Hardy Classes on Infinitely Connected Riemann Surfaces

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

1027 Morisuke Hasumi

Hardy Classes on Infinitely Connected Riemann Surfaces

Springer-Verlag Berlin Heidelberg New York Tokyo 1983

Author

Morisuke Hasumi Department of Mathematics, Ibaraki University Mito, Ibaraki 310, Japan

A M S Subject Classifications (1980): 30 F99, 30 F 25, 46 J 15, 46 J 20, 31A20, 3 0 D 5 5 ISBN 3-540-12729-1 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-12729-1 Springer-Verlag New York Heidelberg Berlin Tokyo This work is subject to copyright.All rights are reserved,whetherthe whole or part of the material is concerned,specificallythose of translation,reprinting,re-use of illustrations,broadcasting, reproductionby photocopyingmachineor similar means,and storage in data banks. Under § 54 of the GermanCopyright Law where copies are madefor other than private use, a fee is payableto "VerwertungsgesellschaftWort", Munich. © by Springer-VerlagBerlin Heidelberg 1983 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210

PREFACE

The purpose on i n f i n i t e l y Already "Hardy print

connected

in this Classes

on R i e m a n n

Surfaces"

analytic

when

H a r d y himself,

disk

J. E.

Speaking

roughly,

sively

in the case of the unit

as well as simplicity. or not,

has

drawn m u c h

Opposed

to this,

our

laid

stress

made

discussed

their

amount

have

been

now be

in the

growth

of func-

The theory

of these

G.

i~ the work

Szeg~

among

of l i t e r a t u r e studied

from the cradle

in

years.

debut

the mean

[15].

appeared

should

its f o u n d a t i o n

growing

classes

which

subsequent

F. and M. Riesz,

disk

topics.

monograph

most

of

others.

in this inten-

for its i m p o r t a n c e

The case

of f i n i t e l y

oonnected

attention

and e n j o y e d

considerable

our k n o w l e d g e

connected

classes

deals m o s t l y

bearing

on our present

analytic

methods

of c o m p l e x class

for example.

theory

theory thus

of f u n c t i o n

surfaces,

planar

progress

it,

still

an i n d e p e n d e n t of general

so we should finitely

this

in

theory

from G a m e l i n ' s

the r e a c h

field

"For w h i c h

known

class

can one get a f r u i t f u l

classes?" at least,

In the present to this

[I0]

surfaces, and

and balls.

is not yet very well

surfaces

partial

connected

book

of the new t h e o r y

of p o l y d i s k s

question:

of H a r d y

try to give an answer,

in the

and the a b s t r a c t

grown to form the core of the n e w l y -

surfaces

Riemann

direct

functional-

applications

classes

in the

of Hardy

not have m u c h

the case of i n f i n i t e l y

basic

small

theory

downwards,

as e v i d e n c e d

lies b e y o n d

Riemann

open

sion of the c l a s s i c a l we will

Hardy

study as in the case

begin with

connected

has

algebras,

1950's

successful

including

created

to be r e l a t i v e l y

The c l a s s i c a l

disk and does

From

their

Nevertheless,

as I u n d e r s t a n d

structure

problem. found

seems

surfaces.

with the unit

have

function

born t h e o r y

needs

Hardy

some r e l a t e d

years.

case of i n f i n i t e l y

Hardy

classes

was

and still

area.

that

in his pa p e r

Littlewood,

classes

Heins,

seen d u r i n g

Hardy

of functions

A n d we now have a large

recent

natural

G. H. H a r d y

on the unit

classes

and

of H a r d y

we have a b e a u t i f u l by M a u r i c e

we have

recognized,

in 1915,

surfaces

series

It is t h e r e f o r e

As g e n e r a l l y

very useful

is to give an account

Riemann

Notes

on some new a d v a n c e s

literature tions

notes

open

Lecture

in 1969.

placed

of these

The and

of inextennotes

question.

IV

Our

idea for a t t a c k i n g

any nice

surface

But what

do we mean

as most

should

promising

(abbreviated in Chapter

to

V.

in the

class

perhaps

motives.

result

class

An open

if every

bounded

holomorphic

of the unit results

almost

every

converges (D) origin

by M.

reason

here

of P a r r e a u - W i d o m will

[70]

[52]

in 1971

which

in

from very

later

why we are

following

type

be given

Parreau

definitions,

in the

functions.

turned

interested

fundamental

of this

following, positive

Green

where

measurable

R

denotes function

other

R

space

has

origin

to the given

boundary

function

along

Almost

Green

every

converges

to a point,

indeed

for

R

(D) means

that

PWS's

for w h i c h

in detail

in the M a r t i n

vertex

behavior

general

at almost

to g e n e r a l i z e

consists

the direct theorem.

the

Cauchy

whose

theorem--(DCT)

Concerning

these,

line.

A

every

of

with r e s p e c t on

solution

boundary

A of the

function

line.

class

maps

prob-

for any PWS. of surfaces

solution. the

following:

point from

of

More-

in R

A

can be

can be

disk.

Cauchy-Read

of two statements,

for

Green

measure

answer

so as to have

of a n a l y t i c

R

which

the B r e l o t - C h o q u e t

has a p o s i t i v e

as in the case of the unit

It is p o s s i b l e

Green

affirmative

first

problem

that

on

boundary

the usual

measurable every

along

from any fixed

and the h a r m o n i c

almost

relevant

R.

every

b~,

(D) can be r e f i n e d with

issuing

Furthermore,

form the

this

almost

R

in p a r t i c u l a r

properties

solution,

on

any b o u n d e d

[5]) has a c o m p l e t e l y

Stolz regions

Cauchy

with

in

lines

is m e a s u r e - p r e s e r v i n g

to the data along

and the b o u n d a r y

respectively,

~ ÷ b~

on the one hand,

statement

The t h e o r e m

Z say

on the other.

statement

genus

line

nice

a limit

of Green

has a unique

0

if and

a PWS. on

function

problem

type

has n o n c o n s t a n t

We list most

boundary

It seems

analysed

R

many

from any fixed

infinite

(E)

over

surfaces.

on the

(see Brelot

defined

inherit

issuing

converges

the

bundle

problem

for the point

The

kind

connected

harmonic

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

A

is of P a r r e a u - W i d o m

line

line

0

Dirichlet

R

complex

sections.

to the Green m e a s u r e

verse

surfaces

The D i r i c h l e t

any b o u n d e d

over,

says that

to put f o r w a r d

The d e f i n i t i o n

The main

surface

disk or f i n i t e l y

Every

(C)

lem

and

of h o l o m o r p h i c

by H. W i d o m

different

same.

simple

we wish

introduced

is e x p r e s s e d

flat

surfaces

in the

(B)

on

first

independently,

Riemann

unitary

Moreover,

R

of R i e m a n n

following).

They used the

family

The c a n d i d a t e

was

of surfaces

is very

of Widom:

(A) only

an ample

is the class

"PWS"

out to be e s s e n t i a l l y in this

carry

by this?

This

1958 and also, different

the p r o b l e m

theorem

refinements for

we first

to PWS's. are called,

short--and have:

the

in-

V

(F)

The

inverse

The c o n v e r s e (G)

If

of b o u n d e d

R

holomorphic

the

utmost As we rect

type.

refinement shall

direct

there

theorem

Cauchy

we have:

(H)

Every

PWS

This,

expectation a PWS can Cauchy

state a n o t h e r

H~(R)

cussed

if

R

exist

genus

planar

of

(R)

R,

surfaces

Cauchy

formula--is

H

is a PWS.

characterizes

of

theorem--an

not always

valid.

for w h i c h

the di-

PWS's

for w h i c h

some

say about

algebra

the

fact

theorem without like

as an open

answer,

in this

is r e g u l a r

in the sense

PWS.

sub-

The

contrary

connection

satisfying

look

interesting

H~(R).

an a r b i t r a r y

has a n eg a t i v e

A curious

with

homeomorphieally

Banach

two things

R

in the

space

to the

is that

the direct

independent.

Finally,

notes.

Riemann

surfaces

tively,

PWS's

Apart

our last r e s u l t

out,

surfaces

shows

relevant

problem

subsets

this

we

that plane

of

of plane

are good regions

theory,

on

R

then

in the

R.

of PWS's

to be dis-

book c o n t a i n s

a detailed

regions

all that h a p p e n

in the c a t e g o r y which

functions

properties

from PWS's,

almost

can h a p p e n

are

on compact

about

of the c l a s s i f i c a t i o n As it turns

of p o t e n t i a l

of all h o l o m o r p h i c

convergence

been t a l k i n g

in these

account

the points

the direct

can be studied

of the

PWS's

the corona

of u n i f o r m

classes.

we have:

if the set

fact:

is dense

We have

space

have.

These

If a PWS

topology

almost

of i n f i n i t e

is all one can

one m i g h t

satisfy

separates

can be e m b e d d e d

for general

theorem.

(I)

R ideal

however,

problem

and

if and only

there

for PWS's

First

set in the m a x i m a l

PWS's

Namely,

surface

integral

while

PWS.

true.

fails.

problem

results.

corona

theorem

exist

holds,

theorem

The c o r o n a

R

On the other hand,

of the Cauchy

see,

Cauchy

on

t h e o r e m holds

Cauchy

for any

is also

Riemann

functions

Cauchy

inverse

Parreau-Widom

t h e o r e m holds

statement

is a h y p e r b o l i c

then the inverse Thus

Cauchy

of this

in terms

of Hardy

in the c a t e g o r y

of plane regions.

in some

sense

of

Intui-

or other.

can be as i l l - b e h a v e d

So

as one

can imagine. In w r i t i n g faces

of P a r r e a u - W i d o m

behaved

surfaces

It is hoped this

t h e s e notes

feeling

study

that

type

been

probably

our d e s c r i p t i o n

somehow

aims

or other. adaptation

faces.

It also

general

nor too special.

at finding

~ust refleet

led by the

of Hardy

I wish

of the e x i s t i n g some new facts

The present

notes

personal

pages

knowledge

will

of well-

justify

is that

our

of R i e m a n n

are n e i t h e r

are not c o m p l e t e interest

the sur-

is concerned.

to note

which

that

family

classes

in the f o l l o w i n g One thing

the a u t h o r ' s

feeling

form the widest

as far as the t h e o r y

is not a mere

sense but

I have

too

in any

in the field.

sur-

VI

At all event I hope that our effort w o u l d help not only extend the theory of H a r d y classes but also increase our k n o w l e d g e of R i e m a n n surfaces in general. The p r e r e q u i s i t e s

f o r ' r e a d i n g these notes are the f u n d a m e n t a l s of

a d v a n c e d complex function theory and some k n o w l e d g e of f u n c t i o n a l analysis.

As for the function theory~

we assume that the r e a d e r has some

a c q u a i n t a n c e with the facts to be found in Chapters Ahlfors and Sario

I, II, Iii, V of

[AS] and also in the first four chapters of the book

[CC] by C o n s t a n t i n e s c u and Cornea. Chapter i of Hoffman

As for the f u n c t i o n a l analysis,

[34] may be useful,

if not sufficient.

We now comment on the contents of the present notes. theoretic prerequisites

are sketched

The f u n c t i o n -

in Chapter I without proof.

order to deal r e a s o n a b l y with Hardy classes on m u l t i p l y - c o n n e c t e d surfaces,

we rely on two concepts:

and Martin c o m p a c t i f i c a t i o n . III, respectively.

multiplicative analytic

These are e x p l a i n e d

Chapter IV contains

In open

functions

in Chapters

II and

preliminary observations

on

H a r d y classes, where the b o u n d a r y b e h a v i o r is our p r i n c i p a l concern. The main body of this book begins

in Chapter V.

