r t i c u l a r

completeness

<~,TI>

÷ <~,T>

~=en=(o,...,O,1,0

[ [l(gn) I<~. T h e n n=1

I

of 1

we

obtain

V ~ 6 1 ~. T h u s

e=(1,1

i+~.

an e l e m e n t = ~(f~)

<~,T>

.... ) to o b t a i n choose

for

T n = l(gn),

V~61 ~ . so t h a t

) to o b t a i n

l(gn)= n=1

= I (f) • QED.

3. C h a r a c t e r i z a t i o n

We

Theorem

fix a r e d u c e d

preparations which

Hardy

in S e c t i o n s

characterizes

the

of M m e a n s

already

3.1

been

that

The

M is c o m p a c t

A2)

If O ~ f n 6 R e L ~ ( m )

A3)

If E ( n ) 6 Z

there

exist

in I V . 4 . 5

subsequent

AI)

then

too

Result

situation

2 we

properties.

H is n o t

considered

THEOREM:

algebra I and

situation

by a list of e q u i v a l e n t ness

and M a i n

(H,~)

can n o w p r o v e

on

that M is c o m p a c t The

basic

small.

idea

Recall

(X,Z,m).

the

After

following

the

theorem

in ~ ( R e L 1 ( m ) , R e L ~ ( m ) ) is that w e a k

that

the

compact-

situation

has

and V I . 6 . 3 .

properties

are

equivalent.

in o ( R e L 1 ( m ) , R e L ~ ( m ) ) .

with

with

fn+O

then

0(fn)÷O.

X=E(O)mE(1)m...mE(n)mE(n+1)=...

h 6 H + and O < c =

+~ such

that

Re h > c

n

=

and m ( E ( n ) ) + O on E ( n )

n

(n=O,I,2,...). BI)

If T6(L~(m)) " is s i n g u l a r

and

TIH

If T6(L~(m)) " is s i n g u l a r

a n d T]H

is s t r i c t l y

continuous

then

TIH=O. B2)

is weak, c o n t i n u o u s

then

T[H=O. B3)

(of F . a n d

into

~=Gm+T

that

TIH=O.

with

M.Riesz G6LI(m)

type) and

Assume

that

~ 6 ( L ~ ( m ) ) " is d e c o m p o s e d

T6(L~(m)) " s i n g u l a r .

Then

~IH=O

implies

178

Proof: E(n)CE

AI)~A2)

with

e(XE(n))÷O

is

in VI.6.3.

A2)~A3)

Consider

X=E(O)mE(1)m...mE(n)DE(n+I)m... after

A2).

Choose

numbers

a sequence

and

gn>O

with

of

m(E(n))+O. En÷O

and

subsets

Then

then

numbers

>0 w i t h n= F

k

[ l a n (\0 ( X E ( n ) ) + S n ] / < ~ n=

From

IV.3.10

and

Re~(Un)

V£M

then

~(Re

and ~

IV.2.5

we

Q(XE(n))

Vn)Vdm

obtain

+ en . Let

= Re~(Vn)

=

[

I=0 Since the

there

Beppo

at

least

theorem

O(IP

- Re

follows

Vnl)

from

one

[ ~ = =. n= I n

functions us p u t

Un6H

Vn:=

V6M

us t h a t

Re

VI.3.8

- Re

that

v n)

P6E.

which

= S(P

A3)~BI)

Assume

Choose

that

E(n)6l

such

that

T(f)

then

for each

- Re

we

after

V.4.1.

BI)~B2) to prove is o f

the

Thus

and that form

the

<

all

~.

whole

of

T

V6M. T h u s

Vn)Vdm

+ O

h:=

for

any V6M.

P + i P ~ 6 H +.

And

(n=O,q,2,...).

singular

If n o w

XE(n )

for

t+O we

are

linear

~(f)=Sfvdm

If

X

and

[ o ~ I X E (i i= ~ "'

c n o n E(n)

Vf£L~(m).

Vf6L~(m).

B2)~B3) each

For

XE(n )

n uI >

=~oal Re

f :

f T(~-~)=O

un ~

O~P6L(m),

with

TIH

strictly

X=E(O)DE(1)m...mE(n)mE(n+1)m...

f

hence

Re

h£H + and

and Cn~O

are

contin-

m(E(n))+O as

i n A3)

have

T

and

>O o n some

all

In p a r t i c u l a r

T 6 ( L ~ ( m ) ) " is

with

= E(fXE(n)) t>O

i

n ~ o;I =: i=0

Re h = P >

uous.

vn =

is

for

n Re h = P => Re

that

aoUo+...+~nUn6H.

v n + P to

Vn)Vdm = l i m R e ~ ( V n ) n~m

= @(P

such

~lRe~(u I) __< 1 [_0~i O(×E(1))+~I

function

tells

~PVdm = l i m f ( R e n~m

that

It

is

Levi

and

)

IIf [I -

<

-

IIT 1 1 + t C n

in particular

deduce

obvious.

from

2.1

B3)~AI)

f£H that

then

96(ReLY(m))

Vf6ReL~(m)

for

some

of

• with

V6M.

Vt>O

T(f)=O.

In view

functional

f ~-~6H

IV.4.5

we

have

~a°=eIReL~(m)

179

L e t us

fix

such

a functional

low t h e d e f i n i t i o n = Re~(u)

Vu6H.

Vf6L~(m).

Then

~ I H = ~.

From

O~F6LI(m)

and

bitrary

V6M

follows

that

since

of o

We extend

} to L~(m)

~ is L ~ ( m ) - n o r m the

and

apply

~IH

B3)

= 0 and

~=Fm

of the

and

section

= 9(Re

which

u) =

f) + i ~ ( I m

f)

and

satisfies

9 = Fm + T w i t h

T 6 ( L = ( m ) ) ". We 9 - Vm =

T(1)

fix

an ar-

(F-V)m + T.

= 0 which

implies

It

that

therefore

F6M.

we

t h a t M is c o m p a c t

assume

fol-

and ~ ( R e

1.1 w e o b t a i n

functional

in p a r t i c u l a r

remarks

Sup

and positive

functional

to the

Hence

remainder

v i a ~(f)

theorem

singular

the

that ~ ~

continuous

decomposition

a positive

T is p o s i t i v e .

For the

9 6 ( R e L Y ( m ) ) *. F r o m

in IV.2 we k n o w

T =0

QED.

in

O(ReL1(m),ReL~(m)).

3.2 C O R O L L A R Y : near

functional

Proof:

We h a v e

continuous. tinuous.

Assume I:H+¢

Since

Now

ly c o n t i n u o u s GmlH.

to s h o w

that

G6LI (m)

3.3 T H E O R E M :

Assume

G 6L1(m) n

I:H÷~

3.1.BI).

must

be w e a k ~

I is L ~ ( m ) - n o r m

decompose

and T6(L~(m)) " singular.

t h a t M is c o m p a c t

(n=1,2,...)

there

exists

Proof:

In v i e w

a function

is s t r i c t l y

it a f t e r

Then

TIH

It f o l l o w s

con-

!.1

in-

is s t r i c t -

that

IIH =

2.2 the

Let K O consist

IV.l.l.iii).

from

so

of the the

for all

f6H.

functions

fundamentals

Thus

REFORMULATION:

that

3.2

we

can

Assume

L I ( m ) / K ° is w e a k l y

can

i: l(f) be

applied.

h6K w i t h of

= lim f f G n d m

the

space

3.3

t h a t M is c o m p a c t complete.

as

Vf6H

QED.

f h d m = O.

Banach

reformulate

sequentially

Let

f6H.

such

functional

that

in o ( R e L 1 ( m ) , R e L ~ ( m ) ) .

that

for all

G£LI(m)

= ffGdm

continuous,

= K± c L ~ ( m )

3.4

of

be s u c h

exists

lim S f G n d m

Then

2 that

then

• A li-

it is w e a k , c o n t i n u o u s .

continuous

Section

TIH = O a f t e r

lim f f G n d m

o from

iff

QED.

functions

Then

from

I to 1 6 ( L ~ ( m ) ) " a n d

and h e n c e

in o ( R e L 1 ( m ) , R e L ~ ( m ) )

continuous

a strictly

H = H we k n o w

extend

to I = G m + ~ w i t h

t h a t M is c o m p a c t

is s t r i c t l y

Then

theory,

(LI (m)/Ko)" ± and

Ko

= H

follows.

in O(ReLI ( m ) , R e L ~ ( m ) ) .

180

Notes

The main

result

open problem sults we

refer

direction

to

in H E A R D stract

function

the

unit

HAVIN uses

disk

Then

GLICKSBERG algebra

dim N < ~

[1967]

[1970],

has

been due

[1973]

invented

proved

and S z e g 6

use

and

rem

is

3 is

close

adopts to the

from BARBEY-KONIG

[1974].

this

[1976].

[1976] proofs

one.

full

due

Our The

final

An a p p l i c a t i o n

proof

ab-

a highly 3.3 in

[1972]

due

in B A R B E Y the

and

[1975]

the p r e s e n t AMAR

in

weak

[1973]

[1952].

f o r m of S e c t i o n s is

in

of it H A V I N

the H e w i t t - Y o s i d a

3.4

It

to A M A R

assumption

Instead

of

and

non-transparent.

follow

of

in

the notion

theorem

to H E W I T T - Y O S I D A

proof

an re-

for c e r t a i n

[1972]

under

transformation.

idea.

results

of M O O N E Y

under

been

situation

to r e p l a c e

The

convergence

latter

theorem

initial

[1972]

subsequent

by K 6 n i g

in B A R B E Y

of the G e l f a n d

disk

is r a t h e r

the

strict

situation,

the decomposition

treatment

[1972]

as d o e s via

Partial

in o r d e r [1967].

has

form classical

In B A R B E Y - K O N I G

to K A H A N E

construction

present 1.1

[1967].

in M O O N E Y

proof

situation

for the u n i t

The

(unpublished),

uses

disk

evolution

BARBEY-KONIG

situations.

c o m p a c t n e s s a s s u m p t i o n for M. The the d e c i s i v e

its

is an i n d e p e n d e n t r e s u l t

3.3 w a s

disk

For

situation

the K a h a n e

unit

to the u n i t

time.

in K A H A N E

construction

[1973].

[1973]. the

are

convergence

complicated

some

to P I R A N I ~ N - S H I E L D S - W E L L S

3.3

[1967],

of s t r i c t

3.3 r e l a t i v e

for q u i t e

The theo-

2 and

in P E L C Z Y N S K I

Chapter

Logmodular

The p r i n c i p a l 3.5:

If

(H,~)

Densities

(H,~)

is a s m a l l e x t e n s i o n

Since

it is not a p r i o r i

t a n t to h a v e c e r t a i n situations

as well.

(H,~)

be a H a r d y

(H,~)

results

is r e d u c e d

representative

for n o n - r e d u c e d

property

functions

theorem and if

3 then H = H.

it w i l l be i m p o r -

a fixed Hardy algebra

The s m a l l e x t e n s i o n

algebra

NL:= real-linear

functions

Hardy algebra situation

(H,~)

is in c l o s e re-

(=densities)

with which

We i n t r o d u c e Vu6HX},

(=densities),

and

span(ML-ML) = {c(U-V):U,V6ML

that M J c M L c M .

subset

situation.

L I (m) :logI~(u) I = / ( l ° g l u l ) V d m

the c l a s s of l o g m o d u l a r

It is c l e a r

in the s e n s e of S e c t i o n

that

intermediate

with dimN<~

Densities

ML:= {O~V6

closed

(H,~)

is the m a x i m a l i t y

situation

starts.

I. L o g m o d u l a r

Let

of

Thus w e a s s u m e

l a t i o n to the l o g m o d u l a r

chapter

Hardy algebra

clear

w h i c h n e e d n o t be r e d u c e d .

the c h a p t e r

and S m a l l E x t e n s i o n s

a i m of the p r e s e n t

is a r e d u c e d

IX

cReLl(m).

A l s o M L is a c o n v e x A n d the e q u i v a l e n c e

a n d c>O}.

and o ( R e L l ( m ) , R e L ~ ( m ) )

r e m a r k VI.6.1

on M c a r r i e s

at once o v e r to ML.

N e x t we i n t r o d u c e The

functional

a close

~:ReL~(m)÷~

relative

is d e f i n e d

~(f) = I n f { - l o g l ~ ( u ) i:u6H x w i t h We list some i m m e d i a t e Sup f Vf6ReL~(m). B is s u b a d d i t i v e , iv) L e m m a defined

Here

properties, S~ o

e:ReL(m)÷[-~,~].

to be

-loglu 1 ~f}

V f 6 ReLY(m).

i) Inf f ~ e ( f ) ~ ~(f) ~ o ( f )

results

= e(f)

from IV.2.5 via exponentiation,

b u t is not c l a i m e d

IV.2.3 permits

to be

of the f u n c t i o n a l

to i n t r o d u c e

to be s u b l i n e a r , the f u n c t i o n a l s

iii)

ii)

B is isotone.

B°,6~:ReL~(m)÷~,

182

It f o l l o w s

~

that ~ ~

= l i m ~ ~{tf) t+O

= Sup~ t>O

6~(f)

= l i m ~ ~(tf) t+~

= I n f ~ 8(tf) t>O n

~

0

=~ 0 =@. A n d

B~

6 ( l o g ] u I) = logI~(u) I V u 6 H ×. It f o l l o w s 1.1 REMARK:

I

B°(f)

Let V 6 R e L 1 ( m ) .

Then

$(tf),

is s u b l i n e a r that

/fVdm~

~(Reu)

B~(f)

Vf6Reh=(m). and isotone,

v)

= ~(Reu) = Re~(u)

Vf6ReL~(m) ~/fVdm~

Vu6H. ~(f)

V f 6 R e L ~ ( m ) ~ V6ML.

Proof:

Direct

verification

1.2 P R O P O S I T I O N : ML+¢

and

Proof:

Assume

as for IV.2.1.i)

that M is o ( R e L 1 ( m ) , R e L ~ ( m ) )

6~(f) = M a x { / f V d m : V 6 M L } This

on MJ and ~

follows

in IV.4.5.

The next result

QED.

compact.

Then

Vf6ReL~(m).

from IV.4.5

via

1.I as the c o r r e s p o n d i n g

result

QED.

is a f u n d a m e n t a l

1.3 A H E R N - S A R A S O N s>O t h e r e e x i s t s

and IV.2.4.

step.

LEMMA:

Assume

that ML ~ ~ and d i m ~ L

a constant

c(s)>O

such that

<~. T h e n to e a c h

00

8(f)

< ~ + C(S)6

V O < f 6 ReLY(m) .

(f)

We s h a l l n e e d the s u b s e q u e n t result while

from number

two lemmata.

theory which

the s e c o n d one

is r e p r o d u c e d

is a s i m p l e

fact

The f i r s t one is a w e l l - k n o w n for the sake of c o m p l e t e n e s s ,

from l i n e a r algebra.

!L4 DIR____~ICHLETSI___MULT_____ANEO____USAPPRgXIMA_____TIO_NNLEMMA: TO a I ..... an6 ~ and r6~ there

exist numbers

q6~ and Pl ..... pn6Z

such that

lqaz-p~l<~(Z=1 ..... n)

and q ~ r n .

P r o o f of 1.4: L e t us s u b d i v i d e

the cube Q : = { x = ( x l , . . . , X n ) : O ~ x ~ < l

(~=I ..... n) }c~ n into the r n s u b c u b e s (i=1,...,n)}

via the r n l a t t i c e

(~=I ..... n). For each s65 then

Q ( u ) = { x = ( x I ..... x n) :~(uz-1)<=xi<~u£

points

u:(ul,...,Un)6~n

(sa I ..... S a n ) m o d u l o

with ui=1,...,r

~n is in Q and h e n c e

183

in one of the Q(u). It follows that in the set of the rn+1 numbers n there is at least one pair of numbers s
O,1,...,r

.... sa n ) mod S n a n d Thus

(tal,.,.,tan)mod sn

for I ~ q : = t - s ~ r n we have numbers

< ~ (~=I . . . . n) as required. r 1.5 LEMMA:

are in the same subcube Q(u). Pl ..... Pn6$

such that

lqal-pzl<

QED.

Let W be a vector space and TcW such that W = linear span

of T. Assume that ~I,..O,Mn£W~

are linearly

exist ~I,...,~n£W • linear combinations such that <~i,u~>=~iZ Proof of 1.5:

independent.

of ~I,...,Mn

(i,~=1,...,n).

It suffices

to show that there are Ul,...,Un6T

that det(<~i,u~)i,i= I .... ,n+O. But this assertion via induction.

Then there

and Ul,...,Un6T

has an obvious proof

QED.

Proof of 1.3: i) We fix F6ML and e>O and prove the existence constant

c(e)>O

VO
6L~{m)

with

finn

(loglHXl) ±.

In case that N=NN(iog]H×i) i this is clear with c(~)=I as above then /fFdm=e(f)~8(f).

(logiHXi)ImNL linear span of

with dim IN/N A(logIHXi)li=:n6~.

loglH×IcReL~(m) and

T:=logIHXl

inclusion

(ReL~(m))~÷W~:~IW.

We start from functions

arly independent fl,...,fn61OgIHXl

since

Thus we can assume

cW ~ via the canonical

We apply

for the that N~N n

1.5 to W:=real-

and consider

N/Nn(1oglH×I) ±

NcReL1(m)c(ReL~(m)) • and the restriction

over NN(logIH×I) ± and obtain

FI,...,Fn6N functions

such that ]Gifgdm=6ii(i,g=l,...,n).

N = (Nn(logIHxl) ±) ~ where

of a

such that

~(f)__<s+c(s)/fFdm

f6ReL~(m)

such

real-linear

the sum is direct and hence L1(m)-norm

It follows

span(G 1 .....

bounded:

which are line-

GI,...,Gn6N

and

that

Gn) ,

there is a con-

stant c>O such that n ItlI ..... itnl ~cIIG + ~ItZGZIIL I =

Consider

now O ~ f6L~(m)

VG6NN(IoglH×I) I and t I ..... tn6 ~. (m)

with f ± N N ( l o g l H × l ) ±. For q6~ and Pl .... 'Pn 65

we put g:=plf1+...+Pnfn 6 logIH×i

and obtain

184

8(f) ~ 5(qf) ~ 8(qf-g)

= Sup / ( q f - g ) V d m VEM

+ 5(g) ~ @(qf-g)

+ /gFdm : Sup/(qf-g)(V-F)dm VEM

B u t for V E M w e h a v e V - F E N w i t h L 1 ( m ) - n o r m ~ direct

+ /gFdm

+ q/fFdm.

2 so that from the a b o v e

sum decomposition

n V - F = G + [ t£G£ w i t h £=I It f o l l o w s

G£Nn

(log]HXl)

I

and I t l l

Itnl <_2c.

.....

that

/(qf-g) (V-F)dm :

n [ t£/(qf-g)G£dm £=I

n = ~ t £ ( q / f G £ d m - p £ ) < 2 c n M a x Ig/fG£dm-p~[ V VEM, £=I I < £
for all q 6 ~ and Pl '''''Pn 6 ~. If n o w r E ~ is p r e s c r i b e d

after

1.4 the n u m b e r s

qE~

and Pl ..... P n 6 $ can be c h o s e n

lq/fG£dm-p£1 < 1 ( £ = I ..... n) and q < r n. T h e n = for

s>O we h a v e

desired

ii)

(m). F i x

a function

I 6H x thus linear

take

an r=r(E)EN

with

with

c(~):=(r(S))

n.

N o w we can p r o v e

Let O ~ f E L exists

to

extimation

the a s s e r t i o n

and

then

f~loglvl

-loglul ~ t f

the

as above.

(f)+6. T h e n there

1

and - ~ logI~(u)i~B

and ~logl~(v)I
Thus

obtain

the same c ( e ) > O

6>O and take t>O such that ~ 8 ( t f ) < B u6H × w i t h

so that

B(f) < 2cn + rnffFdm. = r

2cn<~ r

with

then

(f)+@. F o r

Now O<~loglv[Ereal

-

s p a n of loglHXl c (loglHXl) ±± c ( N A ( I o g l H X I ) ~ ± so that f r o m i)

we o b t a i n ~(f) < B(lloglvl) < S + C(S) / l ( l o g l v l ) F d m = s + ~ l o g l ~ ( v )

< S+C(S)

For

~+O the a s s e r t i o n One of the m a i n

follows.

consequences

[

(B~(f)+6] .

QED. of 1.3 is t h e o r e m

p r o o f we n e e d the fact t h a t a c e r t a i n

result

I.7 below.

In its

f r o m S e c t i o n VI.6 r e m a i n s

185

true without

the reducedness

!±6 REMARK: plies

assumption.

Let F6M. Assume

that ~(fn)+O(condition

that if O~fn6ReL~(m)

VI.6.2.~)).

then

/ f n F d m ÷ O imI Then NcNAF(ReL~(m)) L (m) (con-

dition VI.6.4.vi))o Proof: reduced. tion

We know from VI.6.4 Recall

(H~,~)

from IV.I.11

on

that the assertion the associated

(Y(X),EIY(X),m!Y(X))

i) We claim that ~.(f!Y(X))=~(f) IV.1.10-11

is true when

(H,~)

reduced Hardy algebra

which can be formed for each

for all f£ReL(m).

is

situa(H,~) .

In fact, we see from

that ~,(flY(X))

=

Inf {-logI~o.(u) I : u6H, with -loglu I >__flY(X)}

= Inf{-logl~(u) I :u6H with -loglu I ~ f on Y(X)} = Inf{-logl~(u) 1 :u6H with -log[u I ~ f} = ~(f). ii) We conclude

from i) that condition

iff it is fulfilled that condition

Thus the desired 1.7 THEOREM:

of ML is

~cF

for

(H~,~).

VI.6.4.vi)

Assume

carries

that dim N<~.

for some constant

In particular:

Furthermore

is true for

implication

VI.6.2.~) (H,~)

is fulfilled

it is obvious (H~,~)

to

(H,~)

from IV.I.11

iff it is true for

over from

for

(H~,~).

(H,~). QED.

Let F6M be such that each member

c>O. Then F is an internal

If F6ML is an internal

point of M.

point of ML then it is an inter-

nal point of M as well. Proof:

i) From the proof of VI.6.1.i) ~ i ' )

1.2 we conclude L~(m). 8(fn)÷O

ii)

that there

If O~fn6L~(m)

from 1.3.

is a constant

with ffnFdm÷O

In particular

NcNnF(ReL~(m))LI(m)=NnF(ReL~(m)). In conclusion will be needed 1.8 REMARK: log

then

e(fn)÷O,

applied to ML and from

c>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)

-

<~ then NL = N-LReL] {m) = (logiHXl) ±,

(NL)I=(loglH× I )±±: r e a l - l i n Proof: to prove

i) Is clear (IogIH×I)±cNL

1.7 it is an i n t e r n a l f6NReLI(m)=N But then

from

f±loglHXl

and F6ML

imply

it remains

with

In order

so that

after

If now fE(loglHXl) ± then U6M and c>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-I1uzIi
linear

Assume

2.2 REMARK: exist

which

Assume

then S is d i s c r e t e , is o b v i o u s ) .

that S is not d i s c r e t e .

n ( £ ) + ~ and IIn(£)u£[I÷1. We can pass

that n ( £ ) u z ÷

= (P-~-[n n(£) (1)u~6S c o n v e r g e

of S,

u~6S w i t h u ~ ÷ O or [IuiiI÷o. I n ( Z ) 6 ~ w i t h n(£)+--------1
for f i x e d t>O take the n u m b e r s

lows that u6D(S)

For S c V

points

assume

or p(~) n(Z) - n ( £I)

subspace

(the c o n v e r s e

u6D(S).


space.

span of S,

linear

a subsequenceand

<

vector

If D ( S ) = { O }

points

that Iiu£II<1. Take the n u m b e r s

In fact,

It w i l l

as well.

