Lecture notes in mathematics

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
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© 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~: IzlPoisson

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~Rhence

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=
f(rz)

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

= /S S

The

to 1

_<

f(f P(-~z,s)d~(s))P/q(/ p(~z,s)If(Rs)I Pd~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
< ~.
OLP(1)

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 = <Fl>: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=<Fl>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 = <HI> 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 OS
S

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=<Hl>

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=
in v i e w of iii),

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=<Fl>.

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=<Fl>.

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£Rsequence

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=<Fl>

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

@:[-~,~]ữÂ

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]:Ox
=

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

'


10

and hence

For

l i m s u p l F ( s ) - a I _<_ 2 Sup{I~---~)-al :Os+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 Oabove

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
= 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
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)
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) : <FI> 6 H o I ( D ) }

A(D)

:= {F6C(S)

H~(D)

the b i j e c t i v e

correspondences

c {<FI> :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

: <FI> 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~><GI>

(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=<Fl>.
f(z)h(Rz)

Then F86an(D)

For O~R
and <FS>=<FI><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)=<Hl>(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~)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=<Fl> 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=<Fl> 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=<Fl>.

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-

n

=

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=<Fn ~>, gn =

with Gn:=FnF6H~(D).

=<Gn~>ÊHoI~(D) ã Then

IfnI<1 and fn ữI 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
[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
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 =<F l > 6 H a r m
n
n
n

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