There,

the d e f i n i t i o n

of surfaces of P a r r e a u - W i d o m type is given after Widom. by means of r e g u l a r i z a t i o n , valent to Parreau's,

that this d e f i n i t i o n

p r o b l e m for PWS's

gebra of

L~

(see

(B),

(see (D)).

(F) is e s t a b l i s h e d by using Green lines and, as an

it is shown that

H~

is a m a x i m a l w e a k - s t a r closed subal-

on the Martin boundary.

(DCT) is p r e c i s e l y stated.

Next, the direct Cauchy t h e o r e m

We prove a w e a k e r v e r s i o n of (DCT), which

is valid for any h y p e r b o l i c Riemann surface, ation.

(C)) and then solve the Two types of Cauchy theo-

and i n v e r s e - - o n PWS's form the main theme of Chapter VII.

There, the statement application,

On the other hand,

t o g e t h e r with some applic-

(DCT) itself fails sometimes.

not be seen until we know something about invariant are studied in Chapter VIII. submodules of

But this can-

subspaces,

subspaces of

LP--on the Martin

C o r r e s p o n d i n g to the known results

for the case of

the unit disk, we c o n s i d e r two p r i n c i p a l types of invariant which are called doubly invariant and simply invariant, As for doubly invariant

which

In Chapter VIII we c l a s s i f y closed H ~-

LP--(shift-)invariant

b o u n d a r y of a PWS.

any PWS.

fundamental

In Chapter VI we discuss the D i r i c h l e t

p r o b l e m on the space of Green lines rems--direct

is e s s e n t i a l l y equi-

we present a d e t a i l e d proof of Widom's

t h e o r e m m e n t i o n e d in (A) above.

Brelot-Choquet

A f t e r showing,

subspaces,

respectively.

subspaees the situation is r a t h e r simple for

But the so-called Beurling type t h e o r e m for simply invariant

subspaces is not always valid for PWS's.

It is proved in fact that the

VII

Beurling

type

nection,

examples

types

theorem

is v a l i d

in C h a p t e r

of constr~etion:

Myrberg

type)

planar

PWS's

the third

for w h i c h

gives

same c h a p t e r we first bolic

prove

the

first

(DCT) (DCT)

fails

the

statement

(DCT).

classification (E) will not

are three

appendices

if

(DCT)

PWS's

the

is.

(G), w h i c h

of plane

theorem

a couple

just

by using

and a list of r e f e r e n c e s ,

PWS's

among

Hardy

IX hyper-

on PWS's

problem

classes.

in Chapter

which

and

In the

of c o n d i t i o n s

sketched

(of

In C h a p t e r

XI we solve Heins'

regions

but

three

of

holds;

is false. (I).

characterizes

in Chapter

con-

genus

a family

theorem

(H) and

collect

be proved

yields

but the corona

the c o r o n a

In this

We give there

of i n f i n i t e

second

statements

and then

Finally

statement

defines

holds;

for w h i c h prove

surfaces,

to

concerning

PWS's

we also

Riemann

equivalent

the

for w h i c h

if and only

X m a y be interesting.

The

VI.

There

is by no means

ex-

haustive. My i n t e r e s t was v i s i t i n g this

I am i n d e b t e d

It was

like

classes

to thank

after

thesis

Professor

when

primitive

the notes

written

thanks

expand

for a series

Metropolitan

notes

are due to P r o f e s s o r s and made

who

my k n o w l e d g e

Most

Sakai

some v a l u a b l e

and

remarks

from Professors

to w h o m

I wish

is my first

teacher

couragement

have r e m a i n e d

Mito,

Ibaraki

July,

1983

this

on the r e s e a r c h

book

who

Year

chapters topics

in The have

were reat Tokyo

Particular who o r g a n i z e d

I have much

benefited

and H. W i d o m

and

my appreciation.

to P r o f e s s o r

level

right

classes.

5, 1982.

Z. K u r a m o e h i

to express

thus

New d i s c o v e r i e s

S. Yamashita,

suggestions.

by very h e l p f u l

like to d e d i c a t e

of H a r d y

of the main

of July

study

I would

its B e u r l i n g

then.

I

For

L. Carleson,

I gave on the present

Dr. M. Hayashi, I would

to a serious

surfaces.

during

while

L. Kelley.

showed me the thesis

was w r i t t e n

the week

M.

and J.

to P r o f e s s o r

Institute

subsequently.

during

Riemann

aroused

in 1962-64.

H. H e l s o n

led me in 1972

thanks

to deepen

of lectures

University

lectures

that

first

at B e r k e l e y

Bishop,

L. A. Rubel,

of these

here was

connected

I also owe

I was able

version

E.

[45],

me to the M i t t a g - L e f f l e r

1976-77,

made

treated

of C a l i f o r n i a

on i n f i n i t e l y

its completion.

invited

subject

to P r o f e s s o r s

C. N e v i l l e ' s

of Ha r d y

the

in the

the U n i v e r s i t y

and w h o s e

Zir6 Takeda, inspiration

who

and en-

with me as fresh as ever.

Morisuke

Hasumi

CONTENTS

PREFACE

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CHAPTER

I.

§i.

§2.

§3.

Topology

Riemann

2.

The

Homology

3.

The

Fundamental

Classical

SURFACES:

Surfaces

Groups

Potential

Theory

. . . . . . . . . . . . . . . . .

4

. . . . . . . . . . . . . . . .

4

. . . . . . . . . . . . . . . . .

S

6.

Potential

Theory

Differentials

9.

Cycles

Definition

Class

F

and

Riemann-Roch Cauchy

9

Subclasses

on

9

. . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

Compact

Bordered

Surfaces

. . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

II.

MULTIPLICATIVE

The Line

3.

Existence

Cohomology

Bundles

and

of

Basic

5.

0rthogonal

Group

of

Structure

. . . . . . . . . . . . .

23 23

Analytic

Sections Functions

Functions

.

31

. . . . . . . . . .

33

. . . . . . . . . . . . . . . .

Definition

2.

Integral

3.

The

33 36 38

COMPACTIFICATION

Compactification i.

28

. . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . MARTIN

14 17 22

. . . . . . . . . . . . . . . . . . . .

Decomposition

12

. . . . . . . . . . . . . . .

Multiplicative

Harmonic

ii

FUNCTIONS

Functions

Holomorphic

Structure

4.

ANALYTIC

Analytic

First

2.

III.

its

Theorem

Kernels

Multiplicative

CHAPTER

7

. . . . . . . . . . . . . . . . . . . .

and

Differentials

ll.

Notes

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

I0.

Lattice

i 2

Problem

Dirichlet

i

3

The

The

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

5.

Basic

REVIEW

. . . . . . . . . . . . . . . . . .

Functions

8.

QUICK

Group

Superharmonic

7.

A

. . . . . . . . . . . . . . . . . . . . . . .

4.

i.

51.

RIEMANN

Exhaustion

CHAPTER

§2.

of

OF

1.

Notes

§i.

THEORY

iii

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Representation

Dirichlet

Problem

39 39

. . . . . . . . . . . . . . . .

40

. . . . . . . . . . . . . . . . .

43

X

§2.

§3.

Fine

Limits Definition

5.

Analysis

Covering

IV.

§3.

Some

Harmonic

Harmonic

the

Unit

Definitions

3.

Boundary Some

5.

The

on

. . . . . . . . . . . . .

Functions Measures

Disk

Behavior on

B-Topology

RIEMANN

57

. . . . . . . . . . .

59 63

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

of

Riemann Hp

and

Multiplicative

Surfaces hp

. . . . . . . .

Functions

Analytic

. . . . . .

Functions

and

Basic Widom's

3.

Regularization

Definitions

Widom's

of

on

5.

Proof

Necessity

of

Widom's of

(I)

Regular

of

. . . . . . . . . . .

Parreau-Widom

Type

....

. . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

Theorem

6.

Review

Principal

7.

Modified

Green

8.

Proof

of

Sufficiency

9.

A

Direct

(II)

. . . . . . . . . . . . . . .

Operators

Functions

. . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

Consequenoes

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . GREEN

The

Dirichlet

i.

Definition

2.

The

The

Space

3.

The

4.

Limit

Green

73

83 83 85 86 90 90 95 99 99 102 iii 117 118

LINES Problem of

Dirichlet of

73

TYPE

. . . . . . . . . . . . . . . .

Subregions

66

75

. . . . . . . . . . . . . . . .

Surfaces

64

82

. . . . . . . . . . . . . . . . . . .

Theorem

Analysis

VI.

Properties

Characterization

4.

Few

PARREAU-WIDOM

Fundamental

2.

of

OF

64

74

. . . . . . . . . . . . . . . . . . . . .

SURFACES

i.

of

50 57

. . . . . . . . . . . . . . . . . . . . . . . . . . .

V.

49

. . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

Results Hyperbolic

Results

Definitions

CHAPTER

§2.

of

on

Classes

Notes

§i.

of

Classical

4.

Proof

Behavior

CLASSES

Classes

Basic

Proof

49

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

HARDY

2.

Hardy

Limits

. . . . . . . . . . . . . . . . . . . . . . . . . . .

i.

CHAPTER

§2.

Maps

Preservation

Hardy

Fine

Boundary

7.

Notes

§i.

of

Correspondence

CHAPTER

§2.

of

6.

Notes

§i.

. . . . . . . . . . . . . . . . . . . . . . . .

4.

the

Star

Lines

of

Green

Lines

. . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . on

Regions

Green

Space

Lines

Problem

Green

along

on

Green

Lines

a

Surface

of

Parreau-Widom

Type

119 119 121 124

. . . . . . . . . . . . . . . . .

124

. . . . . . . . . . . . . . . .

129

XI

§3.

The

Green

5.

Convergence

of

6.

Green

and

7.

Boundary

Notes CHAPTER §i.

§2.

§3.

VII.

Statement Proof

The

Direct

3.

Formulation

4.

The

§2.

of

of

. . . . . . . . . .

132

. . . . . . . . . . . . . . .

132

Boundary

13S

Analytic

Common The

Theorem Results

Cauchy

IB

. . . . . . . . . .

Maps

. . . . . . . . . . .

140 143

. . . . . . . . . . . . . . . . .

144

. . . . . . . . . . . . . . . . . .

144

. . . . . . . . . . . . . . . . . .

145

Theorem of

the

Cauchy

. . . . . . . . . . . . . . . . .

Condition Theorem

of

151

. . . . . . . . . . . . . .

iSl

Weak

152

Type

. . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

Weak-star

7.

Maximality

Inner

of

Factors

Ha

155

. . . . . . . . . . . . . .

155

. . . . . . . . . . . . . . . . . .

Orthooomplement

of

156

H~(dx)

157

. . . . . . . . . . . . . . . . . . . . . . . . . . .

VIII.

SHIFT-INVARIANT

Preliminary

Generalities

2.

Shift-lnvariant

Invariant

Subspaces

Subspaees Invariant

4.

Simply

Invariant

S.

Equivalence

160

of

on

the

Unit

Disk

160

. . . . . . .

162

. . . . . . . . . . . . . . . . . . . .

167

Subspaces

. . . . . . . . . . . . . . .

167

Subspaees

. . . . . . . . . . . . . . .

169

( D C T )a

. . . . . . . . . . . . . . . . .

177

. . . . . . . . . . . . . . . . . . . . . . . . . . . CHARACTERIZATION

Inverse

Cauchy

i.

Statement

2.