Let us fix a n o r m If" II on E(S).

we can a s s u m e

-< n ~

topological

the l a r g e s t

Assume

it

Then there exists I

real H a u s d o r f f

linear

so that D ( S ) c S c E ( S ) 2.1

of its own.

subgroup we define

E(S):=

that m e a n s

some i n t e r e s t

section.

< ~I< n (~£ ) "

to

some u6S w i t h lluIl=1. We c l a i m that p ( £ ) 6 ~ w i t h p(1)-1

Then En(£) ~ _ ~ ÷ ~. I Hence

to l u so that 1 u6S or u6tS is a c o n t r a d i c t i o n .

that S is d i s c r e t e

<

the p ( i ) u £ :

for e a c h t>O.

It fol-

QED. and d i m E ( S ) = : n 6 ~ .

Then t h e r e

u I ..... Un6S such that S = $ u 1 + . . . + $ u n. In this case E ( S ) = ~ U l + . . . + ~ u n.

In p a r t i c u l a r Proof:

Ul, .... u n are l i n e a r l y

L e t us fix a n o n z e r o

al,...,ar6S

form a m a x i m a l

determinant

linearly

We have S c ~ a 1 + . . . + ~ a r and h e n c e

independent. function

independent

on E(S).

subset

i) Let

of S. T h e n r~n.

E ( S ) c ~ a 1 + . . . + ~ a r. Thus r=n,

and we see

that E ( S ) = ~ a 1 + . . . + ~ a n. ii) We c l a i m that there e x i s t s Ul,...,Un6S

such that

a fixed maximal

linearly

Idet(ul,...Un) I is m i n i m a l .

linearly

={tla1+...+tnan:O~t

a maximal

independent

independent

To see this s t a r t w i t h

subset al,...an6S

I .... ,t n ~ I} ~ E ( S ) .

subset

and f o r m A : =

T h e n A is compact,

so that ANS

188 is c o m p a c t

and d i s c r e t e

and h e n c e

finite.

Consider

n o w the c o o r d i n a t e

functionals n

~I .... '~n 6 (E(S)) Then ~z(S)c~

: u=

is an a d d i t i v e

~ ~£(u)a£ Z=I

subgroup with

V u£E(S). I£Mz(S).

of the f o r m u = P l a 1 + . . . + P n a n + V

with pl,...,Pn6$

+~z(ANS).

of ANS

m(i)6~.

Thus

the f i n i t e n e s s

It f o l l o w s

with pl,...,Pn6$.

that each u6S

implies

S i n c e e a c h u6S is

and v 6 A A S we h a v e ~ z ( S ) = $ +

that ~£(S) = ~ $

is of the form u = m - T ~ P l a l

F r o m this we see that

for some ...+

for each c o l l e c t i o n

Pnan

Ul,...,Un6S

we h a v e

det(ul '" "''Un) It is o b v i o u s

= i(I) ...m(n) p

det(al ..... an)

for some p65 "

t h a t we can c h o s e u I ..... Un£S w i t h m i n i m u m

[det(ul,...,Un)I>

>O as w e h a v e claimed. iii) tained

Let us fix Ul, .... Un6S w i t h m i n i m u m in ii).

It rer0ains to s h ~ t h a t S c ~ u ] + . . . + $ u n and hence S = $ u 1 + . ~ . + $ u n.

To see this w r i t e v=tlu1+...+tnUn

u6S in the form u = P l U 1 + . . . + P n U n + V

with O~tl,...,tn<1.

2.3 REMARK: Un6S

Assume

..... u n)

that v=O and h e n c e

the e q u a t i o n

implies

that

the a s s e r t i o n .

that d i m ( E ( S ) / D ( S ) ) = : n 6 ~ .

Then

and

QED.

t h e r e e x i s t Ul,... ,

such that S = D ( S ) + $ u I + . . . + $ u n. In this case E ( S ) = D ( S ) + ~ u I + . . . + ~ u n.

In p a r t i c u l a r Proof: =D(S) ~ G . It f o l l o w s crete. hence

It follows

with pl,...,Pn6$

T h e n v6S and h e n c e

det(ul,...,uz_1,v,uz+1,...,Un)=tidet(ul t~O(i=1 ..... n).

Idet(u I .... ,Un)l> O as ob-

U l , . . . , u n are l i n e a r l y

L e t GeE(S)

independent

be an n - d i m e n s i o n a l

linear

F r o m D ( S ) c S we h a v e S = D ( S ) + ( G N S ) that D ( G N S ) c G N D ( S ) = { O }

Furthermore

E(S)cD(S)+E(GnS)

of d i m e n s i o n = n. Thus

G D S = $ u 1 + . . . + $ u n and h e n c e 2.4 C L O S E D

SUBGROUP

subspace

so that 2.1 tells implies

such t h a t E ( S ) =

w i t h GNS a c l o s e d

m u s t be = G

Ul,...,Un6GNS

S = D ( S ) + $ u I + . . . + $ u n. The a s s e r t i o n Let S , T c V be c l o s e d

Then E(SND(T))=E(S)ND(T).

subgroup.

us that GNS is d i s -

that E(GNS)

f r o m 2.2 we o b t a i n

LEMMA:

SeT and d i m ( E ( S ) / D ( S ) ) < ~ .

o v e r D(S).

additive

and

such that follows.

QED.

subgroups

with

189

Proof:i)

We have

D(S)cE(SND(T))cE(S)ND(T)mE(S). E(S)

dim E(SND(T)) D(S) so that O ~ , q , n < then

~ with

ScE(S)cD(T).

is p+q=n.

:p, dim E ( S ) N D ( T ) p+q~n.

We assume

that p+q
that q~1 or

E(S) and dim ~

:: q

The a s s e r t i o n

So we assume

Let us put

is o b v i o u s

and hence

(p+1)+q~n

if q=O since

n~1.

and shall

=: n,

The a s s e r t i o n obtain

a contra-

diction. ii) We i n t r o d u c e

B:=E(S)DT

which

is a c l o s e d

D (S) mE (SAD (T)) cE (S) ND (T)

subgroup

with

E(S)

II

II

D(B) c B c E ( B ) . T h e n we apply

2.3 to o b t a i n

Vl, .... Vq6B

linearly

independent

over D(B)

with B

= D(B)

+ S V 1 + . . . + ~ V q and E(B)

and X l , . . . , X n E S

S = D(S) Now x i E S c B

linearly

+ ~v1+ .... ~Vq,

over D(S)

+ S x 1 + . . . + $ x n and E(S)

= D(S)

with

+ ~ x 1 + . . . + ~ x n.

can be w r i t t e n

q j xz = yz + j~18zvj= Consider

independent

= D(B)

the vectors

6J:=(~

+q~n we can find ~ o n z e r o are o r t h o g o n a l

with

yz6D(B)

and

8~ 6 $(Z=I ..... n).

..... B~)6$nc~n(j=l ..... q).

vectors

to each o t h e r

i:=(e%

:=

n ~

and o r t h o g o n a l

n i

,£=1 =

to ~ 1 . . , ~ q

We see that

+

that

1

~ < 1•, B 3 >•v j=l

ui6SND(B)=SND(T)cE(SND(T)).

w h e n we can show that

with respect assume

q i yz+j~

,%=1

n[ ~ liY l £=1

(p+1)+

..... e~)6~ n(i=O,1 ..... p) w h i c h

to the usual s c a l a r p r o d u c t in ~n. And we can of course ~°,~1,...,~P6~n. iii) C o n s i d e r then ui

In v i e w of

Uo,U I .... ,Up are

=

]

i Z ( i : 0 , I , .... p). !1o.zy

Z We have

the d e s i r e d

contradiction

linearly

independent

over

D(S).

But

190

for 1o,11 ,... ,ip6~ we have P

n

i=O If this is 6 D(S)

liui =

P

~ ( ~ li~Ix~ Z=I i=O

•

then the c h o i c e of Xl,... ,x n i m p l i e s P li~.ii = O

(~=I ..... n)

or

l=O But

0

1,...,~p

independent It f o l l o w s

are n o n z e r o

as m e m b e r s

and p a i r w i s e

of the v e c t o r

that Io=11 = . . . = i p = O

independent

that

P ~ I a i = O. i=O i

orthogonal

and h e n c e

linearly

s p a c e ~n o v e r the s c a l a r

f i e l d ~.

s o that U o , U 1 , . . . , u p

are i n d e e d

linearly

o v e r D (S) . QED.

3. S m a l l E x t e n s i o n s

Let

(H,~) be a H a r d y a l g e b r a

situation.

f i n e d to be a H a r d y

algebra

situation

t h a t HcH and ~=~IH.

The e n t i t i e s

An e x t e n s i o n

of

(H,~)

is de-

(H,~) on the same m e a s u r e s p a c e such

which

come

from

(H,~) w i l l be w r i t t e n

M,N, .... 3.1 REMARK: ii) M = M ~ N = N . converse

Let iii)

(H,~)

is r e d u c e d

i) and iii)

V 6 M then V - F 6 N = N .

are obvious,

Assume

(H,~).

the s u b s e q u e n t

Then

(H,~)

Proof:

is r e d u c e d

(H,~). (H,~)

Then

i) M c M and NoN.

is r e d u c e d

(but the

is reduced.

properties

and E~=E ~. iv) i) ~ i i )

Let

(H,~) be an e x t e n s i o n i) M=M.

ii) N=N.

H=H.

is in 3.1. ii) ~ i i i )

follows

is d o m i n a n t

from E~=

o v e r X. T h e n

so that H c H and

QED.

remark which

of an e x t e n s i o n

To s e e ~ fix F6M. F o r

are e q u i v a l e n t ,

and H=(E~+iE~)N(H~F) i from VI.4.5,

iv) ~ i) is obvious.

W e add o n e f u r t h e r definition

(H,~)

obvious.

QED.

. iii) = iv) T a k e an F £ M c M w h i c h

H=(E~+iE~)D(F~F)± h e n c e H=H.

that

The e q u i v a l e n c e

N ± and E =N

ii) ~ i s

Thus V = ( V - F ) + F 6 M .

3.2 REMARK:

iii)

of

then

n e e d n o t be true).

Proof:

of

(H,~) be an e x t e n s i o n If

shows

can s o m e t i m e s

t h a t the r e q u i r e m e n t s be r e l a x e d .

in the

191

3.3 REMARK:

Assume

be a s u b a l g e b r a w i t h ~=~IH. Thus

(H,~)

Proof: cativity. bounded

that M is q(ReLI (m),ReL~(m))

w i t h HcH and ~ : H + ~

If ~ is L ~ ( m ) - n o r m is a H a r d y a l g e b r a

If ~ is L ~ { m ) - n o r m We p r o c e e d

continuous situation

and h e n c e

linear

Let HcL~(m) functional

then it is w e a k . c o n t i n u o u s .

whenever

H is w e a k *

closed.

c o n t i n u o u s then II~!1:I from the m u l t i p l i -

as in the p r o o f of V I I . 6 . 4 :

linear extension

A is p o s i t i v e

compact.

a multiplicative

Let A : L ~ ( m ) + ~

be a

of ~ w i t h IIA!I=I. T h e n !fAIl=A(1)=± i m p l i e s real.

Thus

for fEReL~(m)

that

we obtain

u6H w i t h Re u > f ~Req0(u) : ReA(u) = A(Re u ) > A (f) ,

and h e n c e

eo(f)~A(f).

for some V6M,

It f o l l o w s

and h e n c e

An e x t e n s i o n

(H,~)

f r o m IV.4.5

the a s s e r t i o n .

of

(H,~)

In this c o n n e c t i o n

3.4 REMARK: an e x t e n s i o n i)

(H,~)

observe

Assume

of

(H,~).

Vf6L~(m)

QED.

is d e f i n e d

Re H c ( I o g l H × I ) ±± : r e a l - l i n e a r

that A ( f ) = / f V d m

to be a small

span(log~

extension

iff

weak*.

that R e H = I o g l e H I c l o g I H X l .

that

(H,~)

is r e d u c e d

T h e n the s u b s e q u e n t

is a s m a l l e x t e n s i o n

of

(H,~).

and d i m N<~. properties

Let

(H,~) be

are e q u i v a l e n t .

ii) NLc(ReH) ±. iii) MLcM.

iv)

NL=N. Proof: means

i) ~ i i )

We h a v e

d i m N ~ d i m N<~. F6M implies We w a n t

that V6M.

to p r o v e

THEOREM:

is a s m a l l e x t e n s i o n

(H,~)

The

iii) ~ iv)

the s u b s e q u e n t

3.6 C O N S E Q U E N C E : If

Assume

is an e x t e n s i o n

implication

the a s s u m p t i o n

1.9 so that s m a l l n e s s

We have M L ~ ~ from

1.2 s i n c e

Let us fix an F6M~Lc/~nML. For V 6 M L then V - F 6 N L c ( R e H ) ±. Thus

3.5 M A X I M A L I T Y (H,~)

(loglHXl) i± : (NL) ± from

that ReHc(NL) ± or NLc(ReH) 1. ii) = iii)

of

and iv) ~ i i ) maximality

Assume of

that

(H,~)

that

theorem.

(H,~)

is r e d u c e d

and dimN<~.

If

is r e d u c e d w i t h

d i m N < ~ and NL={O}.

then H=H.

3.5 ~ 3.6 is c l e a r

NL={O}

QED.

then H=H.

(H,~)

(H,~)

are obvious.

from

all e x t e n s i o n s ( H , ~ )

3.4 w h i c h of

tells

us that u n d e r

(H,~) are small.

And recall

192

that

3.3 p e r m i t s

proof

of

3.7 LEMMA: (H,~). of M

to r e d u c e

3.5 w i l l

Assume

If F 6 M ~ as

that

i) L e t

F6ML

=NNF(ReL~(m)).

point

implies

there

exists

We n e x t

us p u t To see

F i x any

p(n)a(n)-a(n)
set G ~ : = - G to o b t a i n

t6G.

real

a little

But

assume

t>O and

fulfills

that

a,-a

then

finition

of a

. Thus

G=a$.

To see

that

=ma+x

satisfies

that

a-a

a=a.

Gca$

O~z
one.

that

take

~cG since

6G~a-a

of a. L i k e w i s e

remark

F6ML.

see

ii)

that

In v i e w

since

fn£L~(m)

this

is

then

to s h o w 8~(fn)+O.

point

Re v

of M.

> f and n= n

and hence

Hence

Re~(Vn)=

8~(Re Vn) =

(fn)~6

(Re V n ) ÷ O. QED.

which

is to f a c i l i t a t e

After

this

a=O.

the p ( n ) 6 ~

the

final

and O~a~1.

there with

exist

step

under

as well.

Thus

we c a n a p p l y

whenever

£G a n d

-a < a - a ~ O < a - a

We have

±a6G Then

it m u s t

a>O.

a$cG.

is an m65

z=O and h e n c e

The

i) to G ~

and x
iii)

the dethe d e -

We c l a i m such

it

addition. ii)

or a
and h e n c e

be

x6G

~ O or a~a

there

We

O
and p(n)a(n)+t

indeed


under

p(n)-1
that

a-a

z6G

closed

proves

6G and

x6G.

and

$cG a n d G is c l o s e d

-a 6G and x ~ - a

iv)

a6G

Then

from p(n)a(n)EG

which

consider Since

G is c l o s e d

so that

that

the a s s u m p t i o n s

finition

1.7 we

it s u f f i c e s

O~B

of point

lemma w h i c h p r e p a r e s the r e d u c t i o n step

so t h a t

at a c o n t r a d i c t i o n

an a >0 such

It f o l l o w s

that

a:= I n f { x 6 G : x > O } this

Then

arrive

From

if O ~

Vn6H with

A s s u m e that $ c ~ and I n£~. G = ~ $ for some

a>O.

extension

be done.

a(n)÷O.

So we

1.3

The

is r e d u c e d .

of c o u r s e

that

after

of c o d i m e n s i o n

with

follows

of ML.

F is an i n t e r n a l

- a technical

3.5 w i l l

i) Let

that

And

It f o l l o w s

extensions of

Then

Proof: claim

Also

functions

- after

proof

3.8 REMARK: addition.

1.2.

extablish

3.10 to s m a l l

is r e d u c e d ~ (H,~)

point

3.6.

it is an i n t e r n a l

N o w Re V n 6 R e H c ( I o g l H X l ) ± ± c ( N L ) ±

subsequent

in the p r o o f

be a s m a l l

then

to s h o w

since

of

(H,~)

3.5 a n d

section.

t h a t N c N N F ( R e L ~ ( m ) ) "Ll(m)

~(fn)÷O.

a sequence

= S ( R e Vn) F d m a f t e r

the

(H,~)

1.6 we h a v e

that

in b o t h

of the

of ~

of M as well.

d°(fn)=8(fn)+O

: / ( R e v n) Fdm+O.

Let

point

to p r o v e

After

We h a v e

dimN<~.

be an i n t e r n a l

of d i m N <~ i t s u f f i c e s

iii)

remainder

In p a r t i c u l a r :

F is an i n t e r n a l

/fnFdm+O

the a s s u m p t i o n s

the

is an i n t e r n a l

well.

Proof:

occupy

that

x=(-m)a6a$.

that z:=

193 1

V) We have 16G so that 1=an for some n6~. It follows that G = r $. QED. n 3.9 TECHNICAL LEMMA: Assume that (H,~) be an extension of M

(H,~)

is reduced and dimN<~. Let

(H,~). And let F6M be an internal point of

(so that F>O on X and hence

(H,~) is reduced as well).

i) Let P6E ~ and P 6ReL(m) be its conjugate function relative to (H,~). And assume that P61ogIHXl

but P{E ~. Then there exists an n6~

such that V t 6 ~ : e t ( P + i P ~ ) 6 H ~ t 6 ~ . ii) Let P6E ~ and P~6ReL(m) be its conjugate function relative to (H,~). N p~ N If P 6 ~ then 6~, and hence in particular P+iP~£H.

iv) If (H,~) is a small extension of s p a n ( ~ N (loglH x[)AE~).

(H,M) then E~=E ~ G real-linear

Proof: i) We have et(P+iP*)6H for all t6~. Also there exists a function 6H x of modulus e P. Thus the uniqueness assertion in V.I.3 implies that e ±(P+iP~)6H and hence e n(P+iP~)6H Vn6$. Let now Gc~ consist of the t£~ such that et(P+iP~)6H. follows that G = ~ $

Then G fulfills the assumptions in 3.8. It

for some n6~ as claimed.

ii) From F6M we deduce that HFcK. Thus et(P+iP~)6H implies that et(P+iP~)F6K for all t6~. Now from VI.2.7.ii)

I (et(P+iP~)-111 < I

we have

e ( T + £ ) ( P + i P ~ ) + I P + i P ~ I VOO,

and the second member is 6LP(Fm) V1
(P+iP~)F6K.

In view of VI.6.9.iii)

P+iP ~ = (A+iB)+(a+ib) with A+iB6H#ALI(Fm)

Thus for t~O

this means that

and a,b £

F"

Now P+iP*6H # and hence / ( P + i P * ) F d m = @ ( P + i P *) = @(P) = / P F d m = 0 since PF6N. Also /(a+ib)Fdm=O,

so that ~(A+iB)=S(A+iB)Fdm=O.

Furthermore L°(Fm)cL #

from VI.6.4 and the substitution theorem V.2.2 imply that et(A+iB)6H# and ~(et(A+iB))=l

Vt£~. Hence A6E and B=A ~. Therefore P=A+a implies that

A6E~O~ and hence A=O after VI.6.8.i).

Thus B=A~=O so that P~=b6~ as claime~

194

iii) We k n o w H#+(~+

i~)

from ii)

that HFcK.

tion is d i r e c t

subgroups

D(S)=

topology after

so that H=H+ ( ~ + i~)NH.

subgroup

lemma

and to S:=log]HXl

that H c

The d e c o m p o s i -

which

with

the

are c l o s e d

1.8. We h a v e

D(T)=E ~ after

r e a l - l i n e a r span (loglH×l) • E(S) d~mD~ ~

1.8.ii). after

: 0 Vt>O} = E ~,

Furthermore

1.9.ii),

ReLY(m) < d i m - =

as in the p r o o f

2.4 to V : = R e L ~ ( m )

and T : = l o g l H X l

n tS = {f6ReL~(m) :e(tf)+d(-tf) t>O

and l i k e w i s e

implies

after VI.6.8.i).

iv) We a p p l y the c l o s e d L~(m)-norm

.N l~)

and h e n c e H c H + ( ~ +

Thus V I . 6 . 9 . i i i )

(ReL1(m)) '

dim

- dimN<~,

NI

E ~

of 1.9.ii).

E ( S ) = ( I o g l H X I ) ±I =

and

Thus

from 2.4 we o b t a i n

E((loglH×l )hE ~) = (logIU×]) ±± n ~. The s e c o n d m e m b e r

is :E~ since

in v i e w of the s m a l l n e s s call

from V I . 6 . 8 . i i )

E~ = ReH

assumption.

weak, c ( l o g I H X l ) ±± from V I . 4 . 3

To e v a l u a t e

that R e L ~ ( m ) : E ~

and h e n c e

(log]H×I) n ~ ~ : E ~

the first m e m b e r (N~ N ~~ E).

E~=E~

and re-

Thus

C~n(logl H×

E((IogIHXl ) n E~]= E ~ + r e a l - l i n span ( ~ N ( l o g l H X l ) n E~],

since

the r e a l - l i n e a r span

clear

that the sum is direct.

3.10 R E D U C T I O N (H,~) be a small exists

STEP:

Proof:

3.9. F r o m

that

(H,~)

follows.

is r e d u c e d

(H,~) w i t h H~H. algebra

B:HcBcH

an i n t e r n a l

p o i n t of M as w e l l

3.9.iii)

3.2 we c o n c l u d e

of

complex

i) Let us fix F 6 ~

F is an i n t e r n a l

is f i n i t e - d i m e n s i o n a l .

The a s s e r t i o n

Assume

extension

an i n t e r m e d i a t e

in q u e s t i o n

and d i m N<~.

Then dim~<~.

p o i n t of ~ .

so that we h a v e

After

that ~ N (loglH×]) N E ~ { O } .

Let

And there

such that d i m ~B=

1.

3.7 then

the a s s u m p t i o n s

H<~ we see that d i m ~ . A n d from 3.9.iv) N

It is

QED

of

combined with

Let us fix a n o n z e r o

function

195 N

P 6 ~ N ( I o g I H × I ) A E ~ and let P~6ReL(m)

be its c o n j u g a t e

function

relative

to

Thus

we o b t a i n

an n£~

(H,~).

T h e n P~E ~ a f t e r V I . 6 . 8 . i ) .

such that V t 6 ~ : e t ( P + i P * ) 6 H ~ t

6~$.

ii) Let us p u t G : = e x p [ 2 ~ (P+iP*)I6H. that A : = { u + v G : u , v 6 H }

from 3.9.i)

is an i n t e r m e d i a t e

T h e n G~H b u t G26H.

It f o l l o w s

algebra with H~AcH.

In par-

A t i c u l a r d i m ~<~. iii)

N o w we a p p l y the c o m m u t a t i v e

operator

as in the p r o o f of VII.5.1: E a c h u6H d e f i n e s via :[f] ~ [uf] Vf6A. =

It is o b v i o u s

for u , v 6 H and c6~,

there exists

a W6A,

or u W - e ( u ) W 6 H multiplicative

linear

lemma VII.5.2

as w e l l as < 1 > = i d e n t i t y .

with

It follows

and that

e:H÷~ such that [W]=e(u) [W]

It is c l e a r

functional

6L(A/H)

that < u + v > = < u > + < v > , < c u > = c < u >

W ~ H and a f u n c t i o n

for all u6H.

algebra

a linear operator

that

e:H÷~

e(I)=I.

is u n i q u e

and is a

N o w we h a v e W = a + b G w i t h

a , b 6 H and h e n c e W2-2e(a)W=

It f o l l o w s HcBc_AcH

a2+b2G2+2abG-

that B : = { u + c W : u 6 H

and d i m

B

i) The i n i t i a l

(H,~)

of

(H,~) w i t h H + H

In v i e w of 3.10 we can a s s u m e step is as in the p r o o f

the a s s u m p t i o n s

E~+{O}.

Let us fix a n o n z e r o be its c o n j u g a t e

N = E N~ {O}. F r o m 3.9.i) f r o m 3.9.ii)

algebra with

of 3.9. F r o m function

function we h a v e e

We a s s u m e

the e x i s -

and shall d e d u c e

that d i m

of 3.10.

p o i n t of ML. T h e n F is an i n t e r n a l

we have

ReL(m)

is an i n t e r m e d i a t e

(H,~) be r e d u c e d w i t h d i m N < ~ .

t e n c e of a small e x t e n s i o n a contradiction.

and c6~}

2 - a2+2 l a - e ( a ) I W 6 H.

= I. QED.

P r o o f of 3.5: Let

internal

2e(a)W=b2G

~=I.

Let F 6 M L be a f i x e d

p o i n t of M as w e l l

3.9.iv)

we c o n c l u d e

so that N that ~ n(logiHXl)N

P 6 ~N N (log IH × l)nE ~ and let P~ £

relative t(P+iP~

to

(H,~).

)6H at l e a s t

we see that P ~ 6 ~ and P+iP~6H.

T h e n P~E ~ s i n c e for all t65. A n d

Since P+iP~H

it f o l l o w s

that H = { u + c ( P + i P ~ ) :u6H and c6~}.

ii) We h a v e P , P ~ £ R e H c E dependent

. L e t us s h o w that P and P~ are l i n e a r l y

o v e r E~: F r o m a P + b P * 6 E ~ w i t h a , b , 6 ~ not b o t h = 0 it w o u l d

low that a P + b P ~ = O

N s i n c e E ~ N ~ = {0}, and h e n c e

infol-

that b ( P + i P ~) = (b-ia)P£H

198 or P6H w h i c h iii)

is nonsense.

F r o m ii) we d e d u c e

3.9.iv)

implies

that t h e r e e x i s t

n o t real m u l t i p l e s function

that the d i m e n s i o n functions

of P ( a n d h e n c e

Q and let Q*6ReL(m)

of E~/E ~ m u s t be > 2. Thus

Q 6 ~N N (log IH x I)NE ~ w h i c h

in p a r t i c u l a r + O).

be its c o n j u g a t e

T h e n w e see as in i) that Q{E ~ b u t e t ( Q + i Q * ) 6 H

are

Let us fix such a

function

relative

at l e a s t

to

(H,~).

for all t6~,

and

also Q * 6 ~ and Q+iQ*6H. iv)

It f o l l o w s

that Q + i Q * = u + c ( P + i P * )

w i t h u6H and c = a + i b 6 ~ .

From

Re u, I m u 6 E = n ~ we see that u=O and h e n c e Q + i Q * = c ( P + i P * ) .

In p a r t i c u l a r

Q=aP-bP*

so that b # O a f t e r the c h o i c e

at e t ( P + i P * ) 6 H

at least

for t = n and for t = nc Vn£$

v) The p r o o f has an u n e x p e c t e d

O : O(z)

of Q. So we a r r i v e

for some c o m p l e x

finale.

Consider

c = a+ib w i t h b ~ O.

the e n t i r e

function

V z6~.

= fe z (p+ip*) ( P - i P * ) F d m

For n ~ I we have Dry(z)

We list a s e r i e s

= /eZ(P+iP~) ( P + i p ~ ) n ( p - i P ~ ) F d m

of p r o p e r t i e s .

I) ~' ( O ) = / I P + i p ~ 1 2 F d m > O

2) If z6~ is such that e Z ( P + i P ~ ) 6 H Thus ~ ( n ) = ~ ( n c ) = O -T~=O.

This

for all n65.

2

u(P+iP*)

It f o l l o w s

that

then ~ ( z ) = O

3) T h e r e

is seen as follows: (p+ip ~)

Y z6~.

exist

in v i e w of P F , P ~ F 6 N . J,T6~

such that ~ " - ~ ' -

We have

= u+o(p+ip~)

w i t h u6H and o6~,

= v + T ( P + i P ~)

w i t h v6H and T6~.

(u-T) ( P + i P ~ ) = " 6 H

((P+ip~)2-O(P+iP~)-T]HcH.

so that ~ + O.

so that

(u-T)HcH.

But this m e a n s

~ " ( z ) - o ~ ' (z)-T~{z) : / e Z ( P + i P * ) [ ( p + i p * ) 2 - o ( p + i p * ) - T ] ( p - i P * ) F d m = O

as a b o v e

that

Thus we o b t a i n V z6¢,

in v i e w of P F , P ~ F 6 N .

N o w the e l e m e n t a r y

t h e o r y of o r d i n a r y

differential

equations

tells

197

us t h a t

~ has

one

~(z)

= eeSZ+

~(z)

=

where many

of the

8e t z =

two

forms

I ~ + ~ e ( t - s ) Z } e sz

(~+~z)e sz

~,B6¢ zeros

are n o t b o t h = O . as f o u n d

But

in 2) above.

with

complex

s+t,

with

complex

s,

in e i t h e r This

case

~ cannot

is the d e s i r e d

possess

as

contradiction.

QED.

Notes

The Ahern-Sarason SARASON

[1967a].

the D i r i c h l e t to O ' N E I L L rem

simultaneous

[1968].

1.7 is due

lar d e n s i t y

The

lemma

For

is a m a i n

[1969]

and

in G A M E L I N

is m u c h

shorter.

[1969b].The

of the r e s t r i c t e d [1968][1969]

p.170.

of a u n i q u e

to use

Theo-

logmodu-

theorem

3.5 a p p e a r e d

in G A M E L I N - L U M E R Section

present version

is b a s e d

[1968]

idea

to be due

[1967a].

3.6 of the m a x i m a l i t y

GAMELIN

The

1.4 a p p e a r s

case

in A H E R N - S A R A S O N

see a l s o

is f r o m K O N I G

Its p r o o f

special

from AHERN-

[1969b].

to H A R D Y - W R I G H T

the

in 3.7)

theorem

proof

lemma

settled

[1968],

3.5

[1969],

extract

from K O N I G

point

and GAMELIN

the e l e m e n t a r y

is

approximation

result

version

(up to t h e r e d u c e d n e s s

is an a l i e n a t e d

form

1.4 we r e f e r

to G A M E L I N

restricted

1.3

The present

proof

IV.7.

The

[1968] full

is m o d e l l e d

in G A M E L I N - L U M E R

on the A r e n s - R o y d e n

after [1968]. theorem

Chapter

Function

The p r e s e n t analytic compact

final

functions Kc~ these

R(K)

the s u p n o r m

the class interior

All these cC(K) They

under w h i c h

of the polynomials,

closure

in C(K)

of the rational

of functions

are function

complex

it is natural

the individual

have no holes

(= b o u n d e d

P(K)=R(K)=A(K)

after

R(K)=A(K),

sufficient

with

examples

in the

16P(K)cR(K)cA(K)c

to the abstract

as w e l l

III.

theory

of

necessary

are in terms

The above

polynomial

of ~-K.

conditions

The

those K for problem.

A

It is due

sufficient

condition

that the

from

latter

zero.

one results

more ge-

from the

due to V i t u s h k i n

continuous-analytic

Thereafter

Another

point of K be in the

conditions

had b e e n d i s c o v e r e d theory.

even

theorem.

of holes.

is that each b o u n d a r y

of the s o - c a l l e d

of a p p r o x i m a t i o n

approximation

that K

Then

number

away

and sufficient

~-K).

approximation

a finite

to trans-

of K:

to c h a r a c t e r i z e

as the rational

of K be b o u n d e d

condition

property

examon K

It is clear

it is simple

of the c o m p l e m e n t

problem

as simple

conditions

equalities.

geometric

as the more g e n e r a l

of some c o m p o n e n t

quasi-geometric

intuitive

difficult

become

of K is =~. And

is that K have

of the holes

sufficient

can all be p r o p e r

inclusions

components

renowned

condition

to M e r g e l y a n

essential

are h o l o m o r p h i c

to ask for g e o m e t r i c

the M e r g e l y a n

But it is an e x t r e m e l y

niques

functions

on K in the sense of C h a p t e r

P(K)cR(K)cA(K)cC(K)

Hence

into a very

which

which

algebras

concrete

form P(K)=R(K)

boundary

in C(K)

algebras

iff the interior

neral

of

For n o n v o i d

off K,

that A(K)=C(K)

diameters

algebras

plane.

algebras.

reveal.

which

to the standard

of the c o m p l e x

of K.

The inclusions ples

subsets

Sets

in C(K)

are the most p r o m i n e n t

function

is d e v o t e d

Planar

closure

are s u p n o r m - c l o s e d

and h e n c e

on C o m p a c t

are

w i t h poles A(K)

chapter

on compact

the s u p n o r m

P(K)

Algebras

X

capacity.

via the c o n s t r u c t i v e it has been

realized

part of them can be b a s e d on the f u n c t i o n a l - a n a l y t i c

techthat an

theory

199

of function

algebras

of i m p o r t a n t

from its p o i n t

of v i e w

ry has not yet b e e n place

certain

results

developed

concrete

Mergelyan

the m a i n

methods

to enter

theory.

these

restrict

outflows

combined

within

theory.

from the abstract algebra

will be the direct

of course

We return a measurable

of the abstract

Hardy A l g e b r a

construction consequence

IV.

II.4.5.

as abstract

Theory

situation

space

subalgebra

(X,Z)

~cB(X,Z)

and a complex

Recall

the w e a k

of A relative

topology

III the F.Riesz

of C h a p t e r

AcB(X,Z)

A.I.7.

subalgebra

Note

m£Pos(X,E)

in the w e a k . t o p o l o g y

we c o n s i d e r

the closure

~(L~(m),L1(m))

F r o m the b i p o l a r

t h e o r e m we have

VS£A ± w i t h

Let us note

e<<m.

which

The

that in the compacttheorem

and hence

that

Am: = A m o d m-mw e a k ~

is a complex

for f6B(X,E):

a useful

fix

con-

as well.

representation

A.2.1 implies that e I C ( X ) = o ( C ( X ) , c a ( x , z ) ) = ~ ( C ( X ) , C ( X ) ' ) ~DC(X) = ~ s u p n o r m = A.

For nonzero

II: We which

~=~(B(X,Z),ca(X,Z)).

theorem

situation

of C h a p t e r

to ~ is a complex

continuous

Sfd@=O

algebra

of Chapters

to the aim of the pre-

to the b o u n d e d - m e a s u r a b l e

the constants.

as well.

and

Its purpose

Hardy

theory

image

be adapted

na-

plane.

1.6 and 1.9 can be c o n s i d e r e d

We have A ~ = A II from the b i p o l a r

L~(m)

certain

transformation

in the complex

the abstract

of

ourselves

from the abstract

with

are the C a u c h y

into the function

results

to re-

into the depths

We shall

of Baire measures

will

in order

theo-

type theorems.

I. C o n s e q u e n c e s

closure

the abstract

will be the m a i n F . a n d M . R i e s z

of m a t e r i a l

sent chapter:

tains

tools:

remains

IV-IX

the v e h i c l e

The choice

conceptual

- at least w h e n

The road of transfer

1.3 w h i l e

since

But in spite

the situation

constructions.

c e r t a i n m a i n results

of C h a p t e r s

theory,

are more or less d i r e c t

hitherto

section

is to t r a n s f e r

II-III.

concrete

transformation

The initial

theory

to d e v e l o p

approximation

which

additives.

favorable,

work we do not intend

to those

and b a s i c

few concrete

due to the abstract

is not quite

able

rational

theories

logarithmic

rather

complicated

In the p r e s e n t constructive

tural

with

contributions

technical

c

subalgebra

f m o d m £ Am ~ remark.

200

1.1

REMARK:

Fix

1~_p< ~.

For

f6B(X,Z)

the

subsequent

properties

are

equivalent.

i)

fE~.

ii)

For

each

nonzero

m6Pos(X,Z)

we h a v e

f mod m 6 A m .

iii)

For each

nonzero

mEPos(X,Z)

we h a v e

~LP(m)-norm f mod m 6 A mod m

Proof: then

i)=ii)

is c l e a r

were

an h6Lq(m)

there

contradiction Un6A

such

f6~.

with

above,

fuhdm=O

that

~lUn-flPdlel+O.

Then

ii)~iii)

vu£A For

If this

and ~ f h d m ~ O ,

6EA ± there

~Und8=O

implies

which

are

that

were

false were

a

functions IfdS=O.

Thus

QED.

assertions one has

to c o n n e c t

in the

ii)

Each

quel will iii) iv)

i)

~6~(A)

admits

We t u r n

~6Z(A)

to the

are o u r m a i n

F.and

M.Riesz

Thus

we h a v e we h a v e

serious

which

1.3 M A I N

One

a unique

~ as well.

F o r e a c h ~6E(A) each

remark

as in 1.3.7.

notions are

for ~

all

could

with

clear

also

those

f r o m the

for A. T h e

above.

In iv)

r e s t on IV.1.3.

(A~)±=A I.

be n a m e d

For

the b a s i c

subsequent

to p r o c e e d

1.2 REMARK:

concern.

consequence

continuation

M(~,~)

= M(A,~).

MJ(A~,~)

A fundamental which

Thus

se-

M ( ~ m) = M(A).

of the

functions

s t e p of r e d u c t i o n

can be

M.RIESZ CONSEQUENCE: The

F.and

in the

= MJ(A,~) .

characterizations

II.4.5

6 ~ ( A ~) w h i c h

Z (~})=E(A).

formulated

function

in is the m a i n

as follows.

f6B(X,Z)

is in

it s a t i s f i e s

o)

and

the

to f m o d m 6 A m . iii)~i)

We p r o c e e d

iff

from

ffdS=O

for

furthermore

above

are

all

86AlUM(A) A,

satisfies

the

subsequent

i)

For

ii)

For each

m6M(A) : f m o d m 6 A m .

i')

For

o6M(A)

ffdS=O

conditions

which

in v i e w

all e q u i v a l e n t .

e a c h m6M(A) :ffd@=O

each

Ve£A ± with

@<<m.

there

V@6A ± with

@<<m.

is an m 6 P o s ( X , Z )

with

a<<m

such

that

of the

201

ii')

For each

o6M(A)

there

is an m 6 P o s ( X , Z )

with

o<<m

such

that

f m o d m 6 A TM.

For

the

important

concrete

see t h a t A ± N M ( A ) ^ = O , conditions.

contains

so t h a t w e e n d

At t h i s

construction

point

IV.I.3:

measures

examples

For

up w i t h

it is m o s t fixed

<<m t h a t

of t h e p r e s e n t the

natural

~6Z(A)

chapter

residual

to i n v o k e

and m 6 P o s ( X , Z )

construction

produces

we

shall

equivalent

the d i r e c t

such

image

t h a t M(A,~)

the Hardy

algebra

si-

(Am,~ TM) w i t h

tuation

M = {O~V6LI(m) : V m 6 M ( A , ~ ) } , N c

{f6ReL1(m) : f m 6 N ( A , ~ ) } ,

MJ = {O~V£LI (m): V m 6 M J ( A , ~ ) } , N J c {f6ReL1(m) : f m 6 N J ( A , ~ ) } .

And

if m i t s e l f

reduced

with

of C h a p t e r s VI.4

is 6M(A,~)

16M.

to the

iii) ffd~ ~fudm

For

= Ifdm

=

Ifudm

= Ifdm

vial.

i)~iii) and

2)

for s o m e

the

of the

i)ii)i')ii')

(~(f+u) dm) 2

of

iff

algebra

theory

from Section

functions

in A~.

conditions

V ~6Z(A)

and ~,T£M(A,~),

V m6M(A)

a n d u6A.

V m6M(A)

a n d u6A.

it s a t i s f i e s

(Am,~ TM) is

is e q u i -

1.3.

V m6M(A)

ludm

Hardy results

subsequent

V m6M(A),

is clear: the

16M.

= ffdm

the

and

and a n d u6A.

o)

and

some

iii)

and c o n s i d e r to s h o w

we h a v e

assertion since

image

iii)~ii)

We have

ludm

first

assertion

the d i r e c t

to p r o v e

~£Z(A)

assumption

ffudm

second

after

So it r e m a i n s

Under

each

fudm

is in ~

(Am,~ TM) w i t h

tion

of the

(ffdm) 2

is o b v i o u s

m6M(A,~)

characterization

situation

of the e q u i v a l e n t

iii)iv)v).

½(O+T)£M(A), ii)=iv)

central

we

ffd~

=

ff2dm

Proof:

abstract

use

conditions

v)

f6B(X,Z)

apply

algebra

the

f£B(X,Z)

~(f+u) 2 d m =

conditions

and

next

iv)

Thus

can

the Hardy

To s t a r t w i t h

the

1.4 T H E O R E M : valent

Thus we

IV-IX.

to o b t a i n

then

since

um -

construction,

and v)=ii). the

~-T6A ± and

(Sudm)m

For

iv)~v)

reduced Hardy algebra -9 f:= f m o d m 6 L~(m)

I) U , V 6 M

~ Um,Vm6M(A,~) V u 6 A m.

Thus

<<m.

is tri-

this p u r p o s e

that

V u 6 A m o d m and h e n c e

<<

6 A ± and

fix

situais 6A TM.

= Sf(U-V) d m = O f6A m after

202

VI.4.5°

Under

the a s s u m p t i o n

v) w e h a v e

+

=

(~fVdm) 2 a n d 2)

The most

Thus

f6A m a f t e r V I . 4 . 7 .

consequence

Assume

that

is the s u b s e q u e n t

BcB(X,Z)

~ Sf V d m =

Proof:

If B c ~ ~ t h e n

upon

the

application

1.6 T H E O R E M :

Assume

two

of

that

BcB(X,Z)

subalgebra

(which m e a n s

assertions

1.4 to the

QED.

maximality

is a c o m p l e x

B o A ~ iff A ± N M ( A ) ^ c B ± and M(A) cM(B)

follows

such

as before.

prominent

1.5 T H E O R E M : Then

+2

I) V 6 M ~ V m 6 M ( A , ~ )

+

follow

with

1.2.

The

functions

is a c o m p l e x

AcB.

that M(A)=M(B)).

from

individual

theorem.

converse

f£B.

subalgebra

QED.

with

AcB

that

O) A i N M ( A ) ^ c B ±, I) e a c h ~6Z(A)

has

an e x t e n s i o n

~6Z(B),

2) N ( A , ~ ) c B l for e a c h ~ 6 Z ( A ) .

Then

Bc~.

Proof:

In v i e w

~£M(A,~)

for some

cM(A,~).

From

Theorem Condition

2)

For

first

t h a t M(A)c_M(B).

an e x t e n s i o n

of the

Also

Re B c Re A ~ w h i c h is o f t e n

is o b t a i n e d

to s h o w

difficult

when

theorem

desired

in p a r t i c u l a r

6M(A,~).

~6E(B)

the

abstract

This

Hardy

has

some

that

T6M(B,~)c

theorems.

a unique

to a s s u m e

A much

algebra

requires

some QED.

the D i r i c h l e t

to be v e r i f i e d .

Assume

Mergelyan

~£E(A)

it is s u f f i c i e n t

abstract

IX.3.5.

~6M(B,~).

if e a c h

resembles

and

re-

the o v e r -

condition.

more

theory

powerful contributes

preliminaries

on log-

measures.

fixed

~6Z(A)

we

ML(A,~) :={~6Pos(X,~)

the

Choose

~-TEN(A,~)cB ± and hence

is the

measure

the m a x i m a l i t y modular

2) t h e n

1.6

condition

Condition theorem

1.5 we h a v e

2) is s a t i s f i e d

presentative all

of

~£~(A).

class

of l o g m o d u l a r

introduce

: logI~(u) l = S ( l o g l u l ) d ~

measures

span(ML(A,~)-ML(A,~))={c(~-T) is n o n v o i d .

Thus

for ~,

a n d N L ( A , ~ ) := r e a l - l i n e a r

: ~,T6ML(A,~)

MJ(A,~)cML(A,~)cM(A,~)

Vu6(A~)×},

a n d c>O}

from

in case

1.2 a n d v i a

t h a t ML(A,~)

exponentiation,

203

and ML(~,~)

= ML(A,~)

compact-continuous

from

the definition

situation

of C h a p t e r

itself.

Note

III we h a v e

that

in t h e

ML(A,~)~

in v i e w

o f III.1.1.

The

direct

ures w i t h we have we

that

Proof: (Am,~m).

situation

some

consequences

L e t ~6Z(A)

ML(A,~)

Fix

from

such

that

that

this

fact

contains

ML#@.

Hardy

d i m N(A,~)

From

M(A,~)

satisfies

whenever

abstract

measm£Pos(X,Z)

< ~.

algebra

theory.

T h e n MJ(A,~)

~

~.

any m 6 M ( A , ~ )

Then

the

logmodular

For nonzero

involved.

such (Am,~ m)

fm6NL(A,~)}

and consider

d i m N < ~. H e n c e

L e t ~EZ(A) that

In v i e w

Consider

IX.I.7

we

• EM(A,~)

point

IV.4.5

the H a r d y

shows

that

algebra

situation

{O~V£LI(m):

Vm£MJ(A,~)}

Then

the H a r d y

all

obtain

~ T=Gm with

conclude

1.9 T H E O R E M :

the

can choose

Assume

that

c>O such

that

with

the

BcB(X,Z)

second

that

oE

constant

an e x t e n s i o n

an i n t e r n a l

In p a r t i c u l a r

G~cF

for some VGEM.

abstract

is a c o m p l e x

O) A±DM(A) ^ c B I, has

some

c>O.

point ~=Fm with

d i m N < ~. c>O.

Thus

It f o l l o w s

from that

QED.

that

I) e a c h ~£~(A)

for

(Am,~ m) w i t h

or V ~ c F

G E M ~ G ~ c F ~ T~co.

section

< ~. A s s u m e

is ~ c o

are <<m.

situation

~ Vm~co=cFm

a constant

d i m N(A,~)

< ~ we

Y6M(A,~)

algebra

~ Vm6ML(A,~)

that

of ML(A,~)

of M ( A , ~ ) .

of d i m N(A,~)

of M ( A , ~ ) .