A Mean

3.

Proof

Conditions

of

Value of

the

General

S.

Functions

OF

Theorem

the

Main

Theorem Main

Equivalent

4.

Notes

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

Doubly

IX.

159

SUBSPACES

Observations

i.

The

Martin

of

Theorem

Direct

6.

CHAPTER

Boundary

Lines

THEOREMS

Cauchy

Applications

Notes

§i.

CAUCHY

2.

3.

Martin

the

Behavior

i.

Notes

§2.

the Green

. . . . . . . . . . . . . . . . . . . . . . . . . . .

Inverse

CHAPTER

and

Lines

The

5.

§i.

Lines

Discussion mP(~,a)

SURFACES and

OF

Surfaces

Result

PARREAU-WIDOM of

178 TYPE

Parreau-Widom

. . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

Theorem to

the

. . . . . . . . . . . . . . . Direct

Cauehy

Theorem

.....

. . . . . . . . . . . . . . . . . . . and

(DCT)

. . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

Type

179 17g 183 187 198 198 200 207

XII

CHAPTER §i.

§2.

§3.

§4.

X.

PWS

of

§3.

Infinite

Genus

PWS's

2.

Verification

Plane

of

3.

Some

Existence

Further

of

of

Simple

Embedding

6.

Density

The

Corona

7.

(DCT)

8.

Negative

for

. . . . . . . .

208

Which

(DCT)

Fails

215 217

. . . . . . . . . . . . . . . . . Ideal

Space

. . . . . . . . .

Hardy-Orlicz

223

PWS

227

. . . . . . . . . . . . . . . . .

Theorem:

Positive

Examples

....

. . . . . . . . . . . . . . . . . . .

Classes

i.

Definitions

2.

Some Sets

of

3.

Preliminary Existence

N~

of

5.

Lemmas

6.

Classification

7.

Majoration

Null

233

234 234 235

. . . . . . . . . . . . . . . . . . .

238

. . . . . . . . . . . . . . . . . . .

238

. . . . . . . . . . . . . . . . .

Regions

. . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

by

229

. . . . . . . . . . . . . . . . .

Sets

Plane

227

REGIONS

. . . . . . . . . . . . . . . . . . . .

Lemmas of

Classification

Notes

PLANE

Properties

Class

4.

OF

. . . . . . . . . . . . . . . . . . . . . .

Basic

221 221

. . . . . . . . . . . . . . . . . . .

Corona

CLASSIFICATION

215

. . . . . . . . . . . . . . . . . . .

Maximal

Examples

for

. . . . . . . . . . . . . . . . . . .

PWS

for

Type

. . . . . . . . . . . . . . . . . . . . . . . . . . .

XI.

208

213

(R)

the

Holds

. . . . . . . . . . . . . . . . .

of

H

(DCT)

. . . . . . . . . . . . . . . . .

the

Problem and

Which

TYPE

Type

Lemmas

into

PARREAU-WIDOM

(DCT)

Theorem

of

OF

Parreau-Widom

Properties

5.

Null

SURFACES

Myrberg

Regions

4.

CHAPTER

§2.

OF

i.

Notes

§i.

EXAMPLES

Theorem

. . . . . . . . . . . . . . . . .

Quasibounded

Harmonic

Functions

. . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

247 253 253 256 260 261

APPENDICES A.I.

The

Classical

A.2.

Kolmogorov's

A.3.

The

F.

References Index Index

of

and

Fatou Theorem

M.

Riesz

Theorem on

. . . . . . . . . . . . . .

Conjugate

Theorem

Functions

. . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . Notations

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

262 267 269 272 276 278

CHAPTER

I.

Some basic

THEORY

results

OF RIEMANH

in the theory

here

for our later reference.

them

can be found

Sario,

Riemann

Riemannsoher

§i.

together

Surfaces

Fl~chen,

OF R I E M A N N

We refer

to

[AS]

A QUICK REVIEW

of R i e m a n n

surfaces

They are

with

stated w i t h o u t

complete

proofs

or in C o n s t a n t i n e s c u

referred

TOPOLOGY

SURFACES:

to as

[AS] or

for most

basic

definitions

not be given here,

e.g.

parametric

disk,

number,

surfaces

i.

IA.

Every in

structure regular in

R.

Unless

concerning

conformal

otherwise

etc.

Riemann

structure, In what

stated,

local

follows,

all Riemann

to be connected.

Every

region

and

in

R

creasing

R

in

Any sequence

number

a (sub-)region

(or

to have the c o n f o r m a l

D

in

compact,

R

is called

the b o u n d a r y

of n o n i n t e r s e c t i n g

a gD

of r e g u l a r

surface,

then there

regions

n = i,

Existence R

admits

and hence

regions

exhaustion

of

of a r e g u l a r a locally

that

R

of

analytic

components.

of r e g u l a r

for each

is called

A region

if it is r e l a t i v e l y

{Rn}n~ 1

a regular

IB.

disks

R.

is an open R i e m a n n

Rn+ I

R

is supposed

2,...

in and

R

exists

an in-

such that

CI(R n)

R : Un= I R n.

([AS],

12D)

is called

surface

R

has no compact

sequence

is included

set in

in

of a finite

R\ D

If

open

region

from that of

consists

Theorem.

II~

connected

induced

R

curves,

Ch.

and

R~nder

Exhaustion

domain)

D

intersection

surface.

are assumed

Ideale

SURFACES

which will

a Riemann

in Ahlfors

of

[CC] below.

surfaces,

denotes

either

and Cornea,

variable, R

are c o l l e c t e d

proof but most

R

n

in

R

having

shows

that

this

property

R.

exhaustion

finite

covering

can be r e g a r d e d

consisting

every Riemann of p a r a m e t r i c

as a polyhedron,

i.e.

a tri-

angulated

surface,

Theorem.

K = K(R)

hedron

if and

IC. of

K

(ii) An

Let

K

called every

has

I,

that

{Rn}n~ I

every

K

if

does

not

Riemann

which

P

a single

P

to

and

contour.

of

is the u n i o n

surface.

poly-

surface.

subcomplex

subcomplexes

belong

46A).

is a p o l y h e d r o n

and has

K

permits

Every

2A.

open

K

of

P

is and

n

if

Pn"

Then

the

a canonical

polyhedron

exhaustion.

K ([AS],

Let

R

hedron

K

regarded

Riemann

([AS],

2B.

surface ([AS],

the

looking

into

in p a i r s

= b. × b.

= 0,

I,

23D).

R

region

D

contour.

admits

1-simplex

Ch.

34A).

I,

properties

a regular

a. x b.

bi,

= i

exhaustion

or i n f i n i t e

and

a. × b. i

number

which

R.

sequence

for

in

K

is

gives

the

K

poly-

rise

to

canonical

can be o b t a i n e d

a canonical : 0

by

of this

This

is c a l l e d

HI(R)

surface

we d e n o t e

group

1-simplex

surface

of the g r o u p

1-dimensional

By IC the

homology

oriented

is c a l l e d

the

33B).

polyhedron,

on the

i

intersection

HI(R) I,

HI(R) , w h i c h

A finite ai,

and Ch.

Every

onto

by i

the

a regular

a single

1-dimensional

HI(K)

HI(K).

3

R

be the

Ch.

of

labeled

of

as an o r i e n t a b l e

([AS],

Thus

we m e a n

subregions.

group

as a s i n g u l a r

isomorphism

R has

surface

of c a n o n i c a l

HI(K)

an i s o m o r p h i s m

in

R \ D

be a R i e m a n n

can be r e g a r d e d Let

of

Groups

homology

= K(R).

l

(sub-)region

component

The H o m o l o g y

a × b

Pn+l

a subdivision

consisting

singular R

of

bordered

A finite

(i)

of c a n o n i c a l

I,

29A)

Corollary.

2.

of

polyhedron. if

Ch.

It is a f i n i t e

or c o m p a c t

is i n f i n i t e

}"

be an o p e n

By a c a n o n i c a l such

K \ P {Pn n=l

([AS],

polyhedron.

subcomplex

exhaustion

R

K = K(R)

is a c o m p a c t

of

simplex

Let

by

be an o r i e n t a b l e

sequence

border

= K(R)

R

a canonical

a canonical

Theorem.

Ch.

if

component

increasing

we d e n o t e

is an o r i e n t a b l e

only

is c a l l e d every

which

of c y c l e s sequence

in

by K,

a m. × a . 3

if

i ~ j, d e n o t i n g

by

]

of i - c h a i n s

a

and

b

([AS],

Ch.

I,

3iA). If hedron which

R

is a c o m p a c t

and t h e r e forms

exists

a basis

for

surface,

then

a canonical Hi(K).

K

is a f i n i t e

sequence

Hence,

ai,

d i m Hi(R)

bi,

orientable

poly-

i = 1,...,

g,

= d i m Hi(K)

= 2g.

The number If Cq_l,

g

R

q ~ i, then there exists in

i = i,..., of

is called the genus of the surface

is a compact bordered surface with

Hence,

The number

2C.

a canonical

contours

CO,... ,

sequence

ai, bi,

g, which, t o g e t h e r with all but one contours,

HI(K).

31D).

K

R.

q

dim HI(R) g

= dim HI(K)

= 2g + q - i

is again the genus of

Finally let

R

([ASJ, Ch. I,

R.

be an open surface,

so that

K

is an orient-

able open polyhedron.

Let

closure

is r e g a r d e d as a compact b o r d e r e d surface,

CI(D)

of

D

D

forms a basis

defines a finite p o l y h e d r o n sion to of

K.

K

be a r e g u l a r subregion of

K(CI(D)).

K(CI(D))

So we have a natural h o m o m o r p h i s m of R \ D

it

is a subcomplex

HI(K(CI(D)))

into

HI(K).

are compact, we see that the h o m o m o r p h i s m

is in fact an i s o m o r p h i s m and therefore that fied with a subgroup of

Since the

By applying a suitable subdivi-

if n e c e s s a r y we may assume that

Since no components of

R.

HI(K).

HI(K(CI(D)))

Turning to the region

is identi-

D

itself, we

have Theorem.

Let

D

be a r e g u l a r subregion of

is regarded as a subgroup of with one in In case that

HI(D)

HI(R)

R.

Then the group

HI(D)

by identifying every 1-chain in

D

R. D

is a canonical

subregion of

R, it is seen m o r e o v e r

is a direct summand of the free abelian group

HI(R).

See

[AS], Ch. I, §§31-32 for a d e t a i l e d discussion.

3.

The Fundamental Group 3A.

Let

mental group

0

be a point in

R, which is held fixed.

F0(R) , r e f e r r e d to the "origin"

The funda-

0, is defined to be the

m u l t i p l i c a t i v e group of h o m o t o p y classes of closed curves issuing from 0

([AS~, Ch.

I, 9D).

Every closed curve

as a singular 1-simplex. onto

y

from

the c o m m u t a t o r subgroup

phism of

y

Fo(R)/[Fo(R)]

[F0(R)]

of

F0(R).

By a c h a r a c t e r of an abstract group G

into the circle group

plex numbers of modulus one).

of

HI(R)

F0(R)

([AS], Ch.

G

F0(R)

to its h o m o l o g y class.

Under the natural h o m o m o r p h i s m the h o m o l o g y group

isomorphic with the quotient group

3B.

can be considered

We thus get a natural h o m o m o r p h i s m of

HI(R) , which takes the h o m o t o p y class of

Theorem.

0

is

modulo

I, 33D)

we mean any h o m o m o r -

(= the m u l t i p l i c a t i v e group of com-

The set of all characters of

G

forms

a group group by

with

thus

G*.