We h a v e V E M L

such

each member

~ is an i n t e r n a l

Proof:

such

algebra

Vm6ML(A,~)},

is s u c h

m£M(A,~)

We

the d e f i n i t i o n s

and m6Pos(X,Z)

NL c {f6ReL1(m):

1.8 T H E O R E M :

FEM.

(Am) x f r o m

the

IX.1.

• ~. QED.

M(A,~) Then

connects

from Section

ML c {O<=V£LI(m):

1.7 R E M A R K : and hence

IV.1.3

notion

for ~6Z(A)

< < m the H a r d y

insert

= MJ

construction

respective

( ~ ) ×mod m c

conclude

measures

We

image

the

~6Z(B),

Mergelyan

subalgebra

theorem.

with

AcB

204

*)

d i m N(A,~)

< ~ for e a c h ~ 6 Z ( A ) ,

2) N L ( A , ~ ) c B ± for e a c h ~ £ Z ( A ) .

Then

Bc~.

Theorem neither

Condition

the o b v i o u s nor

measure

weak

adequate

2) is s a t i s f i e d

logmodular all

1.9 has

necessary

point

but which

in p a r t i c u l a r

6ML(A,~).

Also

of c o n d i t i o n we

~) w h i c h

is

are

unable

to e l i m i n a t e .

if e a c h

~6Z(A)

has

it is s u f f i c i e n t

a unique

to a s s u m e

the o v e r -

condition

Re

which

corresponds

Proof: there

c

real-linear

to the

In v i e w

of

~£E(A),

m = 5(I a+T)

definition

with

choose

of s m a l l n e s s

such

that ~6Z(B)

is an e x t e n s i o n

Section

IX.3.

Now

f6NLCReL1(m)

we h a v e

so t h a t

fufdm

= O Y u 6 B m and h e n c e

V u 6 R e B m.

Thus

after

maximality

2. T h e

IX.3.4 theorem

(Bm,~ m) IX.3.5

is a s m a l l implies

Transformation

of M e a s u r e s

The Cauchy

transformation

is d e f i n e d

into

the

defined

view

see

study

=

(u) J[d6~--~

for t h o s e

at o n c e

of its k e r n e l

for t h e

of

for e a c h

Assume

Hardy

IX.3.

that

O6M(A,~)

T6M(B,~), algebra

(Am,~ TM) in the

O6M(A)

and put situation sense

fm6NL(A,~)~-B ± in v i e w It f o l l o w s of

of

of 2),

t h a t N L C ( R e B m ) ±.

(Am,~ m)

sO that

the

QED.

to t r a n s f o r m

the m e a s u r e

@6ca(~)

function

~c: eC(z)

We s h a l l

that

and some

B m = A m.

Cauchy

in S e c t i o n

Bmc_Am.

extension

that

,

to s h o w

(Am,~ TM) is a r e d u c e d

(Bm,~ m) for

g<<m

an e x t e n s i o n

6 M(A,~) . T h e n

d i m N < ~ and

span(logl ( ~ ) × I )

1.3 it is s u f f i c i e n t

is an m £ P o s ( X , E )

for s o m e

with

B

that

z6~ w h e r e

this

function

of the

,

is t r u e

the C a u c h y

algebras

R(K).

eAC(z): = I

~

< ~.

for L e b e s g u e - a l m o s t transformation

all

z6~.

is a b a s i c

In

device

2O5

In the present section we fix a measure @6ca(G) . We need two simple lemmata. Let us introduce the notations

v(u,s)

= {zC¢:Iz-ul<s},

?(U,S) = {Z6~:Iz-uI<s} 2.1 LEMMA: Let

for u6~ and s>O.

(X,Z,o) be a (finite or infinite) positive measure

space. Then for each measurable function f:X+[O,~[ we have ~fda = ~ a([f_>t])dt. X ]O,~[ Proof: If o([f>t])=~ for some t>O then both integrals are =~. So we can assume that o([f>t])< ~ Vt>O. Put M = {(x,t):O
~ 2~L(E)

¥z6,.

Proof: We can assume that OO we obtain from 2.1

~ E r

= i L<{u6E:~>t}> O oo

T.~nv(z,s)

=

~+

dt=

i L({u6E:'u-zI<s})~ O

~nv(z,s)

0

r

r <

L V(z,S).s 2 + O

which is = 2 / ~ ( E )

2.3 PROPOSITION: ii)

L(E)s

= ~r +

L(E),

r for r =

L(E)

. QED.

i) The function 0AC:~+[O,~] is Baire measurable.

N:= {z6~:sAC(z)=~} has L(N)=O.

206

iii) The function @C:~-N~¢ is Baire measurable. iv)

For each Baire set Ec~ we have

fleC(z) IdL(z) <_--feAC(z)dL(z) _< 2/ViT~-IlelI. E

E

In particular eC6L~oc(~,L). Proof: The assertions follow from the Fubini theorem and from the above 2.2. QED. 2.4 PROPOSITION:

eC(z)

is defined Vz6~-Supp(@), where Supp(e)

notes the support of e. In other words:

If UC~ is open with

de-

lel (u)=o

then 6C(z) is defined Vz£U. Moreover 8CIu is a holomorphic function. 2.5 PROPOSITION:

If e6ca~(~)

then 6C(z)

is defined Vz6~ with

Izl

sufficiently large, and eC(z)+O for z+~. If moreover @=fL with f6L~(~,L) then 8C(z)

is defined Vz6~, and 0C:~+~ is continuous and hence uniform-

ly continuous and bounded. Proof:

eC(z)÷o for z~

e=fL with f6L~(~,L) view of

is an immediate estimation. Now assume that

and put E:=Supp(f)c~.

eAC(z) = I ~ L ( u ) E

< 2 const/~(E)<~

It remains to show that 0C is continuous.

l

eC(z)-sC(a)

=

If(u)

Then 8C(z)

z-a (u-z) (u-a) dL(u)

Iz-al = const I lu_zllu_aldL(u) E-V(z,½1z-a[)

is defined Vz6~ in

Vz6~.

If a,z6~ with a~z then

< const

Iz-al dL(u) lu-zl lu-al

Iz-a[ + const I lu-zl lu-al dL(u) EnV(z,½1z-al)

The first integral tends +0 for z÷a after the Lebesgue dominated convergence theorem. In order to estimate the second integral note that I I for u6?(z,~Iz-a I) we have Iz-a I =< lu-zl+[u-al =< --Iz-al+lu-al 2 and hence I < ~Iz-al=lu-al. Thus 2.2 implies that the second integral is ~2zlz-a I . QED.

207

We turn to the q u e s t i o n form 8C

h o w to r e - o b t a i n

It is c o n v e n i e n t

distribution

which

8 f r o m its C a u c h y t r a n s -

to u s e a b i t of d i s t r i b u t i o n

corresponds

to 86ca(~)

theory.

The

is co

[@] The c r u c i a l

point

: <[%],f>

= ffde

Vf6C,(~).

is the s ~ s e q u e n t

special

case of the C a u c h y

formula

A.3.5. 2.6 LEMMA:

F o r f6C1(~)

f(z)

we have

= - ~ ~

(u)dL(u)

In terms of the C a u c h y

transformation

2.7 P R O P O S I T I O N :

[e]

8C6LIIoc(~,L) Proof:

in the d i s t r i b u t i o n a l

<~0 C -. = -

~f dL = - I BC ~-~

F r o m 2.7 a f o r t i f i e d 2.8 P R O P O S I T I O N : function)

then

Proof:

From

that

JeJ (U)=O.

In p a r t i c u l a r

2.7 we see t h a t

eCJu 6 L;oc(U,L)

everywhere 0CJu=o

equal implies

is h o l o -

to a h o l o m o r p h i c that

[G] JU=O or f f d % = O V f 6 C , ( U ) .

Jgj (U)=O. It f o l l o w s

QED. Define

Supp(e)

The remainder of t e c h n i c a l

QED.

to 2.4 can be derived.

CS(@):={z6~:

sAC(z)< ~ a n d eC(z)#O}

of e. T h e n C S ( @ ) c ~ is a B a i r e

Furthermore

Vf6C~(~) .

dL vf6c~(~) . B u t this is im-

theorem.

Lebesgue-almost

18j (U)=O.

2.9 C O R O L L A R Y :

support

IrBC ~ = ~.

converse

of

is

If U C ~ is o p e n such t h a t

(that m e a n s

i(~ )c.

f = - ~\~Z L

sense.

derivative

to s h o w that ffde

that

I ~8 ~ C , with differentiation

a f t e r 2.6 and the F u b i n i

morphic

this m e a n s

The distributional

Thus we h a v e mediate

We have

Vz6~.

unless

£=O.

c CS(B).

of the s e c t i o n w i l l be d e v o t e d

nature

<8>:

set w i t h L ( C S ( @ ) ) > O

the C a u c h y

associated

<8>f =

with

(fe) C - f8 c

86ca(~).

to a n e w t r a n s f o r m a t i o n

Define

Vf6B(~,Baire).

2O8 It follows

that

<8>f(z)

exists

2.10 THEOREM:

and = [f(u-)---f(z)de(u) j u-z

Assume

that f6CI,((~). Then

<e>f = !(~f w\~Z 8cL )c Proof:

The Fubini

sAC(z) <~

V z6~ with 9AC(z)<~.

Lebesgue-almost

everywhere.

theorem shows that

and

I

~dL(v)<~

in

Lebesgue-almost

all z6*.

Supp (f) In these points

<9>f(z)

z6~ both functions

=

which

and we have

!{~f ec~)C(z) = •

=

are defined,

- W\~-~

If(u)f(z)de(u) ]

in question

_

'Pf 0C(V)v!-dL(v) Zjo~(v).~.-~---

_

I

~f

I

(u)

u-z

I(f(u)-f(z) u-z

1[~f,

~]~v~

,

(u-v)

~dL (v) /

1(v_z)dL(v))d~(u)

is = 0 in view of 2.6. QED.

2.11 COROLLARY:

If UC~ is open such that f6B(~,Baire)

in U then <8>f is holomorphic where equal to a holomorphic Proof: with 2.10.

in U( (that means Lebesgue-almost

3) In the general

pact subset of U. Then f = everywhere

(fs)CIu and the assertion

then the assertion

Thus

follows

is =I on V and =O outside of some com-

(f-fh)

+ fh where

(f-fh)IV = O and fh6Cl(~)

I) and 2) show that <@>f is Lebesgue-almost

equal to a holomorphic

function

in V. QED.

In the next section we shall need the subsequent particular quence.

follows

from 2.4 combined

case we fix an open V with ~ compact cU and

choose a function h6C 1 (~) which with fhIV holomorphic.

every-

function).

I) If flU=O then <8>f[U =

from 2.4. 2) If f6Cl(~)

is holomorphic

conse-

209

2.12 C O R O L L A R Y :

Assume

1 ~h t h a t e . . . . ~Z L for some h6C 1 (~) and

f6CB(~) . T h e n

i)

<8>f e x i s t s

ii)

If Supp(h)

everywhere

and is c o n t i n u o u s

is c o n t a i n e d

in some V(a,s)

Uhll~ ( f , ? ( a , e ) ) ,

estimation

ll<9>f I ~ 2s ~

l a t i o n = S u p { I f ( u ) - f ( v ) I :u,v6M} iii)

where

on ~.

then w e have

~(f,M)

denotes

the s u p n o r m the o s c i l -

of f on M.

If Uc~ is o p e n such that

flU is h o l o m o r p h i c

then < 8 > 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+<9>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 i<8>fl

=

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

<9>f

on ~ and tend to 0 at infinity.

fh + <9>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))

<9>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 O
V(an,rn)DV(am,rm)

U V(an,rn) n=1

= ~

whenever

n%m,

has no interior points.

a dense sequence

a1=z I and fix O
a compact

1) The construction

such that

n=[1 rn < I ,

Zp~V(al,rl),

for R(K)~A(K).

We construct

closed unit disk DUS. We choose points (n=I,2,...)

on X. QED.

the section with a famous example

with K°=¢ such that R(K)#A(K)=C(K).

i)

and hence

X K

It follows

via fl (~-X)=O.

on all of ~ after 2.5 and holomor-

of points

Zn£D

(n=1,2,...).

Then take the smallest

put a2=z p and fix O
Put

index p with

Now proceed via in-

214

2) In v i e w exhibit

of K ° = ¢ we h a v e A ( K ) = C ( K ) .

a nonzero

Lebesgue

measure

and N n the

measure (= arc

respective

@6R(K) ±. L e t

length)

outer

@ = n=[INnOn For

{zl>1

theorem

while

oo [ I n= I S

=

then

all

A.3.3.

of i n d e x

z6~

vector

we

denote one-dimensional n and Sn:=~V(an,rn), and N

functions.

so that

I Nn(U)dOn(U) u-z

Define

O%G6ca(K).

terms

We

If

Izl~1

then

after

A.3.4

the other

retain

terms

n~m

P(K)

our

abstract

fundamental

either

R(K)

fixed

under

the

write

for

as is the

are

all =O.

We

can

that

K~

to c o m p r i s e

restriction

the m a i n

lemma

II.I.4.

then

the

second

%C(z)=O

results

Assume

term

member,

for all

t h e n Z _ ~ e 6 R ± as well.

ii)

If @ C ( a ) % O

then

that

case

06R ± and a 6 K

there

functions

exists

f6R of the

case

R=A(K)

R=R(K)

one

in the

poles

o f f K in v i e w

the

explicit

combined

use

depend

that

cases.

with

did not

section

convention in b o t h

to e x p l o r e

without

now we

of the p r e s e n t

are t r u e

If @ C ( a ) = O

while

until

a n d A(K)

on

R denotes

As u s u a l w e

for u6K.

i)

in R. In the

P(K)

isomorphism

While

us a d o p t

which

for R(K)

of ~ and c o n t i n u e

supnorm

= R(K).

Let

in r e s u l t s

= M ( R , ~ u)

Measures

subset

= P(K) IK ~ P(K)

or A(K)

The

the C a u c h y

m~l:

t e r m of the

expect

LEMMA:

Proof:

a certain

It f o l l o w s

4.1

dense

are =0 a f t e r

last

representing

compact

the

theories

M(R,u)

member

z£V(am,rm)

a n d the

R(K)cA(K).

since

3.2 w e h a v e

second

e6R(K) ±. QED.

the

algebras

mention

of the

m is = 2 ~

and h e n c e

I u1_--~ N ( u ) d d ( u ) . S

n

4. O n the a n n i h i l a t i n g

the

- NO

R(K)#A(K)

d and o

on S = ~ V ( O , I )

normal

to p r o v e

z6~ w e h a v e

@C(z)

If

In o r d e r

this

takes

of the e q u a t i o n

such

that

an m £ M ( R , a )

form

f = f(a)

follows the

@AC(a)<~.

with

+

m<<e.

(Z-a) F w i t h

f r o m the A r e n s

rational

functions

F6R

are

l e m m a 3.5 P f = ~ with

215

P[a)

f = ~

L e t us p u t

+

(Z-a)

the

done. with

last

And

If

equation

4.2

@C(a)%0

then

In the

ii)

sequel

Proof:

f 6 R in q u e s t i o n

= f(a)Sdo

for all

then

= f(a)@C(a).

f6R.

we obtain

CS(@) the

In the

from

is #@ a f t e r

subsequent

case

II.I.6

@C(a)=O

a measure

2.9

we

are

m£M(R,a)

and contained

fundamental

06R ± is s i n g u l a r

shall

combine

supports

in K

consequence.

to all m6M(R)

i) L e t

a£X.

For each

a6K

with

Supp(m)C

with

a6S.

4.1 w i t h

m6M(R,a)

then

con-

a6Supp(m).

of K O w h i c h

such Put

an e l e m e n t a r y

in q u e s t i o n .

discrete

a n d m6M(R,a)

of ~ - S u p p ( m )

above

component

m#6 a a n d G A S u p p ( m )

I) F i x

the

of the m e a s u r e s

a6K ° and G be the

with

component

If

we

on the

Let

functions

Q=Q(a)+(Z-a)H.

8=0.

4.3 REMARK:

m6M(R,a)

then

Thus we obtain

sideration

+ SFd8

P=P(a)+(Z-a)G,

QED.

CONSEQUENCE:

Supp(@)

the

is t r u e

case

0 6 R ± is n o n z e r o 3.7.

For

= f(a)Sd~

in the

m<<0<<%.

after

P(a)H% , with

- ~

~:= l @ £ c a ( K ) .

Sfdo

Thus

~ (G

then

that

contains

a~Supp(m).

0=6a-m

a. F o r e a c h

~GcSupp(m).

L e t S be

and consider

the

the C a u c h y

transform eC:

Then

subset

for

a6X

K ° which

see

on

O n the

is i m p o s s i b l e contains with

Id~u.~_(~)

that

other

which

a£SNG we

conclude

that

can assume

~G6~S.

that

proves that

conclude

In fact,

Since Vz6S-{a}

8C(z)=O

that

is c o n n e c t e d

from S c G - GASupp(m)

8C(z)%0

hand

follows

a and a s s u m e

G - GNSupp(m)

V z£(¢-Supp(m))-{a}.

(~-Supp(m))-{a}.

conclude

06RIcR(K)±.It

connected Since

z+a w e

of S-{a}.

3.7 s i n c e

we

z-a1

@C is h o l o m o r p h i c

eC(z)÷~

ular

eC(z)

and h e n c e

2) L e t

GnSupp(m)

that

ScG

in v i e w

for x 6 ~ G

on some

S A X = #.

now G be the is d i s c r e t e .

and hence

there

so t h a t

that

and

discrete after

In p a r t i c -

component Since

of

S c K ° is

S c G - GASupp(m).

of t h e d i s c r e t e n e s s

c ~ - Supp(m)

X n ~ S u p p ( m ) or Xn6S,

is c o n n e c t e d

except

V z 6 ~ c (~-Supp(m))-{a}

SN~=~ i).

S-{a}

assumption

S = G - GNSupp(m).

are X n £ G w i t h x£S and hence

Xn+X, x6~S

We

a n d we

since

x~S

216

in v i e w

of x~G.

= ~Supp(m) the

assertion

discreteness o£M(R,a) hence

We

introduce Tr(S)

V06S.

Proof: exists

I)

some

in T r ( S ) N X Tr(S)DX a(n)+a one

Let

o6S w i t h

then

Thus

Note

Scca(K) exists

For

a band

z6K

4.1.ii)

be

a6Supp(a).

2) Let

a band

some

that

In fact,

fix

Then

if then

a6Tr(S)NX ~:=

and

[ 2-nmn n=1

component

G n of K ° a n d m n # ~ a ( n )

i) w e h a v e

a(n)

6 Supp(mn)

c Supp(o)

from

4.3.ii).

b(n)

= ta+(1-t)a(n)

case

t a k e b(n)

a6Supp(o).

c Supp(o)

is t r i v i a l . Since

of a(n) . T h e n

after

In c a s e

a(n)6G n but

6 ~G n c S u p p ( m n )

instead

8) c

is d i s -

there

is d e n s e

each

n~1

we h a v e

iii)

4.3.i). we

a~G n there

c Supp(o)

In c a s e

~G n c

exists

a point

for s o m e O < t <=1 .

the m o d i f i e d

ii)

conclude

sequence

In t h i s

+a as w e l l .

QED.

the

above

4.4 w i t h

subsequent

4.5 T H E O R E M :

CS

situations

a(n) 6 s o m e

Supp(8)NX

a6Tr(S)NX

6 S. F o r

iii)

the

formulated

# ~.

[ 2-nlonl6S with n=1 choose a(n)6Tr(S) with

G n of K O a n d m n = ~ a ( n )

combine

the

M(R,z)NS

o:=

component

We

define

{an:n=1,2,...}

a(n)6some

to o b t a i n

where 2) a n d

c Supp(o).

point

ii)

Hence

from

that K°QSupp(o)

Tr(S)DX

to e a c h

an6Supp(an)

us

such

d6S w i t h

a(n)6X,

Supp(mn)

we

for w h i c h

can be

i)

6 Supp(mn)

(1-m({a}))o

~G c S u p p ( o )

Scca(K)

of t h o s e

that

to p r o v e

and mn6M(R,a(n))NS.

In c a s e

=

a~Supp(m)

in v i e w of t h e

and m = m({a})6 a +

notation.

if O n 6 S w i t h

three

case

e6R ±.

there

c Supp(o).

In the

If a 6 S u p p ( m )

c K to c o n s i s t

It s u f f i c e s

and

of the

a(n) c

Then

ii).

QED.

can be void.

for all

~G c $S c $ ( ~ - S u p p ( m ) )

= Supp(m)-{a}.

as well.

4.4 P R O P O S I T I O N : crete

in 2).

O<m({a})<1

one m o r e

Tr(S)

that

us n o w p r o v e

shown

Supp(o)

= Tr(S,R)

T r ( { e } v)

shown

3) L e t

assumption

satisfies

Of c o u r s e

we have

has been

~G c S u p p ( m )

trace

c

Thus

c Supp(m).

Assume

= CS(O)NX

rather

that

the

fundamental

precise

86R ± s u c h

= Tri{8}~)NX.

lemma

consequence

4.1

information.

that

K°DSupp(e)

is d i s c r e t e .

Then

217

Proof: c

I) W e h a v e

T r ( { 8 } v) a f t e r

S={8} v w e

see

cSupp(8).

Combine

The

above

restriction In v i e w

Supp(8)

4.1.ii)

after

all

these

considerations to c o m p a c t

is s u p n o r m

facts

lead

to the

isometric

2) W e h a v e

3) F r o m

4.4

for s o m e o6S

to o b t a i n

sets Y with modulus

2.9.

above.

t h a t T r ( { 8 } v ) NX is c S u p p ( o )

of the m a x i m u m

f ~-> flY

c CS(8)

as r e m a r k e d

the

the

on A ( K ) .

Thus

idea

to f o r m the

c A(K)

c C(K)

restriction we h a v e

to

QED.

a n d in p a r t i c u l a r

principle

c

and hence

assertion.

fundamental

XCyCK

CS(8)

applied

the

to X i t s e l f .

map C(K)÷C(Y): supnorm

iso-

morphisms

1 6 P(K)

c R(K)

I 6 P(K)IYcR(K)IYcA(K)IYcC(Y).

It

follows

hence

are

should

that

the three

function

n o t be c o n f u s e d

one v e r i f i e s

that

Let

the

us a d o p t

The

restriction

the

spectrum

P become

duced

An

convention of c o u r s e

c ca(Y),

immediate

(Wilken):

We

with

be

tion

preserves

son p a r t II~-~<2

dealt with the

the e q u i v a l e n t

is the

in the

next

identical

two

each

GI(P,~)

iff Y=K).

or R(K)

space

or A(K).

P'=(PIY)"

associated

we h a v e

and

with

(PIY) I =

reduction

re-

will

be

chapter.

subsequent

specialization

of

section.

of S e c t i o n

on the The

consists

= GI(PIY,~).

4.2 the

I.

compact

Gleason

part

restriction

decomposition

of the

on Y i t s e l f :

we have M(P,~)CProb(K) This

and

(which

of the p r e s e n t

remark

time

closed

III

results

Y with

theme

under

XcYcK.

which

considera-

of E ( P ) = Z ( P I Y ) : the G l a d of the ~ 6 E ( P ) = ~ ( P T Y )

But

after

III.4.4

with

we have

M(P,~) v = M(P,~) v and M(PIY,~) v =

characterizations

in v i e w

Banach

of p l c ca(K)

in v i e w

part

P(K)

of m e a s u r e s

c Prob(Y).

final

characterizations

= M ( P I Y , ~ ) v. T h e s e in fact

the

defined

either

dual

supnorm

a n d A(K) IY=A(Y)

( R I Y ) ± D M ( R I Y ) ^ = O ' for e a c h

Gleason

indeed

the

classes

Instead

an i m p o r t a n t

of ~ 6 Z ( P ) = E ( P I Y ) so t h a t

the

is e v i d e n t

4.6 T H E O R E M

will

P denotes

for ~ 6 Z ( P ) = Z ( P I Y )

application

of w h i c h

conclude

But

to o b t a i n

are

of C h a p t e r

R(K) IY=R(Y)

= M(P,~)NProb(Y)

in o r d e r

importance

but

that

smaller: and

sense

p(y)cR(Y)cA(y)c-C(y)

preserves

Z(P)=E(PIY).

to M ( P I Y , ~ )

decisive

with

algebras

on Y in the

P(K) IY=P(Y)

drastically

= P±Nca(Y)

restriction

algebras

above.

appear

to b e d i f f e r e n t

but

are

218

5. On the G l e a s o n

We retain either =E(R)

R(K)

Parts

the

fixed

or A(K).

Note

the G e l f a n d topology

which

after

III.4.5

each G l e a s o n

sets

part

for R(K)

R for

decomposition

of K=

the G l e a s o n

part

map K÷E(R) :u~%0u is continuous

therefore

coincides

each G l e a s o n

and hence

For u,v6K we have

part

= G I ( R , ~ u) denote

of E(R)

Thus

K%~ of ~ and the n o t a t i o n

the G l e a s o n

let GI(R,u)

union of c o m p a c t

REMARK:

subset

the i d e n t i f i c a t i o n

topology

of Kc~.

countable

5.1

that

compact

and A(K)

We consider

for R. As before

of u6K.

for R(K)

with

part of K for R is a

a Baire

set.

ll~u-~viIR(K).~Ii~u-~vlIA(K)..

is the union

in

the m e t r i c

Therefore

of one or several

Gleason

parts

for A(K).

5.2 REMARK: for R. Thus

Note

that there

can contain

Proof V(a,~)cG [1950]

Each

of 5.2: then

components

It follows

The next results

5.3 REMARK:

Proof:

f(a+6~)-f(a)i

C~:GI(R,a)

tion is transitive.

of K ° and a6G.

~ ~iu-al

< 2 Vu6V(a,@)

since

part

for R

p.145).

If 6>0 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 << ½ ( ~ a + m ) 6 M ( R , a ) .

exists

some o6M(R,z)

It follows

that

For each m6M(R,a)

then

QED.

5.4 PROPOSITION: Supp(m)c-~.

[1969]

QED.

o << 6a-m and hence w i t h

z6Gl(R,a).

show that one Gleason (see GAMELIN

part

~PcX.

that

ll~u-~a[ I ~ ~lu-a[

that

which of K °

l]fIl~1 the H . A . S c h w a r z

implies

=

in some G l e a s o n

for R satisfies

Let G be a component

for f6R w i t h

Vol. I p.137)

Thus we have

of K ° is c o n t a i n e d

part PcK

are examples

several

if(u)-f(a),

with

component

each G l e a s o n

Let a6K and P:=GI(R,a).

Furthermore

m6M(R(P),a).

219

Proof: Supp(m) cond

The assertions

assertion

(6a-m)C(u)= that

note

The

first

unit

Let

disk

if m=6 a so t h a t w e

that

From

that

Combine

4.4

this

5.3.

Thus

to the

that m(P)>O.

As

P:=GI(R,a)=D Thus

=S.

to the

se-

3.7

implies

assertion

an e x a m p l e

from

that take

II.4.6

while

m(P)=O.

first

assertion

There

exists

in 5.4

4.4.

and P : = G I ( R , a ) .

applied

to S : = M ( R I X , a ) V c c a ( X ) C c a ( K )

c Supp(~).

J<<m. the

after

support

a£K

the

( @ a - m ) A C ( u ) < ~ and h e n c e

be i m p r o v e d

fact

m%6 a. T h e n

To p r o v e

QED.

claimed

converse

assume

5.3.

an m 6 M ( R , a )

= ~NX.

Tr(S)AX

and after

cannot

elementary

Let

that

and

a £ D we h a v e

a weakened

on the

Supp(m)

Proof:

For with

2.9

u6CS(~a-m)cP

m6M(R(~),a).

in 5.4

K = DUS.

us i n s e r t

implies

e v e n be

is 6M(R,a)

depends

06S w i t h such

otherwise

assertion

5.5 P R O P O S I T I O N : such

clear

u6~-~

and hence

a n d it c a n n o t

m:=P(a,.)l

which

that

O since

6a-m6R(P)±

m(P)=1, the

are

c Supp(6a-m ) c CS(@a-m ) c p after

Now

final with

Tr(S)

5.6 P R O P O S I T I O N :

this

each

the

Gleason

= P per

4. It f o l l o w s

result.

part

we o b t a i n

some

for e a c h m 6 M ( R I X , a )

= Gl(R,a)

in S e c t i o n

to o b t a i n

For

is c S u p p ( m )

= GI(RiX,a)

remark

5.4

And

definitionem

that ~nXcSupp(m)cX.

QED.

PcK

for R the

closure

P is

connected.

Proof: closed S=~. Then

Assume

S,T with Choose

t h a t P is n o t

a6PNS,bEPNT

Supp(o),Supp(T)c~

characteristic

I = XS(a) which

Another

obtain

function

= SXsdO

consequence

the

GI(R,a)

subsequent

PQS,PAT

and m e a s u r e s

Then P = SUT with ~ ~ since

XS is ER([)

= d(S)

and

to o<
of has

5.3

theorem.

in v i e w

of

O = Xs(b)

3.8.

nonvoid

~ PeT ~ [cT

after

such

that

5.4.

Now

It f o l l o w s

= fXsdT

a<
that

= T(S),

QED.

is of p a r t i c u l a r

positive

PAS=~

o6M(R,a),T6M(R,b)

and d 6 M ( R ( [ ) , a ) , T 6 M ( R ( ~ ) , b )

is a c o n t r a d i c t i o n

is % { 6 a } t h e n

connected.

S A T = ~. We h a v e

Lebesgue

importance:

measure

after

If M(R,a) 2.9.

We thus

220

5.7 T H E O R E M :

Note

For

i)

Gl(R,a)

ii)

L(GI(R,a))

that

quences

iv)~iii)

5.3

as e x p l a i n e d from

Thus

we

see

above,

iv)

M(RIX,a)

trivial

In the

Recall

that

the

parts

i)~iv)

Choose

as o b v i o u s

iii)~ii)

b 6 GI(R,a)

and ~ 6 M ( R I X , a )

P={a}

series

Each

parts

PcK

parts

which

which

are P={a}

a countable

of the

the

section

associated

with

86R ± a d m i t s

II.4.4

the

[

we

conse-

follows

= GI(RIX,a)

such

that

from differ-

T<
for R can o n l y have

a6X

L(P)>O,

Then

of

be of two a n d the so-

and M ( R , a ) = { 6 a } .

of n o n t r i v i a l

consider

is the

band

a6X we h a v e

expansion

with

number

for R. T h e b a s i s

e :

the

final

And

parts.

collection

part

F(R)

of S e c t i o n

b ( P ) = ~ 6 a. T h e r e f o r e

Thus

and in v i e w

86R ± simplifies

[

with

~epI I

~,

<

P6F(R) nontrivial

8 p : = < b ( P ) > e 6 R ±.

5.8 C O N S E Q U E N C E : R=C(K).

M(R,a)={6a}

ii)

The

All

subsequent

Gleason

parts

assertionS

P6F(R)

are

are e q u i v a l e n t . trivial,

iii)

K°=@

and

Va6X=K.

5.9 T H E O R E M :

There

nontrivial

exists

Gleason

i)

Pc_E (P) c~,

ii)

E(P)~E(Q)

iii)

if a6P

part

a correspondence P6F(R)

= ~ whenever

then m(E(P))=1

a Baire

which

associates

set E(P)c_K

P#Q, for all m 6 M ( R , a ) ,

such

of

as fol.

representation

8p

of

II.4.

b(P) :=M(R,u) v f o r any u£P.

of the m e a s u r e s

P6F(R) nontrivial

each

# {6a}.

and

nontrivial

parts

for P£F(R)

for t r i v i a l

i)

# {~a }.

fulfilled

the G l e a s o n

at m o s t

remainder

Gleason

where

are

are o b v i o u s

T6M(RIX,b)

so-called

can be

lows:

M(R,a)

i)ii)

a n d ii)~i)

a and t h e n

The

called

4.6

> O.

iii)

are e q u i v a l e n t .

~%6 a. QED.

types:

all

# {a}.

properties

5.2.

ent

there

subsequent

for a£K O p r o p e r t i e s

of

Proof:

course

a 6 X the

with

that

221

iv)

for each 86R l we have

I°

(gp) c (x) =

Proof: parts.

@p = XE(p ) e and

V x6K w i t h 0AC(x)< ~.

@C (x)

if x6P

We can assume that there are at least two n o n t r i v i a l Gleason

I) For each o r d e r e d pair of different nontrivial P,Q6F(R) Q

we

P

form disjoint Baire sets Ep,EQ c K such that

o(E~)

= I

V u6P and a6M(R,u),

T(E~)

= I

V v6Q and T6M(R,v).

To achieve this choose a6P, b6Q and apply III.2.1 cPos(K)

and T(V)=I VT6M(R,b).

Thus we can take U=:E~ and V=:E~.

pcE~ since u6P ~ ~u6M(R,u) P6F(R)

to M ( R , a ) , M ( R , b ) c

to obtain disjoint Baire sets U,VcK such that d ( U ) = 1 V d 6 M ( R , a )

define E(P)

Q#P in F(R)

= 6u(E~)=1

or u6E~.

to be the i n t e r s e c t i o n of the E~ for all n o n t r i v i a l

(the number of w h i c h is at most countable),

P. Then the E(P)cK are Baire sets which For nontrivial P6F(R) some o6M(R,u).

Thus

then @p 6 b(P)

10pl (K-E(P))=O.

6 shows that 0(B)=0p(B)

exists some d6M(R,x)

w i t h o<<%p6b(p)

4) Let 86R ±.

= M(R,u) ~ for any u6P so that 8p << Then the above series expansion of

V Baire sets BcE(P).

(6p)AC(x)< ~ as well.

i n t e r s e c t e d with

fulfill i)ii)iii).

It follows that @p = XE(p)8

as claimed in iv). To prove the last assertion in iv) eAC(x)< ~. T h e n

2) We have

3) For each nontrivial

If

fix x6K w i t h

(@p)C(x)~O then after 4.1 there

so that x6P. Thus

(ep)C(x)=O if

x~P. Then the above series expansion of 0 shows that eC(x)=(ep)C(x)

for

x£P. QED.

6. The L o g a r i t h m i c T r a n s f o r m a t i o n of Measures Capacity

and the L o g a r i t h m i c

of Planar Sets

The p r e s e n t section is to prepare the proofs of the W a l s h - t y p e

theo-

rems of Section 7. The first half is d e v o t e d to the logarithmic transformation of Baire measures

in the complex plane.

Its properties

corre-

spond to those of the Cauchy t r a n s f o r m a t i o n except that its d e f i n i t i o n has to be r e s t r i c t e d to measures of compact support. ient to define

It is thus conven-

222 ca,(E)

Of c o u r s e

ca,(E)

is d e v o t e d planar The

= {@6ca(~) : Supp(e)

= ca(E)

to a s h o r t

sets

defined

In the case tended

discussion

transformation

The s e c o n d h a l f of the s e c t i o n

of the l o g a r i t h m i c

is d e f i n e d

for t h o s e

sense

z6~ w h e r e

to t r a n s f o r m

we d e f i n e

-~<eL(z)<~.

eAL(z):=

eL(z)

the m e a s u r e

For the

Vz6¢

in the ex-

first h a l f of the s e c t i o n we fix a

eEca,(~) .

F o r each b o u n d e d

.l'lioglu-zlldL(u)

<__ , / ~ ( E )

Baire

set Ec~ we h a v e

+ L ( E ) { y 1 + Sup I u - z I )

Vz6¢.

uEE

Proof:

Take

an R>O w i t h E c V ( z , R ) .

I := ~lloglu-zl IdL(u) E =

~

=

L({u6E:loglu-zl~t})dt

first

from 2.1 we have

+

S

integral

L({u6E:loglu-zI~-t})dt

+ fI L ( { u C E : l u - z [ ~ s } ) ~ - -ds 0

is ~L(E) l o g M a x ( I , R ) ~ L ( E ) R .

The s e c o n d i n t e g r a l

can

be e s t i m a t e d

c 2 d__ss + SL(E) I sa£ s 1 =< S~ s s = ~~ C 2 + L ( E ) I o g O c In the c a s e L ( E ) ~ z

put c=I to o b t a i n < 3 ~ ½L(E).

(~L(E)) I/2 to o b t a i n ~ ½L(E)

6.2 P R O P O S I T I O N : N:=

l~t})dt

]o,~[

= f L ( { u ~ E : E u _ z E > s=) d S 7 I

for e a c h 0
Then

~ L({u6E:llog[u-zl ]0,~[

]0,~[

ii)

of

~lloglu-zl Idlel (u)<~.

= ~log[u-z[de(u)

E

put c =

potential

in the n e x t section.

= floglu-zlde(u),

eEPos,(~)

6.1 LEMMA:

The

for n o n v o i d Ec¢.

into the f u n c t i o n

eL:eL(z)

measure

if E is compact.

ab-ovo

to the e x t e n t w h i c h w i l l be n e e d e d

logarithmic

eEca,(¢)

is compact- and CE}

i) The

function

+ /~L(E).

eAL:~+[O,~]

{z6~:eAL(z) =~} has L(N)=O.

.

In the case O < L ( E ) ~ z

QED.

is B a i r e m e a s u r a b l e .

223

iii) The function iv)

8L:~-N÷~ is Baire measurable.

For each bounded Baire set Ec~ we have

II@L(z) IdL(z) < ~oAL(z)dL(z) E E

In particular

<

(~

i

+ L(E)(~ + Suplu-zl) uEE z£Supp (O )

)~6~.

0L £ Lloc(¢,L).

Proof: The assertions above 6.1. QED.

follow from the Fubini

theorem and from the

6.3 PROPOSITION: eL(z) is defined Vz6~-Supp(0). In other words: If Uc~ is open with I01 (U)=O then 6L(z) is defined Vz6U. Moreover @LIu is a harmonic

function.

6.4 PROPOSITION: Moreover eL(z)

0L(z) is defined Vz6~ with

@L(z)-@(~)loglzl÷O

Izl sufficiently

for z÷~. And if @=fL with f6L~(~,L)

is defined Vz6~ and 0L:~÷~ is uniformly

large. then

continuous.

Proof: The first assertion is obvious. I) Fix R>O with Supp(O)cV(O,R) and take z6~ with IzI>R. The calculus inequality X~! ~ log x ~ x - 1 V x > O implies that

llogtl

- ~ll ~T?~R

= lYXogll- ~lde(u) l =-Fa'T~_~IIoII. < R

leL(z)-e(¢)logizll

This proves the second assertion. and put E:=Supp(f).

VuESupp (0 ) ,

2) Suppose now that e=fL with f6L~(@,L)

After 6.1 then

oAL(z) = llloglu-zlllf(u)

ldL(u)

cflloglu-zlld~(u)<~ E

3) Next we show that 0L is continuous. @L(z)-0L(a)

Fix a6@ and take z#a. Then

= f(loglu-zi-loglu-al) f(u)dL(u), E

leL(z)-0L(a) I < c

f

(Iloglu-zl I+lloglu-al I)dL(u)

V(a,21z-a I)

+ e

f

z-a

llogll- u_--qlldL(u)-

E-V(a,21z-a I)

vzc~.

224

After

6.1 the

first integral

4~Iz-a I

is <

t e n d s + O for z÷a. A n d the s e c o n d

integral

+ 4zlz-a[ 2 + 2Ozlz-al 3 w h i c h is

O = ~Fz(U) dL(u)

Observe

with

Fz(U)

that O ~ F z ( U ) ~ 1

the s e c o n d i n t e g r a l

and F z ( U ) ÷ O

to s h o w t h a t

continuous

on c o m p a c t

@L is u n i f o r m l y

Vr>O.

In o r d e r to r e - o b t a i n

Cauchy

formula

F o r f6C~(~)

3) and z~loglzl the a s s e r t i o n

formula VI.5.9

6 •6 P R O P O S I T I O N :

which

E:=Supp(f)

We have

transform

to the

Vz6%.

as e a r l i e r

with

to @Eca.(~)

:= the of

is

= S@LAfdL

Vf6C~(~) .

Vf6C~(~).

can be applied.

But

This

is c l e a r once

for S : = S u p p ( @ )

that

a fortified

L

sense.

derivative

theorem

[8]

and w i t h d i f f e r e n t i a t i o n

I ! ell8l{(z/~(E) + L ( E ) ( ~ + S u p l u - z I) < ~. u£E z6S F r o m 6.6 we o b t a i n

@L we i n v o k e

corresponds

[@] = I__A 2~ (@L),

= <sL,Af>

we see f r o m 6.1

con-

f = ~((Af)L)

corresponds

that the F u b i n i

4)

f r o m I). QED.

this m e a n s

Thus we have to s h o w that ffd8 = ~ f s L A f d L we k n o w

is u n i f o r m l y

transformation

The d i s t r i b u t i o n a l

Hence

follows.

2.6.

8L 6 L l o c ( ~ , L ) in the d i s t r i b u t i o n a l Proof:

u*a.

But @L is u n i f o r m l y

follows

which

"

we h a v e

of the l o g a r i t h m i c

distribution

lu-al>2]z-a{

continuous.

I f(z) = ~-~ S ( I o g l u - z l ) A f ( u ) d L ( u ) In t e r m s

for

The a s s e r t i o n

@ f r o m its l o g a r i t h m i c

representation

transformation

6.5 LEMMA:

Hence

lu-al~21z-al]

for z-~a for all c o m p l e x

sets c ~ a f t e r

on { z 6 ~ : I z I ~ r }

the f u n d a m e n t a l

~aall

llogI1-

tends +O for z+a as well.

It r e m a i n s

tinuous

=

for

converse

QED.

to 6.3 as in S e c t i o n

2.

and

225

6.7 PROPOSITION:

If UC~ is open such that eLIu 6 Lloc(U,L ) is har-

monic

(that means Lebesgue-almost

tion)

then

everywhere

191 (U) = O. In particular

6.8 COROLLARY:

equal to a harmonic

eLIu = 0 implies that

func-

161 (U) = O.

Define LS(e) := {z6~:eAL(z)< ~ and 6L(z)#O} the loga-

rithmic support of e. Then LS(8)c~ is a Baire set with L(LS(e))>O less @=0. Furthermore

un-

Supp(e)cLS(e).

The remainder of the section is devoted to the logarithmic

capacity.

It is defined to be cap(E)

= exp( Sup \e6Prob,(E)

Inf 0L(z)~ / z6~

V nonvoid Ec~.

Thus O~cap(E)~ ~, and AcB implies that cap(A)~cap(B). there is an important equivalent low is an impressive 6.9 THEOREM:

definition.

For compact sets

The equivalence proof be-

application of our Hahn-Banach

version A.2.3.

For nonvoid compact Ec~ we have

cap(E) = Inf{llP[IE/n: P polynomial with P(z)=zn+... where

(n=1,2 .... )},

II.~E denotes the supnorm on E.

Proof: Write I for the Inf in question.

I) For e£Prob(E)

and P:

P(z) = zn+... = (Z-Ul)...(z-u n) we have z6~Inf 8L(z) ~ ~I k=1 ~ 0L(uk) = ~I(k=1 ~ loglz-ukl)d0(z) = iloglP(z)

tl/nde(z)

~ l o g l I P I I E1/n .

Hence log cap(E) ~ log I. 2) The set T:=~! 1oglPl: P polynomial with P(z)=zn+... (n=I,2,...) c USC(E) fulfills u,v6T ~(u+v) 6T. Hence after A.2.3 there exists a measure o6Prob(E) log I = Inf logllPIl /n P =

Inf

f6T

~fdq

=

Inf

with

Inf Max ~ loglP(z) I = Inf Max f(z) P z6E f6T z6E I SloglP(z ) ido(z)

P

< Inf ~loglz-uldq(z) = u~¢

= Inf oL(u) ~ log cap(E). u~¢

QED.

226 6.10 COROLLARY: For nonvoid compact Ec~ we have cap(E) = cap(~E). 6.11 EXAMPLE:

cap(V(a,R))

= cap(~?(a,R))

= R for all a6~ and R>O.

Proof: Write E:=V(a,R).

I) For P:P(z)=z-a we have cap(E)~IIP~E = R. R. zn is a polynomial with Q(O)=R n. then Q:Q(z)=P(a+ ~)

2) If P:P(z)=zn+... It follows that

11P lIE =

Izl~ ISup IP(a+Rz) I = Izl =ISup ]P(a+Rz) I = Izl =ISup IP(a+ ~)R znl Sup

=

[Q(z) I =

I~1=1

Sup

IQ(z) I > Q(O) = R n.

I~1~1

=

Hence cap(E)~R. QED. 6.12 LEMMA: Assume that ~:~+~ satisfies for some c>O. Then cap(~(E))~ccap(E)

l~(u)-~(v) ]~clu-vl Vu,v6¢

V nonvoid Ec~.

Proof: For P:P(z)=zn+...=(Z-Ul)... (Z-Un) put Q:Q(z)=(z-~(Ul))... (z-~(Un)). Then IQ(%0(z)) 11/n =

n l~°(z)-~°(Uk) [ I/n =< c

I/n and hence ,,,,I[QII~(E)

n

iZ_Uk[

I/n = alp(z) ll/n vz6~,

]lq

~ ,PII=/n.~ The assertion follows. QED.

The above lemma will lead us to an important estimation for the logarithmic capacity. 6.13 LEMMA: There exists an E>O such that cap(E)~E£(E)

for all non-

void compact EC~, where i denotes one-dimensional Lebesgue measure on ~. Proof:

I) Let E=[a,b] with a
t6~ and ~(z)=~(Re z) for z6¢. Then ~(E) is the unit circle S. And 2~ I~(u)-~(v) I = l e x p ( 2 ~ i ~ ) - l [ ~ ~-L-~[u-v[ Vu,v6~ and hence Vu,v6~. Thus 2T 6.11 and 6.12 show that 1=cap(~(E)) ~ ~-i--~cap(E) or cap(E)~sZ(E) with g=l/2w.

2) Fix a compact set Ec~ with Z(E)>O. Define ~:~÷~ by ~(t)=

= Z(ED]-~,t])

for t6~ and ~(z)=~(Re z) for z6~. Then ~ is monotone in-

creasing on ~ and fulfills O ~ ~(v)-~(u) u
= i(En]u,v]) ~ v-u Vu,v6~ with

l~(u)-~(v) [~lu-v [ Vu,v6~.