The

F0(R)*.

Let

such

§2.

4.

4A.

Let

s(z)

on

a

such

D

s

4B.

exists D;

and a

on

(ii)

on

an

s3

for

D, t h e r e

z

D

and

s(a)

=> ~

120

V

S

(i)

with

s

in

S

exists

an

s'

in

V; and

s

S.

A collection

if

-S = {-s:

tions.

The

character

surface

and of

every

F0(R).

reason

surface

An

extended

if

(i)

if

with -s

S

s E S}

pair

S

such

are

is g i v e n

V

S

has s"

and

with

A

A function

subharmonic.

on

Sl,

~ s3(z)

D

is c a l l e d

s2

in

for

disk

[e i t - z~

V

S

all

there z

with

in CI(V)

dt

a subharmonie

on

D

such

subharmonic

useful

by the

disk.

that

is a P e r r o n very

unit

functions

parametric

s(elt)Re

0

and

of e l e m e n t s

every

of

disk

is s u p e r h a r m o n i c .

superharmonic

and

(iii)

-~
0

for

a every

a.

be a r e g i o n

in

R

statements

are

equivalent:

point

a

There

exists

There

exist

is a r e g u l a r a barrier

and

let

a

boundary

be a p o i n t

point

of

a

with

for the p o i n t

in

~D.

D. respect

to the

D.

(c) harmonic

function

If the then

s

this

: lim H [ f ; D ] ( z ) z÷a

is not

l i m z ÷ a s(z) V

folZowing

(a)

region

which function

for

~ lim sup H [ f ; D ] ( z ) . Dgz÷a

lim H [ f ; D ] ( z ) z÷a A point

if,

22)

~D,

lim sup f(b) ~Dgb÷a If

regular

p.

a

5C. constant Theorem. tinuous

u

on

connected

is a r e g u l a r

A region positive If

D

function

a neighborhood V nD

such

component boundary

D

in

R

of point

V that ~D of

is c a l l e d

superharmonie

function.

is a h y p e r b o l i c

region

on

~D

of

is r e s o l u t i v e .

a u(z)

and ÷ 0

containing D.

([CC],

hyperbolic

in

R, t h e n

([CC],

a positive

p.

as

super-

z ÷ a.

a

is a c o n t i n u u m ,

p.

23)

if it c a r r i e s

every 27)

bounded

a non-

con-

5D.

Let

D

be a h y p e r b o l i c region in

R

the set of all r e a l - v a l u e d bounded continuous is a linear space over

~

p l i c a t i o n of functions. the s u p r e m u m n o r m every

f

in

function

H[f;D]

on

on

on

D.

functions on

ZD.

be This

It becomes a Banach space if we equip it with If(b)l.

By the p r e c e d i n g t h e o r e m

is r e s o l u t i v e and thus determines a h a r m o n i c D.

In view of Th.

Cb(~D,~)

5A, the map

f + H[f;D]

is a

into the space of bounded h a r m o n i c functions

D, which is o r d e r - p r e s e r v i n g , 0

Cb(~D,~)

under the usual a d d i t i o n and scalar multi-

llffl = suPbE$ D

Cb(~D,~)

linear map from

and let

i.e.

For every fixed point

a

is a bounded linear functional on

f ~ 0 in

D

Cb(~D,~)

on

~D

implies

the map with

H[f;D]

f ÷ H[f;D](a)

rH[f;D](a) i ~ JJfll .

By Riesz's r e p r e s e n t a t i o n t h e o r e m there exists a unique n o n n e g a t i v e D ~a' on ~D such that

regular Borel measure,

(i)

H[f;D](a)

= I

f(b) d ~ ( b ) ~D

for every H[I;D]

f

< i

in

Cb($D,~)

with compact support.

on

~D

solutions,

~D(~D)

The inequality

It may h a p p e n that the Dirichlet D p r o b l e m has only the trivia] solution. In that case the m e a s u r e s wa vanish i d e n t i c a l l y for all a in D. When the Dirichlet p r o b l e m has nontrivial

shows that

< I.

the measure

Da

is called the harmonic m e a s u r e

at the point

a

(with respect to the r e g i o n

A function

f

on

Theorem.

~D-summable. a

~D

is r e s o l u t i v e

D).

if and only if it is

If this is the case, then the equation

(I) holds.

([CC],

p. 28) We now slightly extend the d e f i n i t i o n of

H[f;D].

Namely,

be an extended r e a l - v a l u e d

function defined on a subset of

the b o u n d a r y

f0

We call

f

~D

and let

be the r e s t r i c t i o n of

r e s o l u t i v e with respect to

sense d e s c r i b e d above,

i.e.

f0

is

D

if

f0

H[f0;D]

the b o u n d a r y data

6.

H[f;D]

f

to the set a

~D.

in the

in

D.

to denote

of the Dirichlet p r o b l e m for the region

D

with

f0"

Potential Theory 6A.

Let

let

including

is r e s o l u t i v e

D - s u m m a b l e for some ~a

If this is the case, then we again use the symbol the solution

f

R

ga(Z)

In the rest of §2, = g(a,z)

R

denotes a h y p e r b o l i c Riemann surface.

be the Green function for

This is c h a r a c t e r i z e d by the following

R

with pole

a

in

R.

Theorem.

For e v e r y

harmonic

function

(AI) harmonic z

in

in

R

in any p a r a m e t r i c function

V, w h e r e (A2)

for

ga : H [ g a ; D ]

6B. we

a

u V

on

every on

V

exists

satisfying disk

V

subregion

center

the

of

R

positive

super-

conditions:

ga(Z)

with

D

a unique

the

with

such that

is i d e n t i f i e d

a

there

exists

= - l o g Izl + u(z)

open unit with

disk

< i};

we h a v e ([CC],

For a n y g i v e n

finite

positive

regular

f : | g(z,w) JR

d~(w)

Borel

a

f o r all

{Izl

a ~ CI(D)

D.

measure

p.

~

32)

on

R

set

U~(z)

for

z

in

R.

everywhere

We c a l l

infinite.

function

on

Moreover

we have

Theorem. only

if

does

not

A subset

U~

E

of

on

E.

A countable a condition

a subset

A

of

R

condition

holds

on

Theorem.

Let

u

A

If

U~ £ u R.

As

exists

sets

points

on t h e

closed

D

R

if a n d

potential

set

if t h e r e to

is

exists

+~

E ~ A

a

identi-

a polar

(abbreviated

of

Z.

35)

is a g a i n

set

of

in

Every

p.

it is

superharmonie

support

is e q u a l

superharmonic

p.

~, u n l e s s

set

~.

a polar which

a polar

at t h e

q.e.

([CC],

of

([CC],

of p o l a r

be a p o s i t i v e

on

R

closed

quasi-everywhere

except

u

6D.

on

union

if t h e r e

support

is c a l l e d

holds

a potential. everywhere

R

function

the

o n an o p e n

measure.

by

is a p o s i t i v e

outside

the

positive

generated

U~

is h a r m o n i c

intersect

superharmonic

say that

potential

is h a r m o n i c

by a u n i q u e

6C. positive cally

R, w h i c h

D

it t h e

So a p o t e n t i a l

A potential

generated

We

there

z + ga(Z)

set.

to q.e.)

such that

on

the

E. function

support

on

of

R

and

U~

~, t h e n

U~

we h a v e

the

37)

f o r the r e g u l a r i t y

of t h e

Dirichlet

problem

following Theorem. set.

Then

([CC],

p.

6E. u

on

R

Let

D

the

be a r e g i o n

irregular

in

boundary

R

such that

points

of

R\ D

D

is n o t

form a polar

a polar set.

42)

For a subset

E

we

function

define

the

of

R

and

a positive

R[u;E]

on

R

superharmonie to be t h e

function

pointwise

infimum

of p o s i t i v e

u(z) The

on

E.

(resp.

We R uE

function

Let

function

on

(a)

v

H[u;R\E] points

of

is not

R

of

be a c l o s e d

u

set

v = u~.

R\ E

s

on

R

such

that

s(z)

and to

relative in

Then

R,

the

u

on

E

E.

a positive

following

function

u

to

on

superharmonic

hold:

R, w h i c h

except

R.

at the

is e q u a l

irregular

to boundary

R \ E. is the

smaller If

6F.

smallest

than E

Theorem.

Every as the ([CC],

q.e.

positive on the then

is d e t e r m i n e d

Finally

uniquely

u

is c o m p a c t ,

E, w h i c h

R.

function)

and

functions

u~(a) = lim inf R[u;E](z) for all a in m z÷a R [ u ; E ] ) is c a l l e d the b a l a y a g e d f u n c t i o n

is a s u p e r h a r m o n i c

v

(e) on

E

on

(b)

§3.

set (resp.

the r e d u c e d

Theorem.

on

superharmonic

we

state

positive

set

uniquely.

the

for

some

([CC],

decomposition

superharmonic

function

on

R

that

E.

v : U~

sum of a n o n n e g a t i v e p.

superharmonic

p.

nonnegative

theorem

function harmonic

measure

43)

on

of F. Riesz: R

can be w r i t t e n

function

and

a potential

41)

DIFFERENTIALS

7.

Basic 7A.

Definition Let

R

be a R i e m a n n

differential

~

b dye

(resp.

2-forms

local

variable

are

on

= x

z

complex-valued

disks

Ve

and

R

is d e f i n e d c dx dy + lye, "

functions

V6,

surface.

on

for

that

with

each

(i)

z (V)

respectively

(resp.

as a c o l l e c t i o n

), one such

A first

of 1 - f o r m s

parametric

a~

and local

second)

and

(ii)

b

a dx

disk

V

(resp.

for any

variables

order

z~

two

+ with

c a)

parametric

and

z6,

we

have ~x a ~

aB : on

V

8x

~xB

N V 6.

(resp.

+ • ~Y~ D~Z-~6' Such

cdxdy).

a differential A differential

(or a c k - d i f f e r e n t i a l ) , are of class Basic as

follows:

~(x,y)

~ + b ~Y~ b8 = a ~ Y 8 ~YB

k : 0,

(resp.

on

i,...,

R

) = c~(xB'Y~)

is e x p r e s s e d ~

cB

symbolically

is said

if all

the

as

adx + bdy

to be of c l a s s coefficients

Ck

of

C k.

operations

on f i r s t

order

differentials

~ = adx + bdy

are

10

(a)

multiplication

(b)

complex

(c)

conjugate

differentia3:

(d)

exterior

differentiation:

is of class (e) for

b.dy 3

e x t e r i o r product: j = i, 2.

~b/ax

7B.

order

for

Let are

be a first

y

function

assumed

value

of the

each

being

arc.

borhood,

V

R.

result ~

independent arc

on

with

The defined

being

y

integration similarly.

value

again

borhood,

then

the

of line

ease

~3.

3

:

a . d x +

if

dm = 0,

(~f/~x)dx +

follows, Let

arc

variable

and

local

y

arcs

and

~ = a d x + bdy is i n c l u d e d

z = x + iy the

and

in

is

integral

fy m

subdivision

order

coordinate

of s u b a r c s and we

neigh-

yj

so that

The

integral

set

we choose.

by l i n e a r i t y . C0-differential

be a s i n g u l a r

coordinate

is d e f i n e d

number

parametrization

Yj

1-chain

A

and

in a s i n g l e

neighborhood

of the

let

(t))dt,

variables

a finite

of a s e c o n d

~ = edxdy

2-simplex

neighborhood

V

by a d i f f e r e n t i a b l e

with

map

in

local

(t,u)

is

R.