In particular ~ is con-

tinuous. Therefore ~(R) is an interval c~ and hence =[O,i(E) ]. Now ~ is constant on each interval c~ the interior of which does not meet E. Thus

227

~0(E)=~(~)=[O,Z(E) ]. Therefore

from 6.12 and I) we o b t a i n cap(E)_>_

> c a p ( ~ ( E ) ) = c a p ( [ O , ~ C E ) ])>eZCE). QED.

6.14 THEOREM:

There exists an ~>O such that cap(E) ~ si({Iz-aI:z£E})

for all a6~ and all n o n v o i d Ec~ w h i c h are countable unions of compact sets.

Proof: For fixed a6~ define ~:~÷~ by ~ ( z ) = I z - a I Vz6~. that

l~(u)-~(v) l~lu-v I vu,v6~.

It is clear

From 6.12 and 6.13 thus c a p ( E ) ~ c a p ( ~ ( E ) ) =

= cap({Iz-al :z6E}) ~ e£({Iz-al:z6E}) The extension to countable unions

for all n o n v o i d compact sets EC~.

is then immediate.

QED.

7. The W a l s h T h e o r e m

The W a l s h t h e o r e m asserts

that the D i r i c h l e t p r o b l e m is solvable

for

certain compact p l a n a r sets. M o r e o v e r it furnishes the concrete basis for the action of the abstract theory of Section

1.

We retain the fixed compact subset K¢@ of ~ and introduce D(K)

= {f6ReC(X):BF6ReC(K)

with FIX=f and FIK O harmonic}.

For f£D(K)

the function F6ReC(K)

in q u e s t i o n is unique and satisfies

Max F = M a x f and Min F = Min f. It follows K

X

K

closed linear subspace C R e C ( X ) . =ReC(X),

that D(K)

is a s u p n o r m

X

We define K to be Dirichlet iff D(K)=

that is iff the D i r i c h l e t p r o b l e m is solvable

7.1 REMARK:

For each a6K there exists some q6Prob(X)

for K° such that F(a)=

=Sfdn Vf6D(K).

If K is Dirichlet then q is unique =:qa called the har-

monic measure

for a6K. Of course qa=6a if a6X. This is immediate after

A.2.5. For the points u6~ we define L u : L u ( Z ) = l o g l z - u I Vz£K.

Thus LulX6D(K)

with h a r m o n i c extension L u. We define K to be Walsh iff the real-linear span of {LulX:u6~}

is supnorm dense in ReC(X).

let. N o w we can formulate the W a l s h theorem.

Therefore Walsh ~ Dirich-

228

7.2 T H E O R E M :

Assume

that

log c a p ( ~ N V ( a , T ) ) < ~

l i m inf log

Va6X.

T

T h e n K is W a l s h .

The on the nal

proof

result

sion

to be p r e s e n t e d

logarithmic 6.14

serves

of the W a l s h

7.3 T H E O R E M :

capacity

below

to d e d u c e

theorem

Assume

which

log

ponent

COROLLARY: of ~

Proof then small

7.2 the m u c h

more

above Rather

results the

fi-

tractable

ver-

follows.

< ~

Va6X.

log T

the

! i m inf in

assumptions

7.3

is in the b o u n d a r y

is true w h e n

is

~ has

of s o m e

a finite

com-

number

of

c~ with

of the left

true

for the s o - c a l l e d Iz-al<1}

]0,1[

are to e x p r e s s

implies

Lebesgue is to the

If aEX log

component

endpoint

G of

=0.

:z6~DV(a,T)}

For

so

QED.

too t h i n n e a r

7.5 P R O P O S I T I O N :

in the b o u n d a r y

intervall

is =I.

in 7 . 2 - 7 . 4

with

the b o u n d a r y condition:

the r e q u i r e m e n t point

the o p e n

fulfill

S d x = ~. We S x a s s u m p t i o n in 7.3.

satisfies

a6X.

the L e b e s g u e

The

that same

the is

set S = {Iz-aI:z6~

claim

that

condition

the L e b e s -

then

Z({Iz-a[:z6~DV(a,T)} )

lim i n f T+O if e a c h

a£X

]O,T[={Iz-aI:z6GDV(a,~)}={Iz-al

set

condition

If a6X

is an o p e n

therefore

c

each

in p a r t i c u l a r

7.3 = 7.4:

~ be n o t

that

K is W a l s h .

{[z-al:z6G} T>O

The

of

Then

open

Thus

the

Z({!z-al:z6~nV(a,T)})

Assume

(which

components).

gue

require

definition.

K is W a l s h .

7.4

that

from

not its

that

lim inf T+O Then

does

but merely

a£X satisfies

< I. log the L e b e s g u e

condition

t h e n K is W a l s h .

229

Proof:

Lemma 2.1 shows that

S dx = ~ Z({x6s:l>t})dt S x = ]o,~[

=

~ i({x6S:x<~})dt

+

]o,1[

I

< z(S) +

f~({xCs:x
=

0

~ 9~({x6S:x<1})dt [I,~[

<

=

=

I

Z(S) + S£({Iz-aI:z£~Nv(a,t)}) ~

t 2

=

0

t2

"

Assume now that the lim inf in 7.3 be >c>I. Then i({Iz-al :z6~nV(a,T)})
of the section

We first reformulate conclude

small O
so that S --~ dx < ~ from the above. S

is devoted

the Walsh property:

to the proof of theorem

From the bipolar

that K is Walsh iff {LulX:u6~}±

QED.

7.2.

theorem we

= {@6ca(X) :eLl~=O}

is =O. Thus

6.7 tells us that K is Walsh iff (*) each e6ca(X) with

with

@AL(u)<~

The decisive

@L(u)=O Vu6~ satisfies

QL(u)=O Vu6X

and Vu6K °.

step in the proof of

(*) is anticipated

in the subsequent

lemma. 7.6 LEMMA:

Let Gc~ be open and a6~G with log cap(GDV(a,T)) lim inf Y+O

Assume

<

co

log

that e£ca.(~-G)

satisfies

0AL(a)< ~ and eL(z)

÷ some ~6~ for z÷a

on G. Then l=0L(a). Proof:

There exist some e~1 and a sequence

Tn+O and log cap(GNV(a,Tn))>~ logarithmic

capacity

log T n. Hence

there exist

of numbers O
8n£Prob.(GnV(a,Tn))

with e~(z)>~

Vz6~. We put S(n) :=Supp(0n)CGnV(a,T n) and S:=Supp(e)c~-G such that

lu-al~c Vu6S.

~ log(Tn+C)

~ c

log Yn

and choose c>O

V z6S(n)

theorem can be applied to obtain

f eL
of the

Then

loglz-u I ~ log(Iz-al+lu-al) SO that the Fubini

with

after the definition

fs0~Cu>de~u).

and u6S,

230

Here the first member tends ÷I for n÷~. Thus we have to show that the second member tends ~@L(a).

In view of

Lebesgue dominated convergence

IGI ({a})=0 this follows from the

theorem once we know that

i)

8L(u) ÷ loglu-a I

for n÷~

ii)

8L{u) >=-~lloglu-al-log

V U6~ with u%a, V u6~ with u%a,

21

iii) @L(u) < c

V u6S.

Here iii) is clear from the above. Consider now u6~ with uCa. Then 8L(u)-loglu-al

:

z-a f log ~u-z de n(Z) : ~ logI1- ~/~IdSn(Z). S (n) S (n)

In the case Tn
~z-a I

z-a ) __
T ~

n

)

,

z-a ( z-a ) ( Tn ) logll- ~_al __> log I-I~-:-~I > log I- lu_a i , and hence log< I- ~ /Tn ~>

=< 8 L (U)-log' u-a I =< log< I+ ~ -Tn /~) .

This proves i). Furthermore

I in the case T n < ~lu-al

8L(u) > loglu-al+log

I-

we have

> loglu-al-log

while in the case ~n => IIu-a [ after the definition @ (u) > e log T n => ~(loglu-al-log

2,

of e n we have

2).

This proves ii). QED. Proof of 7.2: Assume that each point a6X satisfies lim inf log cap(~NV(a,T)) • +O log T

< ~.

We fix 86ca(X) with @L(u)=O Vu6~. After 7.6 then 8L(u)=O vu6X with @AL(u)<~. Thus in view of the reformulation

(~) we have to show that

8L vanishes on K °. Let us fix b6K °. I) From 7.1 we obtain a measure o6Prob(X) with Lu(b)

= loglb-u I = floglz-uIdo(z) X

= ~L(u)

Vu6~.

231

We w r i t e

this

equation

in the

flog-lu-zlao(z) X Thus

for

fixed

a6X

= flog*lu-zld~(~) X

the F a t o u

flog-la-zld0(z) X

form

lemma

applied

2) T h e

above

Ida(z)

1) c o m b i n e d

7.6

vuen.

- log d i s t ( b , X )

shows

that

- logla-b I

~ ~ + log+diam(X)

with

loglu-bl

to u + a on ~ i m p l i e s

~ flog+la-zldo(z) X log+diam(X)

fllogla-zl X

-

=:

that

=:

e,

8<~.

loglu-bl=oL(u)

Vu6X.

Furthermore

f(flloglu-zlldl01(u>>do(z) X X For

o-almost

Fubini

all

theorem

8L(b)

z6X thus

= f(flloglu-zlldo(z))dl011u) X X

0 A L ( z ) < ~ and h e n c e

can be a p p l i e d

= f(floglu-zld0(u))do(z) X X

As b e f o r e theorem

to the P r o b l e m

we

with

fix a n o n v o i d

the

abstract

REMARK:

i)

If u , v 6 the

same

ii)

If u6~ ~ t h e n

Lu6ReR(K).

Proof: u6V(a,~)

o) then

o) F o r e a c h

= fen(z)do(z) X

=

And

the

compact

subset

of S e c t i o n

u6~ w e h a v e

component

Lu

Log<1 ~u a)=

= O.

)

QED.

Approximation

Kc~.

We

combine

the W a l s h

I.

:= l o g l Z - u 1 6 l o g i R ( K ) × l .

G of ~ t h e n L u - L v 6 ReR(K).

is o b v i o u s , i) F i x a 6 G a n d 6 > 0 w i t h u-a I~-C~I < lu-a[ < I v z 6 K a n d h e n c e

Log z - a

7.6.

= f(floglu-zld0(z))de(u X X

of R a t i o n a l

theory

8.1

after

to o b t a i n

= floglu-bldS(u ) = foL(u)d@(u) X X

8. A p p l i c a t i o n

0L(z)=O

~,

<

=

k \ ~ - a/

V(a,~)cG.

Vz6K,

For each

232

w i t h L o g the m a i n b r a n c h {s£~:Re

s > O}, w h e r e

Z-u Log ~ / ~ 6 R(K)

of the l o g a r i t h m

the s e r i e s

convergent

ii)

L o g Ithe s e r i e s

If a6~ w i t h

=

is u n i f o r m l y

=

A:=

on K. Thus

Z-u Z-u and h e n c e L u - L a = loglz_-L-~l = R e L o g ~ / ~ 6 ReR(K).

t a i n i) via c o n n e c t e d n e s s ,

where

in the o p e n h a l f p l a n e

is u n i f o r m l y

We ob-

lal>Max{Izl :z6K} t h e n

k\a/

¥z6K, Z 6R(K) on K. T h u s L o g ( l - ~)

convergent

h e n c e L a = l o g I Z - a I = l o g l a [ + l o g 11- Z I E R e R < K ) .

So w e o b t a i n

ii)

and

from

i). QED.

8.2 P R O P O S I T I O N :

If K is W a l s h

ML(R(K) IX,a) with qa6Prob(X)

Proof:

= ML(A(K)IX,a)

the h a r m o n i c

We k n o w

then

measure

from Section

= {qa}

Va6K,

for a6K.

I that #%ML(A(K) IX,a)

c ML(R(K) IX,a).

H e n c e we h a v e to s h o w that each m6ML(R(K) IX,a) m u s t be =~ a . But for u6~ w e have LulX61og[ (R(K) [x) Xl and h e n c e Lu(a) = fLudm. Thus F(a) = X f f d m V f 6 r e a l - l i n e a r s p a n { L u l X : u 6 ~ } and h e n c e V f6ReC(X), w i t h F6ReC(K) X the h a r m o n i c e x t e n s i o n of f. It f o l l o w s that m=q a. QED. W e turn to the m a i n holes,

that m e a n s

theorems.

~=~,

which

This case is p a r t i c u l a r l y ii)

implies

R(K) 1x c C(X)

has the D i r i c h l e t

8.2 s h o w s

Wilken

theorem

the M e r g e l y a n

t h a t M(R(K) IX,a)

the case t h a t K has no

3.2 is e q u i v a l e n t

transparent:

t h a t ReR(K) IX = ReC(X),

Hence

After

that is the r e s t r i c t i o n

property

i) If BCC(K)

is a c o m p l e x

Vf£B t h e n B=R(K).

We p u t this

subalgebra

P(K) IX = R(K) IX c C(X)

MCPCK)

theorem

that K has no holes.

In p a r t i c u l a r

Ix,a)

T h e n 8.1. algebra

in the s e n s e of S e c t i o n

= {H a} Va£K.

approximation

Assume

to P(K)=R(K) °

7.4 K is Walsh.

4.6 i n t o the a b s t r a c t M e r g e l y a n polynomial

8.3 T H E O R E M :

ii)

We s t a r t w i t h

after

theorem

1.6 to o b t a i n

in the s u b s e q u e n t

form.

Then

with P(K)=R(K)CB

and ~f[l=~flXll

P(K)=R(K)=A(K).

has the D i r i c h l e t

= MCR(K)IX,a)

III.1.

fact a n d the

property

= {Qa }

Va6K.

and

233

Proof:

For

BcC(K)

as in i) w e w a n t

R(K) IX c B ! X c C(X)

c B(X,Baire).

a 6 K = [(R(K) IX) w e h a v e ~:flX ~ From

f(a)

Vf6B

1.6 w e

Theorem a fast

8.3

road

of h o l e s ) .

is i n d e e d

connected which

In the that

abstract

rem IX.3.5

with

curve

Va£K.

t h a t K has

Mergelyan

so t h a t

B = R(K).

QED.

3.10

forms

theorem.

an e > O w i t h

Va6K

with

the p o i n t s

lu-aI~6

diam(G)~

K has

for all

a finite

some O < 6 < ~ . z6~ w i t h

then

u{K

there

after

f6A(K) 8.3.

We

number

some

then

that all

in

compo-

v£G with

Iv-al>6

connects

so t h a t

flK(a)

f6R(K)

are

u6some

which

of ~-K(a)

Hence

claim

Iz-aI>6

and hence

exists

component

2) If n o w

a finite But

the

theorem

Assume

that

i)

is a c o m p l e x

In p a r t i c u l a r

number

of h o l e s

subsequent

final

1.9 and h e n c e

the expenditure

8.5 T H E O R E M :

then B = R ( K ) .

functional

£ A(K) [K(a)

in v i e w

u

~-K(a) c

of the

QED.

R(K)=A(K).

If BcC(K)

for e a c h

~6Z(BIX).

theorem

on G c ~ = ~ - K c ~-K(a)

is = R(K(a)) 3.10.

case

the m e r e

that

approximation

is true w h e n

u is in the u n b o u n d e d

theorem

linear

an e x t e n s i o n

localization

exists

of d i a m ( G ) ~ e > 2 6

a continuous

c A(K(a))

the

In fact,

if u6~-K(a)

a n d v. T h u s

Bishop

there

that

R(K)=A(K).

G of ~. In v i e w

and then

the B i s h o p

I) P u t K(a) :=KnV(a,6)

And

multiplicative

and hence

rational

that

1.6 to the r e s t r i c t i o n s to r e m a r k

= R(K) IX a n d h e n c e

in p a r t i c u l a r

is c o n n e c t e d .

~-K(a). nent

Assume

(which

Then

Proof: ~-K(a)

with

to the M e r g e l y a n

G of K

II~=I

that BIX

combined

8.4 T H E O R E M : holes

a well-defined

on B I X w i t h

conclude

to a p p l y

It s u f f i c e s

K has

is m u c h

n holes

subalgebra

upon

we

can p r o v e

theorem

rests

the m a x i m a l i t y

more upon theo-

higher.

(n=O,I,2,...).

with

R(K) cB a n d

Then

U fll=IflXll V f 6 B

A(K)=R(K). ReC(X)

ii)

d i m N(R(K) IX,a)

~ dim

~ n

Va6K.

ReR(K)! X iii)

M L ( R ( K ) IX,a ) = {qa}

Va6K.

And ~a

is an i n t e r i o r

point

of

M(R(K) IX,a).

Proof: of K. T h e n

ii) 8.1

Choose

points

implies

that

a(1),...,a(n)

from

the h o l e s

G(1),...,G(n)

234

{Lu:U6~ } c ReR(K)

+ real-linear

span{La(1) .... ,La(n)}.

Since K is W a l s h after 7.4 we obtain

ReC(X)

= ReR(K) IX + real-linear

span{La(1) IX ..... La(n) IX},

and hence the second inequality The first inequality

(the case n=O is comprised as well). ± is then clear from N(R(K) IX,a) c ReR(K) IX com-

bined with the usual norm isomorphism ReR(K) IX follows

from 8.2 and 1.8. i) results

( eC/Xl)"

~

. iii)

then

from 1.9 applied to the restric-

tions R(K) IX c BIX c C(X) c B(X,Baire)

as we deduced above 8.3.i)

from

1.6. QED.

Notes

The c o n s t r u c t i v e

theory of p o l y n o m i a l

and rational a p p r o x i m a t i o n up

to the decisive work of M e r g e l y a n and V i t u s h k i n

is p r e s e n t e d in

ZALCMAN

See also V I T U S H K I N

[1968] and GAMELIN

[1969] Chapter VIII.

[1975]. The action of the f u n c t i o n a l - a n a l y t i c

theory of function alge-

bras is a main theme in each of the treatises of B R O W D E R GAMELIN

[1969], LEIBOWITZ

development

[1970] and STOUT

see G L I C K S B E R G

[1972] and the literature

access to the M e r g e l y a n p o l y n o m i a l

[1969],

[1971]. For the subsequent

approximation

cited therein.

stract m e t h o d s is due to BISHOP

[1960] and G L I C K S B E R G - W E R M E R

See the b e a u t i f u l p r e s e n t a t i o n s

in W E R M E R

is due to A H E R N - S A R A S O N

[1967a], G L I C K S B E R G

The f u n c t i o n a l - a n a l y t i c

[1964].

theorem 8.4-8.5 the access [1968]

and GARNETT

[1968].

theory led to an abstract version of the

polynomial

approximation

continuous

situation under the overall

Theorem

[1963].

[1964] and CARLESON

As to the M e r g e l y a n rational a p p r o x i m a t i o n

The

t h e o r e m 8.3 via ab-

t h e o r e m 8.3: it is theorem 1.6 in the compactassumption that A be Dirichlet.

1.6 as it stands and its close relative

1.5 depend on more

elaborate versions of the abstract F.and M.Riesz t h e o r e m and of the a b s t r a c t H a r d y a l g e b r a theory. is in G A R N E T T - G L I C K S B E R G are in K O N I G - S E E V E R

In the c o m p a c t - c o n t i n u o u s

[1967] after G L I C K S B E R G

situation

[1967] w h i l e

1.5

1.4-1.5

[1969]. A similar abstract v e r s i o n of the rational

235

approximation

theorem

tuation

the o v e r a l l

under

due to G A M E L I N - L U M E R essence

of t h e o r e m

The e s s e n c e present since

[1958].

mental

further

LIN

son idea

[1907]

Sections

versions

capacity

3.10 we refer 4.1 has the

4.3-4.5

are perhaps

S e c t i o n VI.3.

for R(K)

when K has

[1967b]

and 5.5 new.

The

Fundamental

a finite

and r e p r e s e n t e d

Section

to

funda-

and 5 . 3 - 5 . 4 , 5 . 6 - 5 . 7

the remarks which

[1959][1960].

on 3.5 is due

[1969]

num-

in GAME-

26.

[1928][1929]

which

is also attri-

proof

in C A R L E S O N

from K O N I G

to o b t a i n

of the W a l s h

use in the

a new ab-ovo

6-7 are

developed

[1968a]

added

[1971]

found

theorem

to

to

transformation

based

15. L e m m a

in q u e s t i o n

theorem

has

ourselves

is due to B I S H O P

to W I L K E N

parts

and STOUT

approximation

for w h i c h we refer

Z(A(K))=K

Section

We have

on the G l e a s o n

is further

III.

The

of the d e f i n i t i o n

of the Cauchy

localization

are due to A H E R N - S A R A S O N

of the l o g a r i t h m i c Chapter

IV.7.4.

is adequate

and its s y s t e m a t i c

theory

5.9 is from G A M E L I N

The p r e s e n t

tain e x t e n d e d

due

of the m e a s u r e s

to L E B E S G U E

[1964].

4.2,4.6

C h a p t e r VI

The W A L S H

plane

[1968]

[1967][1968b].

results

[1969]

buted

F o r the Bishop

theorem

b e r of holes

variants

material

The i n t r o d u c t i o n

and ZALCMAN

on the supports selection

standard

3.6 of the s p e c t r u m

consequences

due to W I L K E N

Theorem which

In these Notes we r e s t r i c t

approximation

identification

[1968]

[1969]

obvious

IV. The

[1969b].

comprehensive

certain

in the complex

functional-analytic

GARNETT

above.

points.

of Baire m e a s u r e s

to ARENS

more

Chapter

si-

It is

measures.

cited

some p a r t i c u l a r

[1969]

is in K ~ N I G

1.8 is in G A M E L I N

is somewhat

2-5 c o n t a i n m u c h

the treatises

that A be h y p o - D i r i c h l e t .

see also G A M E L I N

to consider

of the logmodular

The

[1968],

of t h e o r e m

it permits

1.9 in the c o m p a c t - c o n t i n u o u s

assumption

1.9 as it stands

formulation

Sections

8.5 is t h e o r e m

a soft a n a l y s i s

theorem.

of p l a n a r

[1975]

where

For a d e t a i l ~ d

sets we refer

the Carle-

proof

to TSUJI

of cer-

treatment [1959]

Appendix

The Appendix definitions theory

useful

and

and

we q u o t e

unconventional

of all

sublinear We

versions There

to r e c a l l

functional

of s t a n d a r d

are no p r o o f s ,

and the H a h n - B a n a c h

vector

linear

functionals

iff

space

9(u+v)~8(u)+8(v)

start with

that TeE

9(f-½(u+v))<_O.

analysis,

manner.

theorems except

certain measure

In p a r t i c u l a r

which

when

we

found

we c a n n o t

name

THEOREM

there

specialize

sublinear.

Then

there

such

exists

= Inf f6T

to T = { f }

conventional

9(tu):tS(u)

version

and

o(f)

subsequent

dual

e:E~

for

defined

that

Let

all u , v £ E

o£E ~ w i t h

o~8

there

such

and

to be real

theorem.

G : E ÷ ~ be

to u , v 6 T

to con-

is d e f i n e d

of the H a h n - B a n a c h

sublinear.

exists

f£T w i t h

that

9(f).

and = { - f }

for

fixed

f o r m of the H a h n - B a n a c h

exist

space,

A functional

(CONVEX V E R S I O N ) :

is n o n v o i d Then

Theorem

and E ~ its E-~. and

a powerful

Inf f6T

If we

from

and

in an u n s y s t e m a t i c

a real

1.1 H A H N - B A N A C H Assume

notations

reference.