If vari-

÷ z(t,u),

we d e f i n e

ff A the

in w h a t

0 £ t £ i, t h e n

D

to any

Namely,

0 £ u £ t £ i, t h e n

parametrization.

(=

smooth.

If the

included

coordinate

in a s i n g l e

z = x + iy

R.

if

to be c l o s e d

~ : df

+ b(z(t))

into

independent

extended

it is i n c l u d e d able

when

C 2.

and

local

of b o t h

is not

we d i v i d e

is t h e n

if

Here

Y

of

is said

exact

= x(t) + iy(t),

is in a s i n g l e

the

~a/~y)dxdy

formula

If the

then

yj

f~ = (fa)dx + (fb)dy;

a2bl)dXdy

-

of c l a s s

(a(z(t))at(t)

the

~

f

cl-differential

by the

(alb 2

:

to be p i e c e w i s e

neighborhood t ÷ z(t)

is d e f i n e d

~i~2

be an arc on

order

by

d~ = ( $ b / ~ x -

It is c a l l e d

some

always

a coordinate defined

f:

9" = -bdx + ady;

cl-differential

= ~a/ay.

($f/~y)dy)

1-chains

functions

~ = a d x + [dy;

CI;

A first i.e.

by n u m e r i c a l

conjugation:

being When

we h a v e

m =

ff O £ u £ t £ 1

independent A only

integrals,

is not

.

c v z ) ~

of the

the

choice

included

to a p p l y

dtdu,

a suitable

result

of

local

in a s i n g l e

being

variables

coordinate

subdivision

independent

of

to

A

and neighas

in

subdivision.

11

The integral of

~

over any singular

arity, where each simplex definitions

any

first

8.

The Class 8A.

order

Let

+ Ibl 2) dxdy

cl-differential

on

R

and

any

singular

and its Subclasses

~

be a first order cl-differential.

II~II = II ~ * F I : FI(R)

with

function

co

is a second order cl-differential.

We denote by

2-chain

×,

~*

= (lal 2

Then

We set

= II (lal2+ IbI2)dxdy"

the totality

11~ll < ~.

(2) defines

With these

formula

F

(2)

co

is assumed to be differentiable.

we have the Stokes

for

2-chain is then defined by line-

Then

a norm.

F1

of first order cl-differentials

is a linear space,

Indeed,

in which the

this norm is induced

from the

inner product (3)

(~I'~2)

for

~i' ~2

in

F I.

For a set

the set of conjugate in

FI

(resp.

if

analytic)

We denote by

d~ = 0

differentials

FI

(resp.

8B. face

FI(R)

belongs

the bomder in

C2

R

Theorem. (resp.

the class of closed

is the interior of a compact bordered

sur-

(resp.

F I.

R

is the set of are of class

have the obvious meaning.

C I up

We say

e

and that

with

f = 0

~ on

F~0(R) belongs

~R.

(3), which defines

FI(R) *) in

FI(R)

whose coeffieients

FI(R)

to the class

F~0(R) C

It is

dif-

on

and

DR

inner product

are closed.

exact)

C

f

FI

to the class

first order differentials m

~*

A*

A differential

The set of all harmonic (resp. FI). is denoted by F hi a

Then we use the following notation:

to the border. that

and

A.

co* = -i~. in

F I)

~

we denote by

in

e

belonging

Suppose that

R.

of differentials of elements

if both

and

C

ferentials

A

differentials

is called harmonic

called analytic

= IS ~I~2"

(resp. FI(R).

if

~ • F~(R)

and

~ = 0

along

to

FI~(R) if ~ = df for some eu i The class F (R) is equipped with the the o r t h o g o n a l i t y

F~0(R)) ([AS],

relation there.

is the orthoeomplement Ch. V, 5A)

of

FI(R)*e

12

8C.

If

R

is an o p e n

Riemann

F~0(R)

(resp.

compact

support.

In this

Theorem.

FI e0 in

is o r t h o g o n a l

of

F e0 I *

norm

FIe0 : F~0(R))

FI .

ease

([AS],

8D.

We d e n o t e

(2),

so that

by F

the

F

fundamental

Theorem.

the

If

result

BE. shall

Among

use the

direct

basic

sum in the

(4)

and

formulae

following

to h a v e

hood

define

respect Fe

to the

CI(FI), e

=

Fe0 the

Fh

is the

%

c* '

following Fhl

such

([AS],

decomposition

denotes

F

the

that

Ch.

of

V,

9A)

F, we

orthogonal

= re0

re0 ~{

•

we h a v e

If a f i r s t V

order

Cl(rl).

:

([AS],

Ch.

V,

10C)

C

if a.

cl-differential,

of a p o i n t

a singularity

of

FI = cO with

and D i f f e r e n t i a l s

neighborhood

harmonic)

We

with

there exists a ~i in i Fh can be i d e n t i f i e d .

C

9A.

FI

class

in w h i c h

r

Cycles

F~)

is the o r t h o c o m p l e m e n t

of

space.

the

~ rc

(5)

9.

by

(resp.

F:

r = Fe0

As a c o n s e q u e n c e ,

FI c

for o r t h o g o n a l

later,

space

we d e n o t e FI c

: (Fe*) ± , w h e r e the c l o s u r e and in the space F. We set Fh = F c n

, then

Fh

in

Fc0

concerning

~ E Fc~Fc*

and

completion

is a H i l b e r t

~lll = 0; n a m e l y ,

lJw-

F I* e 6C)

V,

= CI(F ~0 ) , F c : (Fe0*) ± , and orthocomplementation are t a k e n The

then

~

we have

to

Ch.

surface,

set of

~

at

a

a.

is a n a l y t i c

It is c a l l e d

in The

R

say

except

singularity

(resp.

removable

R

is d e f i n e d

a, t h e n

is c a l l e d

harmonic) if

~,

at

in some

~

analytic

deleted

is c o n t i n u e d

in a is said (resp.

neighbor-

to the

point

a

as a c l - d i f f e r e n t i a l . Take

a parametric

consider

a holomorphic

fdz

some

for

local

variable

disk

function

The

set

~a(~)

order

on f

on

about

a point

V \ {a}. V \ {a},

a

in

So we h a v e i.e.

in t e r m s

R

= inf{k:

c k ~ 0}

is i n d e p e n d e n t

of the

[

ckzk,

and call choice

9 =

0 < Iz[ < 1. it the

of local

order

of

variables.

~

at

and

of the

we h a v e

f(z) : We

< i}

differential

holomorphie z

V = {Izl

a.

Moreover,

13

the c o e f f i c i e n t the

C_l , the r e s i d u e

local v a r i a b l e s

and is d e n o t e d

Resa(~) for

any

point

cycle a.

0.

The

ential

on

R

V

by

a

Res a (~). = ~

is

subset

with

In fact,

number

1

removable

of

we h a v e

with

if

and

If

~

only

if

such t h a t

the

Va(~) cl-differ -

singularities exists

to

may be con-

is a c l o s e d

R, t h e n t h e r e

singularities

respect

singularities

of a n a l y t i c

of

independent

y

fact:

with a finite number

T

a, is a l s o

with prescribed

of the f o l l o w i n g

differential

at

winding

at

off a compact

~

= C_l

having

differentials

by m e a n s

holomorphie monic

in

singularity

Analytic

structed

Ch.

y

of

and is

a unique

T - ~ E Fe0

har-

([AS],

V, 17D).

9B. p, q

We c o n s t r u c t

be two d i s t i n c t

Suppose metric

first that t h e s e disk

positive Let

v

V = {Izl

numbers

e ~ i

define

p,

rl, r 2

= d(ev)

we use the r e s u l t T - ~ E Fe0.

q

q

for

and

and

which

c

and

to c y c l e s

an arc

I~lJ,

I~2[

of the f u n c t i o n let

e

E 0

on

and c h o o s e

log{(z-~2)/(Z-~l)}

R \ {Izl

on

< r2}.

= 0

R

such

F u r t h e r we

otherwise

y

in

p

for

R.

Then

~

with

differential

not p a s s i n g

T = 2~i(c x y).

I

Let q.

< r I < r 2 < i.

be a C 2 - f u n c t i o n

Izl < I, and

1-cycle

R. to

~ = { ( z - ~2 )-i - ( z - ~ l ) - l } d z

by s e t t i n g

for any

on p

in a s i n g l e p a r a -

~2 = z(q),

in 9A to find a u n i q u e h a r m o n i c

Thus,

joining

are c o n t a i n e d

satisfy

and

rI £

related c

~i = z(p),

branch

{IzI £ r I}

a differential

R

Set

r I < Izl < i

on

Izl < rl,

in

< i}.

be a s i n g l e - v a l u e d

in the a n n u l u s that

some d i f f e r e n t i a l s points

and

q, we h a v e

([AS],

Ch. V, 19C)

Y If metric

p,

j = i,..., within Tj T and

q

disk,

and

c

n, so t h a t

a parametric

for e v e r y

are not n e c e s s a r i l y

t h e n we d i v i d e

j

is a h a r m o n i c

each

number

of s u b a r c s

cj,

ej

joins a p o i n t

Pj-I

to a p o i n t

pj

For e v e r y

P0 = p

and set

differential

T - ~ E Fe0.

in a single p a r a -

into a f i n i t e

disk with

as a b o v e

contained

c

with

1-cycle

and

Pn = q"

~ = [j ~j

and

singularities y

We d e f i n e T = [j ~j.

o n l y at

not p a s s i n g

p

~j

and

any of the

and

Then q, pj's,

we h a v e

I

T = I "f

The d i f f e r e n t i a l

y T

Q = [ I ~. = [ 2~i(c. x y ) = J Y J j is seen to d e p e n d

2~i(cx7).

o n l y on the h o m o l o g y

class

of

c.

14

We w r i t e

T(c)

~(c) Then,

in p l a c e

¢(e)

= ~ ( c ) + ~--~-~. at

p

The

We

can

We

with

e

has

iT--~*)/2.

differentials

only

-i

set

: (T--~+

analytic

¢(c)

construction

our

works

~(e)

definition

Suppose

Moreover,

to

we

~(c)

residues

by l i n e a r i t y .

C F h.

and

are

see that q

is a cycle.

T(c)

~(c)

differentials

extend

belongs

Further

on

R

two

singularities,

i

respectively,

and

and

T(c)

simple and

~(e)

holomorphic.

above

1-chains c

and

and

is e v e r y w h e r e

resulting

T.

: (T(c) + i T ( c ) * ) / 2

both

poles

of

Fe0 ,

that

of

T(c), has

~

belongs

in the

~(c)

%(e) this

since

T(e)

even

and

case

are %(c)

no

p = q

compact

to

F h N Fc0

the

and

to a r b i t r a r y

case.

has

then

holomorphic.

~(e)

singularities

is the

and

everywhere

if and

Then

support

¢(c) and

only

E Fa

since

if

and T(e)-

(= Fh0 , by d e f i n i t i o n ) .

By s e t t i n g

~(c) = !~T(e), we get the Theorem.

following If

harmonic

c

is a c y c l e

differential

on

o(c)

R, t h e n

in

Fh0

there

such

exists

a unique

real

that

(~,~(c)~) : Ic for

each

~

in

F . e

(6)

If

c

(~(c),~(y)*)

and

y

= I

are

~(c)

any

cycles

on

= c x y.

R, t h e n

([AS],

Ch.

V,

20)

Y iO.

Riemann-Roch 10A.