Functionals

L e t E be

basic

theorems

calculus

in a p p l i c a t i o n s .

1. L i n e a r

t~O.

fundamental

advanced

a convenient

sist

is to i n t r o d u c e

linear

functionals

f6E

t h e n we o b t a i n

theorem: a6E * w i t h

Let

9:E~

o~8.

the be

Further-

more

{o(f) :06E ~ w i t h

We q u o t e

another

important

1.2 H A H N - B A N A C H Assume

that TeE

9(f-(u+v))~O. ~(f)~O

THEOREM

is n o n v o i d

If @ ( f ) ~ O

d~@}

=

[-8(-f),9(f) ]

consequence

of the

(CONE V E R S I O N ) : and

Vf6T

such

then

that

there

Vf6E.

convex

Let

9 : E + ~ be

to u , v £ T exists

version

there

o6E ~ w i t h

1.1.

sublinear. exists ~9

f6T w i t h

such

that

Vf6T.

The most

familiar

f o r m of the H a h n - B a n a c h

theorem

is the e x t e n s i o n

237

theorem:

Let e:E÷~

SeE w i t h ~ I S quence

of

the

sublinear

There and the

above

E ~ its

(relative

e a c h ~ 6 S ~ on a l i n e a r

form

to r e a l

(real or complex)

in t e r m s

hicle which but

special

situation:

linear

over

important

space,

%:E+~

important

and

ture

functions

and with

theorem will VcB(X)

define

be

modified

be

the

vector

let II.II:E+[O,~[ Then

real

space be

a semi-

e a c h ~ E S ~ on

a

I~I~!I-II can be e x t e n d e d

natural

to f o r m u l a t e

situation

vector

real-linear

each

for

vector

other

this

to II-II - T h e

is the

to verve-

subsequent

space.

linear

functionals

ReV:={Re

supremum

~:E+~

= Re~(x)

functionals with

order

complex-linear are

in o n e - t o -

VxEE,

is p o s i t i v i t y .

structures.

to c o n s i s t

s e t X, w i t h norm

in b o t h

of the b o u n d e d

its n a t u r a l

(= s u p n o r m ) . a real

We

Instead

shall

complex-

pointwise

The basic

and a complex

do n o t we

struc-

equivalence

version.

For

f:f6V}.

Let VcReB(X)

a real-linear

the

via

defined

on the n o n v o i d

formulated

Then

Vx6E:¢~.

spaces

to B(X),

If.If the

1.4 T H E O R E M : ~:V÷~

aspect

ourselves

in b o t h

E" of E r e l a t i v e

complex

a complex

the

with

to c o n s i d e r

is v a l i d

and

SeE w i t h

space

to the

(x) = ~ ( x ) - i ~ ( i x )

valued

subspace is a c o n s e -

to an a p p r o p r i a t e

(real or complex)

¢~:%0(x)

restrict

This

remark.

L e t E be

correspondence

An

~0.

coefficients).

subspace

of the n o r m d u a l

carries

functionals

dual

or c o m p l e x

a

l~l~II.II . It is of c o u r s e

1.3 R E M A R K :

intend

with

applied

case w h i c h

L e t E be

(real or complex)

some ~ 6 E ~ w i t h

one

Then

to s o m e ~ 6 E *

conventional

is an i m p o r t a n t complex

norm

simple

sublinear.

functional.

with

sion

be

c a n be e x t e n d e d

be

a real-linear

functional.

Then

the

subspace

subsequent

with

16V and

properties

are

equivalent.

i)

If fEV a n d

f>O t h e n ~ ( f ) > O ,

ii)

q0(f) < S u p

f for all

iii)

l~(f)l
for all

1.5 T H E O R E M : #:V+~

be

L e t VcB(X)

a complex-linear

are e q u i v a l e n t .

f6V.

f6v,

be

and ~ ( 1 ) = 1 .

a n d ~0(I)=I.

a complex-linear

functional.

Then

the

subspace

with

subsequent

16V and

properties

238

i)

If f£V a n d Re

ii)

Re~(f)

iii)

I~(f) I
all

iv)

¢(f)6Konv

(= the

<

If V is c l o s e d

f => O t h e n

Re~(f)>O,

S u p Re f for all

f(X)

under

f6V,

complex

a n d ¢(I)=I.

f6V.

and ¢ ( I ) = I .

closed

convex

conjugation

hull)

then

for all

the

f6V.

equivalence

extends

to

i ~)

The and

If f6V and

steps

i)~i ~)

1.5 w h i c h

f>O then

i)~ii)~iii)~i)

in the p r o o f

of

is an i m m e d i a t e

1.6 LEMMA:

#(f)>O,

and ~(I)=I.

in the p r o o f 1.5 are

all

consequence

L e t K c ~ be c o n v e x

of

1.4 a n d

of the

bounded

la-zl ~ suplu-zl Vze¢

i)~ii)~iii)~iv)~i)

conventional,

except

subsequent

~@ and a6~.

implies

that

iii)~iv)

in

lemma.

Then

a6~.

u6K Proof:

Assume

that

a~K.

Icl=1

such

that

of m o d u l u s

O<6

Take

Then

from

:= I n f l u - a I = Inf Re ~(u-a) u6K u6K

an R > O

such

lu-(a+tc) [2 =

that

lu-aI~R

I (u-a)-tc[ 2 =

1.1 we o b t a i n

and h e n c e

Vu6K°

For

t>O

a complex

Re ~ ( u - a ) ~ e

number

c

Vu6K.

then

lu-al 2 - 2 t R e ~ ( u - a )

+ t 2 < R 2 - 2te

+ t 2,

I/2 t =

It f o l l o w s

We

la-(a+tc) i ~ S u p l u - ( a + t c ) u6K

that

conclude

2t~R

with

2 Vt>O which

I ~

( R 2 - 2 t e + t 2)

is a c o n t r a d i c t i o n .

an u n c o n v e n t i o n a l

but

useful

QED.

version

of the b i p o -

lar t h e o r e m .

1.7 B I P O L A R near

subspace0

THEOREM:

L e t E be a real

Then

closed

the

convex

vector

hull

space

a n d FeE ~ be

in the w e a k

of Mc_E is

Konv

M = {u6E:~(u)

~ SupS(x) x6M

for all

~6F}.

topology

a li~(E,F)

239

2. M e a s u r e

Theory

Let

be

(X,~)

algebra ded

a measurable

Z of subsets.

complex-valued

complex-valued sist

functions

measures

of the m e a s u r e s

probability a dual

measures

system

via

the v a r i a t i o n

with

XA the

on

with

that

is a n o n v o i d

consist

on X. D e f i n e

~6Pos(X,Z)

~0,

to c o n s i s t

~(X)=1.

functional

is d e f i n e d

characteristic

function

to con-

of the s o - c a l l e d

B(X,E)

a n d ca(X,Z)

(f,e) ~-> ~fde.

For

form

eEca(X,Z)

to be

SuP{ISfdel:feB(X,~) with Ifl~×A}

=

a oboun-

of the

Pos(X,Z)cca(X,Z)

and Prob(X,Z)

with

the bilinear

set X w i t h

of the m e a s u r a b l e

ca(X,Z)

Z, in p a r t i c u l a r values

181£Pos(x,E)

lel(h)

space,

Let B(X,E)cB(X)

of AcX,

VAeZ,

whence

in p a r t i c u l a r

the

norm

IIeI[ =

In this

lel (x) = S u p { I f f d e I : f £ B ( X , E )

norm

ca(X,Z)

For m£Pos(X,Z) classes

modulo

the o b v i o u s

fines Banach the

spaces

for some

LP(m)

e into

of the n o t i o n

In the measurable

fn£L(m)

I~

After

with

the

For

that

part

section

where

of its B a i r e

define

that

that

is

de-

the

usual

to c o n s i s t

Ifl~e F for some

theorem

of

F6LI (m).

8 is m - c o n t i n u o u s Iml-continuous

this m e a n s

that

in

e=fm

the L e b e s g u e d e c o m p o s i t i o n de de LI 8m = ~ m with ~ E (Iml) a n d

we

to be d e a l t

consider

X is a c o m p a c t

subsets.

we have

L°(m)

decomposition

decomposition

of the

(X,Z)

For example,

we have

The Lebesgue

of p r e b a n d

In L(m)

is w i t h

Radon-Nikodym

on X. W i t h

[fn+O]:={x6X:fn(X)÷O}

sets.

to m),

equivalence

and e x p r e s s i o n s

themselves.

@ < < m is to m e a n

respect

of the

functions

notations

symbol

m-null

%,m6ca(X,E)

8~.

same

functions the

the m - c o n t i n u o u s

part

spade

the

~. F u r t h e r m o r e

the n o t a t i o n

remainder

the o - a l g e b r a

for

to c o n s i s t

complex-valued

use

EZ m o d u l o

continuous

fELl ([ml).

e=em+8 ~ of

can

(Ioglfl)+6L1(m),

sense.

the m - s i n g u l a r

= L(X,~,m)

as for the

set

8,m6ca(X,Z)

the u s u a l

we

of L(m)

with

(= a b s o l u t e l y

L(m)

of m e m b e r s

a measurable

f6L(m)

For

define

precautions

for the m e m b e r s

IfI~]}.

is c o m p l e t e .

m of the m e a s u r a b l e

for a s e q u e n c e

with

We w r i t e

the

is a p a r t i c u l a r

case

with

II.2.

in S e c t i o n

important

Hausdorff B(X,Baire)

space

particular and

Z is

and o b s e r v e

240

that C(X)cB(X,Baire). wise

Pos(X)

is the F . R i e s z linear

2.1 and

of

between

F.RIESZ

theorem

(f,~) ~-> ~ f d ~ the

supnorm

~6ca(X)

1.5.

the m e a s u r e s

As

be

usual

identified,

We do n o t procedure positive

intend

measure

a subset

this

smallest

N o w we

and

combine

Note

the p r o b a b i l i t y

Prob(X)

REMARK: with with

Proof: then

implies

measures

subset

likefact

above

bi-

isometric

iso-

a n d ca(X).

functionals

~6(C(X))"

with

each

II~II=II~II.

functional

their

for

functionals

all

of

sense

~6(C(X))"

f6C(X).

the

standard

recall

USC(X)

in the

the

of u p p e r

extension

extension

of a

semicontinuous

to be

with

F>__f~ ~ > -~

define

the

KcX with

Vf6USC(X).

support

We

then write

Supp(a)

full m e a s u r e

of a 6 c a ( X )

I~I (K)=i~l (X),

to see.

representation

I to o b t a i n

some

functionals

theorem with

efficient

is a s u b l i n e a r

~:ReC(X)÷~

with

#~.

Then

are p r e c i s e l y

the

the H a h n - B a -

representation

functional.

a E P r o b ( X ) . L e t us p r o v e

L e t K c X be c l o s e d

We

~Max

1.4

refinement.

functionals

the p r o b a b i l i t y

theo-

from

are p r e c i s e l y

a certain

linear

see

measures

~

:

o6

o(K)=I.

Let ~:ReC(X)+~

~ O,

let o 6 P r o b ( X )

can

the F . R i e s z

linear

a norm

linear

details

is n o t h a r d

o~Max(.IK)

o6Prob(X).

Max(1-flK)

we

the

correspondence

us m e r e l y class

and

fundamental

that

X K 6 U S C ( X ) iff K is c l o s e d .

that Max:ReC(X)÷~

1.5 t h a t t h e

2.2

the

is d e f i n e d

closed

of S e c t i o n

and

= ~fd~

Let

to the

point

of w h i c h

versions

ReC(X)+~

which

KcX we have

existence

rems.

to d i s c u s s

:= I n f { o ( F ) :FEReC(X)

f f d a = : a ( K ) . At to be t h e

~6ca(X)

measures.

~6Pos(X)

X+[-~,~[

ffda

nach

The

iff ~ is a p o s i t i v e

so t h a t ~(f)

for B a i r e

functions

the

(C(X))"

v i a ~ (f) = ~ f d ~ V f 6 C ( X ) . F u r t h e r m o r e

~6Pos(X)

For

space

the

states

in o n e - t o - o n e

In p a r t i c u l a r

will

which

THEOREM:

are

ca(X) : = c a ( X , B a i r e ) situation

in fact p r o d u c e s

dual

REPRESENTATION

the m e a s u r e s

other

abbreviate

In the p r e s e n t

representation

functional

morphism

Also we

and P r o b ( X ) .

And

a linear

for XK ~ f6ReC(X)

so t h a t

with

be

o(K)~I

~(K)=I.

that o(f)~Max(f[K).

For

functional, we h a v e

and h e n c e f6ReC(X)

QED.

i)

1-o(f)

o(K)=I, then Maxf

ii)

If g ~ M a x ( . I K ) ~ M a x = o(1-f) For

the

converse

- f~(Maxf-Max(flK))XK

241

THEOREM:

2.3

and

such

exists

that

Let

K c X be

to u , v 6 T

o6Prob(X)

with

some

Follows

upon

f6T}cReC(X).

QED.

2.4 T H E O R E M : and such Vf6T

that

then

Proof: some

Follows

upon

f£T}cReC(X).

QED.

In c o n c l u s i o n

we

2.5 T H E O R E M :

there

exists

f > O. T h e n

Proofs:

We

fulfills tension 1.4.

such

the

there

2.1

3. T h e C a u c h y

can be c o n s i d e r e d present

the

that

is n o n v o i d If M a x ( f I K ) ~ O

ffdo~O

subset

Vf6T.

{F6ReC(X) :F

theorem,

subspace

extension

theo-

as b e f o r e

in

such

that

such

to 2.5.

preserves

iii)

for

with

16V and

all O ~ f £ V .

Then

vf6V.

a complex-linear

condition

subspace

Re~(f)~O

that

#(f)=Sfd~

We

can

in

1.4.

assume

the e q u i v a l e n t

with

for

Hence

all

16V a n d fEV w i t h

vf6v.

that ~(I)=I. it a d m i t s

properties

Then

an ex-

i)-iii)

in

QED.

the C a u c h y

of the

to be

details.

= Sfdo

v i a the D i v e r g e n c e

that

version

on K.

the H a h n - B a n a c h

that ~( f)~O _

o6Pos(X)

applied.

Formula

to the

a real-linear such

ourselves

which

It is w e l l - k n o w n appropriate

exists

c a n be

and

such

{F6ReC(X) :F

TcUSC(X)

f~u+v

representation

be

be

equivalent

o:ReC(X)÷~

Then

1.5

1.2

subset

that

o(K)=I

of

functional

restrict

Assume

t h a t ~(f)

L e t VcC(X)

a complex-linear

there

version.

functional

a6Pos(X)

2.6 T H E O R E M :

and

to the

f6T w i t h

with

important

L e t VcReC(X)

a real-linear

Re

1.4

1.1

~.

application

and a complex

~:V+~

¢:V÷~

of

exists

o6Prob(X)

use

another

is n o n v o i d

on K. T h e n

= Inf M a x ( f I K ) . f6T

there

exists

TcUSC(X)

f~½(u+v)

that

L e t K c X be c l o s e d

there

a real

such

that

f6T w i t h

application

to u , v 6 T

r e m to o b t a i n both

%@. A s s u m e

exists

a(K)=1

Inf f f d o f6T Proof:

closed

there

divergence

its n a t u r a l

Theorem

formula

c a n be d e d u c e d

theorem. form.

L e t us f i x a n o n v o i d

We

It t h e n take

bounded

from

appears

an in w h a t

the o p p o r t u n i t y open

subset

to

C~z~=~ 2.

242

A boundary hood

point

u6~G

u c ~ of u s u c h

manifold).

is c a l l e d

that UN~G

Otherwise

u6~G

regular

iff

is a s m o o t h

is c a l l e d

there

curve

a singular

exists

a neighbour-

(= o n e - d i m e n s i o n a l

C l-

boundary

in-

point.

We

troduce

Then

R(G)

regular normal t>O.

vectors

X(G)

The

curve

unique

We

a n d S(G)

u6R(G)

is c o m p a c t

is c a l l e d

at u is s u c h

N=:N(u)

is c a l l e d

that

is c a l l e d

inner,

outer

which

with

u-tN6G

the o u t e r means

~G=R(G)US(G).

iff one

and u+tN~ normal

that

A

of the t w o u n i t

u is

for

small

of G at u. an i n t e r i o r

introduce

X(G)

= {u6~G

: u is outer},

Y(G)

= {u6DG

: u is

inner}.

smooth

curves

with

and Y(G)

are

is c o n t i n u o u s

divergence

is m a d e

defined

: u is s i n g u l a r } .

point

u£R(G)

of ~.

u ~-> N(u)

ment

: u is r e g u l a r } ,

= {uE~G

N to R(G)

this

Otherwise

Then

= {u6~G

S(G)

is a s m o o t h

boundary

Then

point

R(G)

theorem

precise

holds

in terms

for a c o m p a c t

X(G) U Y ( G ) = R ( G ) .

The

function

on X(G).

set Sc¢

true whenever

of the

S(G)

is small.

one-dimensional

This

Minkowski

state-

content,

to be

L({z6~:dist(z,S)~}) • (S) = lim sup ~+O where

L denotes

if S c o n s i s t s

3.1

of r e a l

two-dimensional

of

THEOREM:

a continuous

2~

finitely

Assume

bounded

analysis)

X (G)

many

that

vector

such

that

G satisfies function, divA:G÷~

both

integrals

exists.

measure

(= arc

length)

on X(G).

Observe above

that

in the

theorem

masure

on ~.

Clearly

Y(S)=O

T(S(G))=O.

L e t A : G U R ( G ) + ~ 2 be

differentiable is c o n t i n u o u s .

on G

(in the

sense

T h e n we h a v e

do (x) = S d i v A ( x ) d L ( x ) , G

whenever

the

Lebesgue points.

are

case

Here

T(S(G))=O

fulfilled

~ denotes

one-dimensional

and o ( X ( G ) ) < ~ all

if A is d e f i n e d

Lebesgue

assumptions

a n d C I on s o m e

open

of set

243

Uc~ with

Gc--GcU. F r o m

3.1 we o b t a i n

the G r e e n

formula

via a well

known

argument.

3.2 C O R O L L A R Y : f,g£C2(U) tional

Assume

on some

derivative

open of

that

G satisfies

T ( S ( G ) ) = O and q ( X ( G ) ) < ~. Let 3f GcGcU. L e t ~n(X) d e n o t e the d i r e c -

set U c ~ w i t h

f at the p o i n t

x6X(G)

in the d i r e c t i o n

N(x).

Then

we h a v e

X(G)

G

We derive introduce

from

the

3.1

the C a u c h y

differential

~-~ - ~ ~-~ + y ~-~ A function of real _

f:G+~

analysis)

I Df(z) i ~Y

theorem

3.3 T H E O R E M :

Assume

Then

that

function

and

such

that

3.4

both

= ~

~X

i ~-Y "

Let

f:GUR(G)+~

be

a

on G w i t h

= 2

~-~(x)dL(x),

3.1

exist.

to the v e c t o r

f) a n d s u m the

CONSEQUENCE:

m(S(G))=O.

f is d i f f e r e n t i a b l e

G

integrals

Apply

f, Re

Assume

functions

two e q u a t i o n s .

that

A=(Re

f, - I m

f)

and

QED.

G satisfies

T(S(G))=O

f:GUR(G)÷~

entiable

We

we h a v e

X(G)

Let

~

G satisfies

f(x)N{x)do(x)

Proof:

formula.

is h o l o m o r p h i c iff f is d i f f e r e n t i a b l e (in the s e n s e ~f ~f ~f with ~ = O; in this case f" (z) = ~ ( z ) = ~(z) =

bounded

~~Z f continuous.

B=(Im

the C a u c h y

VzEG.

continuous

whenever

and

operators

and ~ ( X ( G ) ) < ~ .

be a c o n t i n u o u s b o u n d e d f u n c t i o n s u c h t h a t 3f on G w i t h ~-~ c o n t i n u o u s . S u p p o s e f u r t h e r t h a t

f is d i f f e r -

I 3~(x)dL(x)<~. G Then we have

I I x-u f(x) N ( x ) d q ( x ) I ~ I X1_--~~(x)dL(x) 3f f(u) = 2--~ X(G) G

Vu6G.

244

The

subsequent

3.5 S P E C I A L Let

special

CASE:

f6C I (U) on s o m e

f(u)

1 = ~-~ X(

case will

Assume open

that

be

G satisfies

set Uc~ with

!

sufficient

GcGcU.

N(x) d~ (x)

) x-u

for o u r p u r p o s e s .

T(S(G))=O

a n d o ( X ( G ) ) < ~.

Then we have

~

~(x)

dL (x) Vu6G.

Notes

There rems.

is an e x t e n s i v e

The

ancestor

MAZUR-ORLICZ

[1953],

1.1-1.2

and

bipolar

theorem

2.3-2.4

theorem

is

literature

of n u m e r o u s see

also PTAK

follows

on

fortified

versions

KONIG

[1956].

The

above

[1968] [197Ob].

is

from KONIG

[1972].

adapted

from KONIG

[1964].

Hahn-Banach

is a w e l l - k n o w n

The

The

type

theorem

presentation version

presentation

theo-

due

to

of

1.7 of the

of the d i v e r g e n c e

References

P.R.AHERN [1965]

On the g e n e r a l i z e d F.and M.Riesz Pacific J.Math.15(1965) 373-376.

theorem.

P. R. AHE RN and Donald S A R A S O N

[1967a]

The H p spaces of a class of function algebras. Acta Math.117(1967) 123-163.

[1967b]

On some h y p o - D i r i c h l e t algebras of analytic functions. A m e r . J . M a t h . 8 9 ( 1 9 6 7 ) 932-941.

A . C . A L L E N and E.R. KERR [1953]

The converse of Fatou's theorem. J.London Math. Soc.28(1953) 80-89.

Eric A M A R [1973]

Sur un th~or@me de Mooney relatif aux fonctions analytiques born~es. Pacific J.Math.49(1973) 311-314.

R i c h a r d ARENS [1958]

The maximal ideals of certain function algebras. Pacific J.Math.~(1958) 641-648.

Klaus BARBEY [1975]

Ein Satz Hber abstrakte analytische Funktionen. Arch.Math.26(1975) 521-527.

[1976]

Zum Satz von Mooney f~r abstrakte a n a l y t i s c h e Funktionen. Arch.Math.27(1976) 622-626.

Klaus BARBEY and Heinz K~NIG [1972]

Ein G r e n z w e r t s a t z fHr a n n u l l i e r e n d e MaBe. Arch.Math.23(1972) 509-512.

[1976]

Zum Satz von M o o n e y f~r abstrakte analytische F u n k t i o n e n Preprint.

Herbert S . B E A R [1965]

A g e o m e t r i c c h a r a c t e r i z a t i o n of Gleason parts. P r o c . A m e r . M a t h . S o c . 1 6 ( 1 9 6 5 ) 407-412.

[1970]

Lectures on G l e a s o n parts. Lecture Notes V o l . 1 2 1 , B e r l i n

1970.

II.

246

Herbert S.BEAR and Max L.WEISS [1967]

An intrinsic metric for parts. Proc.Amer.Math. Soc.18(1967) 812-817.

Arne BEURLING [1949]

On two problems c o n c e r n i n g linear t r a n s f o r m a t i o n s space. A c t a Math.81(1949) 239-255.

in Hilbert

Errett BISHOP [1959]

A minimal b o u n d a r y for function algebras. P a c i f i c J.Math.~(1959) 629-642.

[1960]

Boundary measures of analytic differentials. Duke Math. J.27(1960) 331-340.

[1963]

H o l o m o r p h i c completions, analytic c o n t i n u a t i o n and the interpolation of semi-norms. Ann.of Math.78(1963) 468-500.

[1964]

R e p r e s e n t i n g measures for points in a u n i f o r m algebra. Bull.Amer.Math. Soc.70(1964) 121-122.

S.BOCHNER [1959]

G e n e r a l i z e d conjugate and analytic functions without expansions. Proc.Nat.Acad. Sci. USA 45(1959) 855-857.

Andrew BROWDER [1969]

I n t r o d u c t i o n to function algebras. New York 1969.

Constantin CARATH~ODORY [1950]

F u n k t i o n e n t h e o r i e . V o l . I. Basel 1950.

Lennart CARLESON [1964]

M e r g e l y a n ' s t h e o r e m on u n i f o r m polynomial approximation. Math. Scand. 15(1964) 167-175.

A l l e n DEVINATZ [1966]

C o n j u g a t e function theorems for Dirichlet algebras. R e v . U n . M a t . A r g e n t i n a 23(1966) 3-30.

Nelson DUNFORD and Jacob T.SCHWARTZ [1958]

Linear Operators.Vol. I: General theory. New York 1958.

Peter L.DUREN [1970]

Theory of H p spaces. New York 1970.

247

Frank FORELLI [1963]

A n a l y t i c measures. P a c i f i c J.Math.13(1963)

571-578.

T.W.GAMELIN [1968]

Embedding Riemann surfaces in m a x i m a l ideal spaces. J . F u n c t i o n a l Analysis 2(1968) 123-146.

[1969]

U n i f o r m algebras. Englewood Cliffs 1969.

T.W.GAMELIN [1968]

and G . L U M E R

Theory of abstract Hardy spaces and the universal Hardy class. Adv.in Math.~(1968) 118-174.

John GARNETT [1967]

A topological c h a r a c t e r i z a t i o n of Gleason parts. Pacific J.Math.20(1967) 59-63.

[1968]

On a t h e o r e m of Mergelyan. Pacific J.Math.26(1968) 461-467.

John GARNETT [1967]

and Irving G L I C K S B E R G

Algebras w i t h the same m u l t i p l i c a t i v e measures. J . F u n c t i o n a l Analysis !(1967) 331-341.

F.W.GEHRING [1957]

The Fatou t h e o r e m and its converse. Trans.Amer.Math. Soc.85(1957) 106-121.

Andrew M.GLEASON [1957]

Function algebras. Seminar on analytic functions.Vol. II, 213-226. Institute for A d v a n c e d Study, Princeton 1957.

I.GLICKSBERG

(see also GARNETT)

[1967]

The abstract F.and M.Riesz theorem. J . F u n c t i o n a l Analysis ~(1967) 109-122.

[1968]

Dominant r e p r e s e n t i n g measures and rational approximation. T r a n s . A m e r . M a t h . S o c . 1 3 0 ( 1 9 6 8 ) 425-462.

[1970]

E x t e n s i o n s of the F . a n d M.Riesz theorem. J . F u n c t i o n a l Analysis 5(1970) 125-136.

[1972]

Recent results on function algebras. Conference Board of the M a t h e m a t i c a l Sciences Regional Conference Series in M a t h e m a t i c s , N o . 1 1 . A m e r . M a t h . Soc. P r o v i d e n c e 1972.

248

I . G L I C K S B E R G and J . W E R M E R [1963]

Measures o r t h o g o n a l to Dirichlet algebras. Duke Math.J.30(1963) 6 6 1 - 6 6 0 . E r r a t a , i b i d . 3 1 ( 1 9 6 4 )

717.

A.GROTHENDIECK [1954]

Sur certain s o u s - e s p a c e s vectoriels Canad.J.Math.~(1954) 158-160.

de L p.

G.H.HARDY and E . M . W R I G H T [1968]

An introduction Oxford 1968.

to the theory of numbers.

V.P.HAVIN [1973]

Weak c o m p l e t e n e s s of the space LI/H ~ I (Russian,English summary). V e s t n i k L e n i n g r a d . U n i v . M a t . M e h . A s t r o n o m . V y p 3.13(1973)

77-81.

E l i z a b e t h Ann HEARD [1967]

A sequential F.and M.Riesz Proc. Amer.Mat. Soc.18(1967)

theorem. 832-835.

Henry HELSON [1964]

Lectures on invariant subspaces. New York 1964.

Henry HELSON and David L O W D E N S L A G E R [1958]

P r e d i c t i o n theory and F o u r i e r series in several variables. Acta Math.9_~9(1958) 165-202.

Edwin HEWITT and Kosaku Y O S I D A [1952]

Finitely additive measures. Trans.Amer.Math.Soc.72(1952)

I.J.HIRSCHMANN,Jr. [1974]

46-66.

and Richard ROCHBERG

Conjugate function theory in weak • Dirichlet algebras. J . F u n c t i o n a l A n a l y s i s 16(1974) 359-371.

Kenneth HOFFMAN [1962a]

Banach spaces of analytic Englewood Cliffs 1962.

functions.

[1962b]

A n a l y t i c functions and logmodular Banach algebras. Acta Math.108(1962) 271-317.

249

Kenneth HOFFMAN

and Hugo ROSSI

[1965]

Function theory and multiplicative linear functionals. Trans.Amer.Math. Soc.116(1965) 536-543.

[1967]

Extensions of positive weak, continuous Duke Math. J.3_~4(1967) 453-466.

Jean-Pierre [1967]

functionals.

KAHANE

Another theorem on bounded analytic functions. Proc.Amer.Math. Soc.1_88(1967) 827-831~

Yitzhak KATZNELSON [1968]

An introduction New York 1968.

to harmonic

E.R.KERR

analysis.

(see ALLEN)

A.KOLMOGOROV [1925]

Sur les fonctions Fourier. Fund.Math.Z(1925)

Heinz KONIG

harmoniques

conjug~es

et les s&ries de

24-29.

(see also BARBEY)

[1960]

Einige Eigenschaften der Fourier-Stieltjes-Transformation. Arch.Math.1_!(1960) 352-365.

[1964]

Ein einfacher Beweis des GauBschen Integralsatzes. Jber. Deutsch.Math.-Verein.66(1963/64) 119-138.

[1965]

Zur abstrakten Theorie der analytischen Math.z.88(1965) 136-165.

Funktionen.

[1966a]

Zur abstrakten Theorie der analytischen Math.Ann.163(1966) 9-17.

Funktionen

[1966b]

Lectures on abstract H p theory. Summer School on topological algebra theory. tember 6-16,1966.

[1967a]

Holomorphe Funktionen von BP-Funktionen. Arch.Math. 1_~8(1967) 160-166.

[1967b]

Zur abstrakten Theorie der analytischen Arch.Math. l_88(1967) 273-284.

[1967c]

Theory of abstract Hardy spaces. Lectures Notes, California Institute Pasadena 1967.

II.

Bruges,

Funktionen

Sep-

III.

of Technology.

[1968]

Uber das von Neumannsche Minimax Arch.Math.1_~9(1968) 482-487.

Theorem.

[1969a]

Abstract Hardy space theory. Papers from the Summer Gathering on functions algebras at Aarhus,July 1969. Aarhus Universitet,Matematisk Institut, Various Publication Series No 9,56-62.

250

[1969b]

Ein abstrakter Mergelyan Satz. Arch.Math.2_~O(1969) 405-412.

[1969c]

On the G!eason and Harnack metrics for uniform algebras. Proc.Amer.Math.Soc.22(1969) 1OO-101.

[1970a]

Generalized conjugate functions in abstract Hardy algebra theory. Journ&es de la Soci&t~ Math&matique de France, Alg~bres de Fonctions,Grenoble, 11-16 Septembre 1970, 36-44.

[197Ob]

On certain applications of the Hahn-Banach theorems. Arch.Math.2!(1970) 583-591.

[1972]

Sublineare Funktionale. Arch.Math.23(1972) 500-508.

[1975]

Ein funktionalanalytischer Beweis des Approximationssatzes Yon Walsh. J.Reine Angew.Math.(Crelle) 274/275(1975) 158-163.

[1978]

On the Marcel Riesz estimation for conjugate the abstract Hardy algebra theory. Commentationes Math.1978.

and minimax

functions

Heinz KONIG and G.L. SEEVER [1969] Gottfried [1966]

The abstract F.and M.Riesz theorem. Duke Math. J.3_66(1969) 791-797. KOTHE Topologische lineare R~ume I. 2.Auflage,Berlin 1966.

Nand LAL

(see MERRILL)

Henri LEBESGUE [1907]

Sur le probl~me de Dirichlet. Rend.Circ.Mat.Palermo 2--9(1907) 371-402.

Gerald M.LEIBOWITZ [1970]

Lectures on complex Glenview 1970.

function

algebras.

Lynn H.LOOMIS [1943]

The converse of the Fatou theorem for positive harmonic functions. Trans.Amer.Math. Soc.5_~3(1943) 239-250.

David LOWDENSLAGER

(see HELSON)

G. LUMER

(see also GAMELIN)

[1964]

Analytic functions and Dirichlet problem. Bull. Amer. Math. Soc. 7_~O(1964 ) 98-104.

in

25';

[1965]

Herglotz-transformation and HP-theory. Bull.Amer.Math.Soc.71(1965) 725-730.

[1966a]

Int~grabilit~ uniforme dans les alg~bres de fonctions, classes H @, et classe de Hardy universelle. C.R.Acad.Sci.Paris S~r.A 262(1966) 1046-1049.

[1966b]

H ~ and the imbedding of the classical HP spaces in arbitrary ones. Function algebras. Proceedings of an international symposium on function algebras held at Tulane University, 1965, edited by Frank T.Birtel,Glenview 1966, 285-286.

[1968]

Alg~bres de fonctions et espaces de Hardy. Lecture Notes Vol.75,Berlin 1968.

S.MAZUR and W.ORLICZ [1953]

Sur les espaces metriques lin~aires Studia Math.13(1953) 137-179.

(II).

Samuel MERRILL [1968]

Maximality of certain algebras H ~ Math. Z.106(1968) 261-266.

Samuel MERRILL,III,and [1969]

(dm).

Nand LAL

Characterization of certain invariant subspaces of H p and L p spaces derived from logmodular algebras. Pacific J.Math.30(1969) 463-474.

Michael C.MOONEY [1972]

A theorem on bounded analytic functions. Pacific J.Math.43(1972) 457-463.

Michael MURMANN [1967]

Zur Theorie der abstrakten HP-R~ume. Diplomarbeit,Universit~t des Saarlandes,Saarbr~cken

Paul S.MUHLY [1972]

Maximal weak~Dirichlet algebras. Proc.Amer.Math.Soc.36(1972) 515-518.

Takahiko NAKAZI [1975]

Invariant subspaces of weak.Dirichlet Preprint.

[1976]

Nonmaximal weak~Dirichlet algebras. Hokkaido Math.J.~(1976) 88-96.

Rolf NEVANLINNA [1953]

Eindeutige analytische Funktionen. Berlin 1953.

algebras.

1967.

252

B.V°O'NEILL [1968]

Parts and o n e - d i m e n s i o n a l analytic spaces. Amer.J.Math.9_OO(1968) 84-97.

W.ORLICZ

(see MAZUR)

A l e x a n d e r PELCZYNSKI [1974]

Sur certaines propri~t~s isomorphiques nouvelles des espaces de Banach de fonctions holomorphes A et H ~. C.R.Acad. Sci.Paris S~r.A.279(1974) 9-12.

S.K.PICHORIDES [1972]

On the best values of the constants Zygmund and Kelmoaorov. Studia Math.44(1972) 165-179.

in the theorems of M.Riesz,

George P I R A N I A N , A I I e n L.SHIELDS and James H.WELLS [1967]

Bounded analytic functions and absolutely continuous measures. Proc.Amer.Math. Soc.1_88(1967) 818-826.

I.I.PRIWALOW [1956]

Randeigenschaften Berlin 1956.

a n a l y t i s c h e r Funktionen.

Vlastimil PTAK [1956]

On a t h e o r e m of Mazur and Orlicz. Studia Math.15(1956) 365-366.

Heydar RADJAVI and Peter ROSENTHAL [1973]

Invariant subspaces. Berlin 1973.

John R A I N W A T E R [1969]

A note on the preceding paper. Duke Math.J.36(1969) 799-800.

R i c h a r d ROCHBERG

(see HIRSCHMAN)

Hugo ROSSI

(see HOFFMAN)

Donald SARASON

(see AHERN)

Luis SALINAS [1976]

G.L.SEEVER [1973]

M a x i m a l i t ~ t s s ~ t z e in der a b s t r a k t e n Hardy-R~ume-Theorie. D i s s e r t a t i o n , U n i v e r s i t ~ t des S a a r l a n d e s , S a a r b r O c k e n 1976. (see also KONIG) Algebras of continuous functions on h y p e r s t o n i a n Arch.Math.24(1973) 648-660.

spaces.

253

Allen L.SHIELDS T.P.SRINIVASAN [1966]

(see PIR~hNIAN) and Ju-Kwei WANG

Weak~Dirichlet algebras. Function algebras. Proceedings of an international symposium on function algebras held at Tulane University,1965,edited by Frank T.Birtel,Glenview 1966,216-249.

Edgar Lee STOUT [1971]

The theory of uniform algebras. Tarrytown-on-Hudson 1971.

Ion SUCIU [1973]

Function algebras. Bucharest 1973.

M.TSUJI [1959]

Potential theory in modern function theory. Tokyo 1959.

A.G.VITUSHKIN [1975]

Uniform approximations by holomorphic functions. J.Functional Analysis 2--0(1975) 149-157.

J. L.WALSH [1928]

Uber die Entwicklung einer harmonischen Funktion nach harmonischen Polynomen. J.Reine Angew.Math. (Crelle) 159(1928) 197-209.

[1929]

The approximation of harmonic functions by harmonic polynomials and by harmonic rational functions. Bull.Amer.Math. Soc.3_~5(1929) 499-544.

Ju-Kwei WANG

(see SRINIVASAN)

Max L.WEISS

(see BEAR)

James H.WELLS

(see PIRANIAN)

John WERMER

(see also GLICKSBERG)

[1960]

Dirichlet algebras. Duke Math. J.2_~7(1960) 373-381.

[1961]

Banach algebras and analytic functions. Advances in Math.l(1965) 51-102.

[1964]

Seminar ~ber Funktionen-Algebren. Lectures Notes Vol.1,Berlin 1964.

David Vernon WIDDER [1946]

The Laplace transform. Princeton 1946.

254

Donald R.WILKEN [1967]

Lebesgue m e a s u r e of parts for R(X). P r o c . A m e r . M a t h . S o c . 1 8 ( 1 9 6 7 ) 508-512.

[1968a]

R e p r e s e n t i n g m e a s u r e s for h a r m o n i c functions. Duke Math. J.35(1968) 383-389.

[1968b]

The support of r e p r e s e n t i n g m e a s u r e s Pacific J.Math.26(1968) 621-626.

for R(X).

KSz8 Y A B H T A [1973a]

Funktionen mit n i c h t n e g a t i v e m algebren. Arch. Math.24(1973) 164-168.

[1973b]

On the d i s t r i b u t i o n of values of functions in some function classes in the abstract Hardy space theory. Tohoku Math.J.25(1973) 89-102.

[1974a]

On b o u n d e d functions in the abstract Hardy space theory. Tohoku Math.J.26(1974) 77-84.

[1974b]

On b o u n d e d functions in the abstract Hardy space theory II. Tohoku Math.J.26(1974) 513-533.

[1975]

On b o u n d e d functions in the abstract Hardy space theory III. Tohoku Math.J.27(1975) 111-128.

[1976]

On the d i s t r i b u t i o n s of b o u n d e d functions in the abstract Hardy space theory and some of their applications. Preprint.

[1977]

M.Riesz's theorem in the abstract Hardy space theory. Arch.Math.2_88(1977).

Kosaku Y O S I D A

Realteil

in abstrakten Hardy-

(see HEWITT)

Lawrence Z A L C ~ N

[1968]

Analytic capacity and rational approximation. Lecture Notes Vol.50, Berlin 1968.

A.ZYGMUND [1968]

T r i g o n o m e t r i c series. Cambridge 1968.

Notation

an(D)

12

A(D)

HarmP(D)

12

an (~) =an (A, ~)

32

Index

3

HolP(D)

12

HOI~(D)

12

N f P N+ N

3 21 60

a (F)

67

HP(D)

A(K)

198

Hol # (G)

16

N (~o)=N (A,~)

Hol + (G)

16

NJ (~o)=NJ (A,q))

b(P)

35

B(X)

237

B(X,X)

H # (D) 239

49 59,60

72 131

ca,(~)

131

CS(@)

225

ca(X,X)

69

POS (X,X)

H+

93

P r o b (X,X)

H

122

Pos (X)

H#

144

23

dP(F)

61

D(S)

187

D(K)

227

E °~

60

S

69 225

226

L°(m)

116,119

L

187

G(u,v)

35

H a r m ~ (G) Hol (G) HoI~(G)

I 1 I

237

1

St(u,v)

49

Supp(o)

240

Tr (S) =Tr (S,R) USC(X)

M (~) =M (A,~)

M

1

214

216

239

22,44

240

V(u,~) , V(u,6)

22

M J (~) =MJ (A,~) H a r m (G)

239

242

M(A)

49

140

227

U

L(m)=L(X,X,m)

G1 (~)=GI (A,~)

240

198

Re

LS(e)

111

E(S)

R(K)

37

L#

L

239 240

P r o b (X)

R K

239

168

I(f)

I

DP(o)

198 217

R p (Fm)

240

202

1

H#

240

C(X)

E

P

H(F)

239

ca(X)

D

P(K)

65

62

181

P(Z,S)

207

cap(E)

62

NL (~o)= N L (A,~o)

60

H,

61

NL

62

I

C,(~)

16

(H,~)

1

CHoI(G)

NJ

H(u,v) Hp

CHarm(G) C

12

44

W=Wf

97

Y(E)

64

6O

MJ

61

ML

181

M L (~) =ML (A,~)

202

Z

14

205

256

~(f)

A (h)

66

o(f)

68

a~(f)

68

a+(f)

116

m $

227

72

45 61 62

q0~

65

XA

239

181

B°(f)

181

B~(f)

181

1

2

Z(A) r(A) A

34

$(f) qa

0 (f) B(f)

~a

104

35

22,44

:*(H)

~=~(B(X,I),ca(X,X))

158

~(f) ~(f,M)

8

a

102

T(T)

34

fR 2 <0>:z ~(z)

T ~T

KV,K ^

A S ~S

28

0=0K+O ~

u ~

<8>f 207 K ~X=~K, K O

149

29

209

156

K ~,~ A K ~K

176

28

:0 ~O K

209

152

~w

199

Am

61,199

F ~F ×

82

@C, @AC

P ~P

112

[0]

210

f ~f!Y 8 L , 0 AL

204

G ~G

207

the u s u a l n u m b e r

209

217 222 242

systems

BI

the a n n i h i l a t o r

Bx

the set of i n v e r t i b l e

of the set B in a d u a l elements

system

of the a l g e b r a

B

22

Subject Index

A b s t r a c t Hardy algebra situation .... r e d u c e d

59

64

Ahern-Sarason

lemma

182

Analytic disk theorem 158 Analytic

function

Annihilator

12,32

22

Arens-Royden theorem

Band

60

(=density)

A n a l y t i c measure

197

29

- decomposition - reducing

29

34

Bipolar t h e o r e m

238 212

Bishop l o c a l i z a t i o n t h e o r e m

Capacity continuous-analytic - logarithmic Cauchy support

207

transformation

-

198

225 199,204

Closed subgroup lemma

188

C o m m u t a t i v e operator algebra lemma Completely

singular measure

Conjugate function Conjugation

108,112

108,111

Continuous-analytic

capacity

Direct image c o n s t r u c t i o n Dirichlet property -

-

-

set

198

61

13,45

227

simultaneous approximation weak •

lemma

124

distribution

function

distribution

theory

dominant

165

35

97 131,207

64

Embryonic mean value theorem Enveloped function E v a l u a t i o n map

160

139

37

182

258

Exponentiation Extension -

small

Fatou

191

theorem

Forelli Full

44

190

7

lemma

set

64

Fundamental

Gelfand

58

lernma

24

topology

57,158

- transformation Gleason - part

(metric)

-- t r i v i a l

220 220

Gleason-Harnack Geometric

mean

Hahn-Banach

Harmonic Harnack

part

49

lemma

36

theorem

algebra

236

situation

--- r e d u c e d

measure

227

function

Hewitt-Yosida

(metric) theorem

decomposition

Hole

198,209

Hull

210

Hypo-Dirichlet

235

Inner function

81,90

Inner spectrum Internal

138

Invariant

subspace

-- s i m p l y

149

-- t h e o r e m

-

theorem

138

149

construction

theorem

function

inequality

212

150

image

Isomorphism

49

(=Lumer spectrum)

function

- point

Inverse

59

64

Hartogs-Rosenthal

Jensen

49

35

-- n o n t r i v i a l

Hardy

60,173

function

62

163

(=density) 15,19,41,64

61

172

104

259

Jensen -

inequality

measure

44

Kolmogorov

estimation

Krein-Smulian

Lebesgue

condition

228

capacity

- support

225

transformation

-

Lumer

Mergelyan

104

theorem

Measurable

space

239

theorem

Nontrivial

12

part

220

209

function

80,81

kernel

I

linear

functional

172,237

28

Hardy

Reducing

28

algebra

band

measure

Restriction,

situation

64

34

Representative

Riesz

233

58

decomposition

-

theorem

theorem

measure

Gleason

Oscillation

Reduced

approximation

approximation

Multiplicative

Preband

152,191

polynomial

- rational

Positive

181

7

spectrum

Poisson

(=density)

202

Maximality

Outer

199,222

theorem

Minimax

235 225

function

measure

Loomis

74

theorem

Logarithmic

Logmodular

1OO,126,144

consequence

Lebesgue-Walsh

-

41,63

universal

function

(=density)

60

22,44 restriction

algebra

F. band d e c o m p o s i t i o n

-- r e p r e s e n t a t i o n

theorem

theorem 240

217 29

232

260

Riesz

F.and M.theorem

.....

modified

14

.....

abstract

31,48

-

M.estimation

Simply

Small

128,145

invariant

Singular

subspace

functional

extension

Spectrum

22,44

- Lumer

104

Strict

191

175

continuous

Subharmonic Sublinear

Support

176

function

131

functional

236

Substitution

map

160

240

- Cauchy

207

- logarithmic

225

Swiss

cheese

Szeg6

213

functional

- situation - theorem

23,61

78 16

Szeg6-Kolmogorov-Krein abstract

Taylor

Trivial

152

Gleason

Value

carrier

Walsh

set

227

theorem

228

Weak ~ Dirichlet Weak-LP(m) Wilken

156

216

Transporter

-

theorem

38,63

coefficients

Trace

149

172

convergence

Strictly

....

14

type

theorem

part

220

102

property 97 217

124

16