Here

Fix a c a n o n i c a l intersection n = i,..., harmonic

we

consider

basis

numbers g

(see

~(A n) m

a compact

of 1 - c y c l e s vanish

2B).

differentials

IB and

Theorem

except

By our

Riemann

An,

Bn,

the

cases

observation

and we h a v e

= A n × B m : @mn'

for all

IA

~(Bn) m

surface

n = i,...,

R

of g e n u s

g.

g,

for w h i c h

all

A n × B n = -B n x A n = i, in 9B m,

~(An),

~(B n)

n

= Bn × A m : -@mn'

are

15

S

A

Theorem.

(a)

a(A n)

and t h e r e f o r e (b) periods,

dim

a(An)

: I

o(B n) B

m

and

O(Bn) ,

= 0.

m

n = i,...,

g,

form a basis

for

Fh

F h = 2g.

dim

F a = g~ and

i.e.

(fA 1 ¢ .....

every fA

~

in

9).

Fa

([AS],

is d e t e r m i n e d Ch.

V,

by

its A-

24A-B)

g 10B.

A divisor

D

on

(7)

R

is a f o r m a l

finite

sum

D = n l a I + --- + nkak,

where

a. are p o i n t s J is c a l l e d a p r i n c i p a l funetion

at the

f,

i.e.

point

a,

in

R

and

divisor

D = (f)

n.

are

= ~aER

Va (f)a'

to be

inf{k:

is d e f i n e d

integers.

if it c o m e s

from

Such

a nonzero

where

Va(f),

c k ~ 0}

a divisor

D

meromorphic the

with

order

f(z)

of

f

=

~k ek(z - a)k" Let

D

divisors) tively. the

R.

The

makes

that

R. by

R

deg

function

We also

is a n o t h e r

(~)

on

for all

R

functions

f

dimension

of

divisor

meromorphic

of the

sueh

belongs

to

Hence

addi-

group

form

(7)

function

sphere,

that

we

deg

on

on see

is a

= (~'/~)

of

D2

L(D)

that

(7)

(f) by

defined

e'/e

E DO.

the

D I- D 2 space

by If

Thus,

(~)'s

belong

class.

integral

if

n.j => 0 For

is i n t e g r a l .

of all m e r o m o r p h i c

is a m u l t i p l e dim D, w h i c h

~

in 9A.

is a m e r o m o r p h i e

canonical

is e a l l e d

if the

differential

of

D.

depends

The

only

complex

on the

D.

divisor

~(D)

class

of the

Riemann

was

then

called

form

by

is d e n o t e d

D

in

differentials

a fixed

class

denote

R

L(D)

we t a k e

Z = (an).

we

~a(~)

differential, (e') - (~)

D

D

principal written

D/D 0 .

of a m e r o m o r p h i e

divisor

divisor

therefore

where

is a m u l t i p l e

of

of the

and

divisor

D

every

DO

ma (~)a'

on

class

For

group

so t h a t

DI

surface

in

group

meromorphic

the

same

divisor

nonconstant

= [aER

A divisor j.

D

meromorphie

and t h e

10C.

quotient

define

~'

covering

every

the

D, of a d i v i s o r

every

(resp.

Abelian

is c a l l e d

deg

Since

divisors

is a free

D/D 0

degree,

for

set of all

DO)

group

a complete

D = 0

formula

every

The

on t h e

funetion

(resp.

[j n..3

the

to one

D O ) be the

D

quotient

surface

is d e f i n e d R

(resp.

on

nonzero if and dim

e

D

we d e n o t e

such

meromorphic only

~(D)

that

(~)

by

~(D)

(~/~0)

= dim

( D - Z).

space

is a m u l t i p l e

differential

if

the

s0

on

is a m u l t i p l e

of

of D.

If

R, t h e n of

D- Z

with

16

Theorem

(Riemann-Roch).

divsor

Z

For

every

D

in

D

and

every

canonical

we h a v e

(8)

dim D = dim

(-D-Z)

- deg

D - g + i. ([AS],

10D. = dim

By s e t t i n g

Fa = g

D = -Z

(see Th.

10A),

69) for

in

canonical

Theorem. pole

There

at a n y

other

Proof.

any

(8)

that

This,

together

with

(8),

implies

function

point

We c a n p r e s c r i b e

in

R.

([AS],

further,

be a n y

dim

~(0)

in

Ch.

J

subset such

from

we h a v e

V,

f

which

has

the

a simple

location

of

28B)

we n e e d

infinite

a I .... , ag

funetions,

fact

27A)

Z = 2g- 2

Z.

as well.

J

the

a meromorphic

al,... , ag

Take

by

f

Let

points

constant + i

of

To p r o c e e d

Theorem. tinct

exists

prescribed

poles

10E.

divisor

using

V,

we have

deg each

(8) a n d

Ch.

J.

1 ~ dim

of

R.

that

Then

dim

Sinee

there

exist

9(al+.-.+ag) L ( - ~Ij~s-

(-~j~l- a~)j : d i m

dis-

= 0.

a.)~o c o n t a i n s ~(~j~l- a~)j + s - g

and t h e r e f o r e s

(i0)

d i m ~(

We now prove nothing first

the

to p r o v e ,

theorem

for

~(0)

then dim

by

_> g - s .

induction.

= F a = {0}.

If

g = 0, t h e n

Suppose

that

there

g > 0.

is

We n o t e

that dim

for

[ a.) j=l ]

each

a

in

g ~ dim

R.

~n fact,

(a 0 - Z)

= dim

(a 0 - Z) - deg

meromorphic the point conformally

( a - Z)

onto

~(a)

~ g- i

if t h i s

were

not

~(a 0) ~ d i m

(-a 0) - g + i = 2.

function a 0.

= dim

f

on

As n o t i e e d the

R

with

in 10B,

Riemann

Thus

sphere

F a = g. there

only

this

the

case

By

(8)

would

implies

that

g

some

dim

a0,

(-a 0) =

be a n o n c o n s t a n t

one pole--a

and thus

for

simple f

should

pole--at

would be

map

zero,

R

a

contradiction. Thus, take

for

a nonzero

s = i, w e h a v e

d i m ~ ( a I)

= g - I.

~i

Since

zeros

in

~(al).

the

of

If ~i

g > i, t h e n w e is f i n i t e

in

17

number

and

J

is i n f i n i t e ,

we can find

a2

Then

d i m ~ ( a l + a 2) ~ d i m 9(a I) - i = g - 2.

gives

us

d i m ~ ( a l + a 2) : g - 2.

f i n i s h t h e proof.

10F.

Let

..., ag aj,

in

91,... , %g R

g.

ii.

Cj : fjdz

Then

Cauchy IIA.

{Aj, Bj: R 0 : R\

and

Kernels Let

R

in w h i e h

the

is c l e a r

of

dim ~(al+...+a (fj(zk))

}l(a2) (i0),

~ 0.

this

this a r g u m e n t

to

F a.

g

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

Take d i s t i n c t

disk

{Vj,

) = 0

z}

points

al,

for each p o i n t

if and only if

~ 0,

z k : z(ak).

on C o m p a c t

Bordered

Surfaces

be a c o m p a c t R i e m a n n

j = i,..., {%gl(Aj

which

one p a r a m e t r i c

det where

with

We h a v e o n l y to r e p e a t

be a b a s i s

and c h o o s e

j = i,...,

J

[]

We add one m o r e r e s u l t ,

Theorem.

in

Combined with

g}

be a c a n o n i c a l

UBj)}.

R0

surface basis

can be a s s u m e d

single boundary

contour

of g e n u s

g

of 1 - c y c l e s

on

and let R.

We set

to h a v e a c a n o n i c a l

form~

is of the f o r m

A I B I A ~ I B ~ I . . . A g B A - I B -I gg g' where

A[I (resp. ] g i v i n g the o p p o s i t e Let an are

tial and Fa

p, q e

in

¢(c) q.

B[ I) is the c u r v e o b t a i n e d f r o m ] o r i e n t a t i o n ([AS], Ch. I, 4OD).

be d i s t i n c t R0

on

joining

R

in 9B imply t h a t

p

q

and

with resides ¢ ( c ) - 91

is d e n o t e d

by

-i

p'

and

p'

A. and B.. Take ] ] an a n a l y t i c d i f f e r e n -

singularities

¢(c).

o n l y at

differential

The p r o p e r t i e s

of

_, r e s p e c t i v e l y - - a n d

has

p

¢i

in

¢(e)

has o n l y two p o l e s - - s i m p l e

is seen to be i n d e p e n d e n t

be a s i m p l e q'.

and

and

B.) by 3

not on any

to find a u n i q u e as

(resp.

poles

at

zero A-

of the c h o i c e

of

c

and

~p,q.

Then

is an a n a l y t i c at

R

and d e f i n e

(~(c) has

¢ ( e ) - 91

= ~p,q, y

So

10A,(b)

Next we t a k e d i s t i n c t

Let

q

the same A - p e r i o d s

noticed

periods.

in

to

as in 9B.

T h e n we u s e Th.

w h i c h has

points p

A. J

p, q, p' , q '

m = Cp,

q,

arc j o i n i n g ~

Let

p

to

is s i n g l e - v a l u e d

differential

q'.

points

Z0

on

R0 \ y

be the t o t a l

and

~ :

q

within

on

in

fz

R0 \ y

R0

and

set

9. R0

and not p a s s i n g

and t h e r e f o r e

with possible

~

singularities

sum of r e s i d u e s

(see 9A) of

only ~m

18 +

o v e r the r e g i o n (resp.

R 0 \ y.

right-hand)

Denoting

b a n k of

2~iE 0 = [ ] Since

¢(z +) - ¢(z-)

e q u a l to periods y'

%

joining

p'

¢~ + r ~R 0

We a l s o

and

~

to

E0 = R e s p , ( } m )

within

q

and

y

~y'

y'

Let

[

We d e n o t e

liB.

by

conjugate

of

R

@P,q

be a c o m p a c t R' (see

function

there

g' p o i n t s

dh

exist

has n e i t h e r

zeros

is

all the A-

if we t a k e any arc

- ~(p')

= [

%p,

y,

arcs on

R0

,q

,.

with

~y =

= I y CP',q''

bordered

= RU DRU~ [AS],

surface with nonvoid

the d o u b l e

Ch.

II,

3E).

In o r d e r to d e f i n e the C a u c h y k e r n e l meromorphic

because

side

Then

I y,

DR.

= ~(q')

be n o n i n t e r s e c t i n g

: q' - p'.

y, we h a v e

R 0 \ y, t h e n we h a v e

,(~)

Theorem.

Let

/3R0 {~ = 0

On the o t h e r hand,

the f o l l o w i n g

and

on

s e c o n d t e r m of the r i g h t - h a n d

So we h a v e p r o v e d

q- p

z

z ) the l e f t - h a n d

(¢(z+) - ¢(z-))w.

see that

+ Res

(resp.

J.y

vanish. q'

z

at the p o i n t

= 2~i, the

2%i fy ~. of

y

by

h

on

R', w h i c h

al,... , ag, nor p o l e s

in

at

of

R, w h e r e

boundary

R

is the

Let

g'

for

R, we t a k e a n o n c o n s t a n t

be the g e n u s

is h e l d fixed. R

Then,

of

R'.

by Th.

10E,

such that the d i f f e r e n t i a l

al,... , ag,

and

such that

d i m ~ R , ( a l + - . . + a g , ) = 0, where Let

~R,(.) {Aj, Bj:

such t h a t IIA.

denotes

the

space

j = l,...,,g'}

R 8' = R' \ { u j g I ( A j U B j ) }

We m a y also a s s u m e

that

dh

as w e l l as all the p o i n t s

Th.

1OA we c h o o s e

fA

9(-)

~n = 6mn

for

a basis

for the s u r f a c e

be a c a n o n i c a l

R 0'

R'

(see 10C).

basis

of 1 - c y c l e s

on

is a c a n o n i c a l

f o r m in the

sense of

contains

al,... , ag,

{~n: n = I,...,

m, n = i .... , g'

all the

zeros

and p o l e s

in its i n t e r i o r . g'}

of

R'

Fa(R')

We fix one p a r a m e t r i c

of

By use of such t h a t disk

m

{Vj, z} terms

of the

we w r i t e det

around

each of the p o i n t s

local v a r i a b l e s

z k = z(a k)

(hj(Zk))

~ 0.

for

Since

chosen

al,... , ag, as

k = i,..., d h ( a k) ~ 0

and e x p r e s s

}n = h n ( Z ) d z g', for

on e a c h

t h e n we have, k = I,...,

by Th.

%n

in

Vj.

If

10F,

g', we see that

19 @j(ak)/dh(a k) : hj(Zk)/dd-~hz(Zk) are well-defined

and that the matrix

$j (ak) ] j,k:l ..... g' is nonsingular.

So a basis

{~ n

:

n = i,...,

g'}

of

F (R')

is deter-

a

mined by the formulas ' ¢:(a~) ~ k:l dh(a k)

:

(li)

$

CJ for

j : i,...,

from

g'.

If we take three distinct

and draw disjoint

such that

2y' = p - p'

is holomorphic

R~ \ y".

p ~ q, q"

shows that

and

we set

z

F(p,q), q

and

by u s i n g

:

(12),

, q

in R~

p,

p'

IIA implies that

,, Sp, ,p'

defined by either

side of this equa-

and holomorphic

is holomorphic

Differentiation local variables

@q,,,q(p) = f(z,q",~)dz

3 Then,

is distinct

within the region

points

7"

in

jointly

of

is locally holomorphic C

which

in the form

R 0' \ y'

in F(p,q)

q ~ p'.

Cq,,q(p)

Denoting by tively,

in

It follows that

and

~y" = q - q", then Th.

p, Sq",q

we see that the function

p

y'

writing this

(12)

when

arcs

and

Sy, ~q,,,q : Sy,, ~p,,p.

tion,

q" E R N R 0'

We now fix a point

al, .. ., ag,.

R~ \ {q"}

~k

F

in

for

p

in

in p

and

with respect q

p

when and

q to

q ~ p, q".

q, respec-

and also

J we h a v e

3

J

fA +IIp •

IA

p'

.

q)d p!

J

=I

A. ~ ]

the last equality vanish.

As

p

1

q" Cp,,p d~ =

I

A. Cp''P = 0, ]

sign being valid because

is arbitrary,

we conclude

all the A-periods that,

of

for every fixed

$p,,p p

in

20

the region

R N R~,

fA ~q~q,,

(13)

'q

=

0.

J

A similar computation

shows that

Ip'

B. ]

'

B. ]

As we shall see below in IIC, an application

'P" of Th.

IIA easily

shows

that

f

(14) for

j = I,...,

g'.

@p, : -2~i @j B. 'P p' 3 Thus we have arrived at the following:

fB. ~q~q"'q :-2~i%j

(15) for

j = i,...,

g'

We finally set ~(p,q) Since

q"

and all

everywhere on

R

on

+i

Cauehy kernel, ~(p,q)

aj

at the point

we claim that,

q.

of the differential (13) follows

=

B. J

in

jointly

and

q ÷ ~q~(p,q)

(14) and

~q(~q"

'q

in

p

p, the function p

q q

has a simple pole

R 0'

in

along

A~~

easily that the period along any

by use of (ii),

~qm(p,q)

is holomorphic

To finish the construction

To see this , we fix

First,

B. J

p ÷ ~q,,,q(p)

for every fixed

the periods Secondly,

%q,,,q(aj)

is holomorphic

Moreover,

is single-valued. from

j:l R,

%q,,,q(p)

p ~ q.

~ ' ~q,,, dh q(aj)~j(p).

-

belong to

R, while

except at

with residue

= @q,,,q(p)

of the q ÷

and compute and A. J

B~. vanishes.

(15) we get

(P)) -

k=l

dh

(ak)mk(P)

~' ~j = -2~i[~j(p)

-

~-~(ak)~ k(p)]

: O.

k=l Hence,

~qW(p,q)

valued function.

has only null periods

and so

Summing up these observations,

which is the main objective

of this section.

q ÷ ~(p,q)

is a single-

we get the following,

21

Theorem. Riemann

There

(BI) tial

exists

surface

on

R

a differential

such

for e v e r y R

with

(B2)

q

only

R,

p

in

and

set

of

q ÷ f(z0, q)

is a m e r o m o r p h i c

with

residue

IIC.

within

the

verify

ourselves region

point,

distinct

b,

fB ~ p ' , p

in

To p r o e e e d

Aj,

Bj

except

B"

b"

A I.

= B \ B'

(i)

vertex

differen-

residue

variable

z

+i;

in a

z 0 = z(p);

having

a simple

let

V

then

pole

at

3B"

B,

at a s i n g l e

R~, (14)

slightly say

and

(ii)

is equal

B \ [b} to

show

~i"

be a p a r a m e t r i c

take

of

to

BI

curve, AI

of the p o l y g o n of

of g e n e r a l i t y ,

move

with

side

loss

p'

does

the

end p o i n t s

first

intersects

= -2~i 'P

CI(V)

We

closed

only

~p,

Without

a simple

left-hand

B

We t h e n

we h a v e

j = i.

B

the

containing

be the

case

we h a v e

further,

closure

B, r e s p e c t i v e l y , and

with

with

R

(14).

to o b t a i n

Then

f its

q

a local

on

equation

f r o m any

R~.

and t h e r e f o r e

such t h a t

is a m e r o m o r p h i c

pole--at

= f(z,q)dz

function

to the

R' \ y'

the c o n d i t i o n s :

entirely

simple

~(p,q)

the

satisfying

lies

bordered

+i.

We now

we r e s t r i c t

on a c o m p a c t

p ÷ ~(p,q)

R, we c h o o s e

neighborhood

p

p

in

one p o l e - - a

for e v e r y

~(p,q)

that

not m e e t small

point B'

arcs b

= b " - b'

and

A'

and

so that

disk

y'

the

B'

of

AI

and

in

= b' - b".

fact

at

all

contained

~B'

F r o m the

centered also

V.

b cycles and

Let

Thus,

B = B' + B"

b'

setting

follows

that

B

We n o w c o m p u t e First

we note

from

y'

'P

the two that

So Th.

the IIA

B'

integrals arc

B"

implies

to deal

a modified deformation the

same

with

polygon

the

V, the

end p o i n t s

such

of arc

that

' R 0. A' A"

number

over

member

polygon

R~

separately. and

is d i s j o i n t

p, ~b',b"" B',

on the

In fact is c h a n g e d

curve

+i

Setting

= (AIXA') + A" , we c o n s t r u c t

with our

other

by m e a n s

is d i s j o i n t

is a c l o s e d Ai

of w i n d i n g

'P

right-hand

in the

b' ~ P ' ' P =

integral

in p l a c e

within

on the lies

B"

that

B" ~ P ' ' P = In o r d e r

'P

to a n o t h e r from respect

hand,

we use

of a c o n t i n u o u s

B'

arc and

A"

with

A' - A "

to the p o i n t

new polygon,

' out of R I,

b".

22

fA~ ~p,,p

analytic

differential

on

residue

+i

and

AI

' A I.

R 0' just by replacing the arc y', we have

by

R'

V

= fAl ~p,,p having

-i, respectively,

A{, A2,... , Ag,.

the cycles

Since

does not intersect

: 0.

Thus,

Let

simple poles and having by Th.

Cb", ' b'

at

b'

IIA applied

be the

and

null periods

with

b"

with

along all

to the polygon

R~, we have

I

B' CP' ,p = I .¥, Cb",b'

and hence

B

'P

B'

B"

Our construction of Cb",b' larities and has null period

'P

'y'

'

shows that @b' ,b '' + ¢b",b' has no singualong A2,...Ag,. As for the period along

AI, we have

f

!

f

AI ((~b',b" + Cb",b' ) =

=f Since the differentials we eonclude

in

that

,

AI Cb" ,b'

A' -A" ~b" ,b'

F a (R') T

f

=

A I @b",b'

~b',b" + ~b",b'

:

f

A~ Cb" ,b'

-2wi.

are determined

by their A-periods,

-2~i~i , as was to be proved.

[]

NOTES Most results and Chapters information results

come from Chapters

i, 3, 4 of Constantinescu can be obtained

in 10E,

from Kusunoki

and Cornea

from sources

10F and in Subsection

[42]

I, II, V of Ahlfors

(see also Hurwitz

[CC],

indicated

and Sario

so that more

respectively.

ii (Cauchy kernels) and Courant

[AS] The

are taken

[35] for this matter).

CHAPTER II.

M U L T I P L I C A T I V E A N A L Y T I C FUNCTIONS

In our later d i s c u s s i o n of Hardy classes an important role will be played by a certain family of m u l t i p l e - v a l u e d multiplicative

analytic functions.

analytic functions having

analytic functions called

These are defined as m u l t i p l e - v a l u e d

s i n g l e - v a l u e d modulus.

The first objective

of this chapter is to give a precise d e f i n i t i o n to such functions. fact, we will define t h e m in terms of two equivalent notions: bundles and characters

of the f u n d a m e n t a l group.

that, on a compact b o r d e r e d Riemann surface, n o n v a n i s h i n g bounded h o l o m o r p h i c observe the order structure

sections.

In

line

We shall then show

every line bundle admits The second purpose

is to

in the space of h a r m o n i c functions,

leading

to the so-called i n n e r - o u t e r f a c t o r i z a t i o n of m u l t i p l i e a t i v e analytic functions.

§i.

M U L T I P L I C A T I V E A N A L Y T I C FUNCTIONS

i.

The First C o h o m o l o g y Group IA.

Let

{Vi: i E I} {~i~}

over

modulus one, with

R

be a Riemann surface.

of V

R, we denote by with values

i.e.

V. N V. ~ ~ l 3

~ij

in

~

of complex numbers of

is a s s i g n e d to every pair

i, j

in

I

6ij<jk : ~ik V inv.3 N V k I ~.

Two l-cocycles

said to be e q u i v a l e n t i

in the group ~

V =

the t o t a l i t y of l-cocycles

in such a way that

(i) if

For every open covering

ZI(v;~)

in

I

such that

{~ij}

{~ij} , in

and

if there exists an element ~j

= 8~l~ij@j

if

Vi AVj

@i ~ ~.

is then defined to be the set of e q u i v a l e n c e classes

in

ZI(v;~) T

The set in

HI(v;~)

ZI(v;~),

which forms a group with respect to the p o i n t w i s e m u l t i p l i c a t i o n , {~ij]{~ij} = {~ij~ij}. Next, take any two open coverings {V~,:

i' @ I ~ }

fining map

~.

of

R

such that

V'

V = {Vi: i E I} is a r e f i n e m e n t of

We may regard the refining map

~

are

for each

and V

i.e.

V' with re-

as a map from

I'

24

into

I

so that

ZI(v;~).

If

the c o v e r i n g to p r e s e r v e

]'

=D V'i' A V!]' ~ ~

V'.

choice

of r e f i n i n g

that

varies

of

with

lB.

Theorem.

Proof.

The p r o o f

definitions. consists in

V of

admits

coverings

V, we m e a n

a finite

Vi(v) N V i ( v + I )

is c a l l e d

obtained order;

from

e.g.

y

of

c

if t h e r e

for e v e r y When

~.

(a) Let

~ E HI(R;~) c

of

~

and

R

case,

faithful

in a s i m p l y R

if

covering

So we r e s t r i c t

elements

c

on

..a m

of

V2,

it is

ourselves

covering,

m-l.

in

V

if it is changing

{VI,V2,V3}.

the On

we m e a n a p a r t i t i o n

say

oi,... , am, of

c

with

c : al..-o m

y, each s u b p a r t i t i o n

so

has an ob-

is said to c o v e r c

say

A c h a i n of

without of

R

arcs,

V

VI,

of e l e m e n t s

~ : I,...,

of a p a r t i t i o n •

(i)

and that e v e r y o p e n

y = {Vi(1),...,Vi(m)}

c = oI

some

connected

By a c h a i n of a g i v e n

of an arc

R.

is s e p a r a b l e ,

is a s u b d i v i s i o n

to

of

(it) for each t r i p l e

Since

we say that of

F0(R)

We b e g i n w i t h

is c a l l e d

of c o n s e c u t i v e

is a d a p t e d

and

group

steps.

~ = {Vi(1) , .... V i ( m ) }

First we d e f i n e

and

with

some of the

a partition

to some s u b d i v i s i o n

limit of this

isomorphic

is f a i t h f u l .

A subpartition

In this

c : ~l..-Om

adapted

of

of a c h a i n

number

A chain

exists

when

is c a n o n i c a l l y

for e v e r y

by r e p e a t i n g

c : a l a 2 - - - a m.

s y s t e m of g r o u p s

y : {Vj(1),...,Vi(m)}

~ @

by a p a r t i t i o n

vious m e a n i n g . c

sequence

into a f i n i t e

that

V" This

group

is i n c l u d e d

{VI,VI,V2,V3,V3,V3}

the o t h e r hand,

if

cohomology

a faithful

which

a subdivision

on the

the f i r s t

sets and

in the proof.

such t h a t V

V

open

it is c o n n e c t e d . R

Moreover,

The d i r e c t

is c a l l e d

into s e v e r a l

V IUV 2 UV 3

has a r e f i n e m e n t

to f a i t h f u l

does not d e p e n d

of the f u n d a m e n t a l

connected

whenever

R.

is seen HI(v;~)

: I(V",V')o I(V',V).

a direct

HI(R;T)

An o p e n c o v e r i n g

R

I(V',V)

I(V',V):

over

~.

is d i v i d e d

not h a r d to see that covering

in

F0(R)*

the u n i o n

{~ij } + {~i'j" }

I(V",V)

of

and

is defined.

a l-eocycle

we get a m a p

forms

The g r o u p

of s i m p l y

subset

relation,

HI(R;T)

group

'

and is a g r o u p h o m o m o r p h i s m .

coefficients

the c h a r a c t e r

{{i,j,}

of

t h e n we h a v e

forms

I(V',V)}

by

I'

Then

o v e r all o p e n c o v e r i n g s

is d e n o t e d

'

in

~(i'),u(j')

V', t h e n we h a v e

{HI(v;T);

be any e l e m e n t

j'

and t h e r e f o r e

see t h a t the m a p

maps

of

system R

.

{~ij} i'

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

We also

is a r e f i n e m e n t

Let

for some

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

÷ HI(v''~), .

V3

= Vu(i, ). ~ ~

t~,j ' = ~ ( i ' ) , ~ ( j ' )

We set

V

A V[

i'

V (i,) N V ~(j')

means

~(Vi,)

V!

an arc

=C Vi(v)

o

is a d a p t e d of

to

al-''o m

y. is

y.

a homomorphism

e 6 F0(R).

~, r e s p e c t i v e l y .

of

Choose

HI(R;T)

into

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

Namely,

{~}

F0(R)~. {~ij}

is a l - c o c y c l e

25

over a covering

V : {Vi: i E I}

from the origin

0.

covers

Let

and

c

is a closed curve issuing

y = {Vi(1),...,Vi(m)}

be a chain of

V, which

c, and set m

(2)

F(e;Y;{6ij})

with

i(m+l)

and

: i(1).

: ~ ~i(~)i(~+l) ~:i

We have to show that the product

(b)

does not depend on the choice of

F(c;y;{~ij})

representative covering

c.

{~ij}.

To see this,

Y1

is a subdivision

If

~ii = i, we see that {Vj(1),...,Vj(m)} to both y therefore

and

F(C;Yl;{~ij})

let

only on

of

y, then,

~ : i,..., m

with

i(m+l)

Next,

: i(1)

and

of

for

V

using the property

c = oi---o m

Vi(v) A V j (v)

~

y, fixing the

be another chain of

= F(c;y;{~ij}).

and if a partition YI' then

YI

vi(~) A V i ( ~ + l ) A V j ( ~ ) n V j ( v + l ) for

depends

~.

e

if

YI =

is adapted

~ : i, .. . , m

and

~

j(m+l)

: j(1).

Thus,

in

view of (i), we get m

F(C;Yl;{$ij))

= ]-F ~j(~)j(~+l) 9:1 m

: ~ [6j(v)i(v)~i(~)i(v+l)$i(v+l)j(~+l) ] ~:i m

= I T ~i(~)i(~+l) v=l Finally,

let

YI

be any chain of

subdivisions

y'

and

tition

c = oi--.o ~

F(C;Yl;{~ij})

YI' of

a covering valent,

c

y

{~i,j,}

of

V

and

{@i":

c.

simultaneously.

V'

Since

{~ij}

Thus we get

in place of of

and

~

associated

{~,j,}

V" = {V'~,: i" E I"}, which

with refining maps

i" E I"}

to which a par-

= F(c;y;{~ij}) ,

F(c;{~ij})

be another r e p r e s e n t a t i v e i' E I'}.

Then we can find

YI' respectively,

= F(c;y';{~ij})

So we may write

V' : {V~,:

and a system

covering

is adapted

there exist a covering

refinement

V and

= F(c;y~;{~ij})

as was to be proved. F(o;Y;{~ij}). (c) Let

of

= F(c;Y;{~ij})"

with values

in

~ T

and

: 8i"$~(i")~(j")@~'~

is a common

~', respectively,

such that

!

6~'(i")z'(j")

with

are equi-

28

for each pair {V~,,

i",

V"

j" }

(i) .... ' i " ( m ) for simplicity, i(v)

in

I"

with

be a chain

V'~,,nv'~,, ~ @.

of

V"

= ~(i"(v)),

covering

i'(v)

Now let

y"

the curve

= ~'(i"(v))

c.

for

: Write,

v = i,...,

m.

Then, B(y") = {Vi(1) .... ,Vi(m)} (resp. ~'(y") : { V i , ( 1 ) , . . . , V i , ( m ) } ) is a chain of V (resp. V') c o v e r i n g the curve. Then, u s i n g the convention

i(m+l)

= i(1)

etc.

as before,

we have m

F(e;{E~,j,})

= F(c;~'(T");IE~,j,})

= ]-m v=l

E~'(v)i'(v+l)

m

]-~ v:l

:

(@i,,(~)~i(~)i(~+l)@i,,(~+l~ I)

m

: T~ ~i(~)i(v+l) v=l : F(e;B(T");{~ij}) This m e a n s {~ij}.

that

F(c;{~ij})

So we may w r i t e (d)

Let

eI

and

If the d e f o r m a t i o n covered

by the

depends

F(c;~) c2

cI

chain,

say

and

..., c k'

issuing

for e v e r y chain.

that

~2 ~

{~ij}, using

is d e n o t e d

in

any

in

F0(R).

then they

are

= F(c2;{).

ci = el,

of c l o s e d

c k' = e 2

are c o v e r e d = F(c~;~) by

~

c{,

and such that, by the

.....

only on the h o m o t o p y

paths

F(c~;~)

class

and

same

~.

~

of

= c.

This

F(~;6). seen,

Thus

HI(R;T).

Given

{qi j} the

F0(R).

~

small,

sequence

in (2) is d e t e r m i n e d

by

As is e a s i l y in

(f)

expression

depends

of

= F(e2;Y;{~ij})

such that

of

so that

c' and e' v v+l F(Cl~ ~) = F(c~;~)

F(c;~)

(e) fixed

0

class

F(e;{~ij}).

is very V

k-l,

Hence,

quantity and

from

v = i,...,

the

on the e o h o m o l o g y of

we can find a finite

It follows

F(e2;~). Namely,

case,

c2

y, of

F(el; ~) = F ( C l ; Y ; { 6 i j } ) In the g e n e r a l

only

instead

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

between

same

: F(c;{~ij}).

This

u n

q.e.

on

En

__> u(z) s

for

q.e.

every

on e v e r y

n

by T h e o r e m

En

6E-(b),

and therefore

on

Ch. E.

I, we

see t h a t

It f o l l o w s

s(z)

that

uR
0

~K , there

~K

so

min{ga,~}

is

exists

such that

min{g(a,z), e} = f g(z,b) dl (b). Setting

z = a, we see that

f dl

f

(4)

By Fubini's

(b) : i.

theorem we have

flf

We note that I

is continuous ting

n ÷ ~

k(z,b)dl

min{~(z,a), ~} g(z,0)

(b) :

and finite throughout in (4) and using

R*

as a function

in

z.

So let-

(i), we get

(5) = I min{~(z,a), ~} dp(z)

g(z,0)

R uE

Since

is lower semicontinuous ~

are shrinking to the point

a

as

f dX

: i

to

eonvergenee

f k(z,a) d~(z)

desired

integral

Theorem

6B, Ch.

Corollary. that

Let

b ~ E*.

as

~ + +~.

(b).

Combining Finally,

these with

unicity of

(3), we get the

B[R

follows

from

[]

b E AI(R) Set

1

theorem shows that the last member of (5) tends

expression. I.

of

~ ÷ +~, we see that

u~(a) £ lim inf f u~(b)dl The monotone

and the supports

and let

E = E* h R .

Then

E*

be a closed

(kb) R E

subset of

is a potential.

R*

such

46

Proof.

By the p r e c e d i n g

(6)

(kb)~(a)

for some p o s i t i v e v(a) kb

t h e o r e m we h a v e

measure

: S k ( z , a ) d~'(z) is m i n i m a l ,

kb(a)

for

we h a v e

~ 0, t h e n we set

that way.

Thus

3C. write

~ = 0

subset

u

e

~' : pI A

with

support

of

does not con-

point

kb

(kb) ~

is a p o t e n t i a l .

harmonic

is a p o s i t i v e b ÷ (kb)~(a)_

be w r i t t e n

function

measure

If

expression

v

cannot

and

Since

0 ~ ~ ~ i.

a contradictory

the c l o s e d

~

We set

on

[]

on

R

A.

is i n t e g r a b l e

in

and

If

E

for e v e r y

and

a

of

R \ E, t h e n

If

a E R \ E

a, then,

and h a v e

R, t h e n

uER(a) = I If

E*.

0 ~ v £ (kb) ~ ~ k b.

be a p o s i t i v e

where

of

(7)

Proof.

Then

and t h e r e f o r e

Let

by

for some

In fact,

u = fA kb d~(b), a E R

a E R.

v = ~k b

so t h a t the e x t r e m e

Theorem.

is a c l o s e d fixed

b

k(z,a)dD(z)

supported

dv = ~-id~'

= ~ k(z,a) d~(z).

t a i n the p o i n t

p

f

=

is an i n t e r i o r u~(a) and

as we h a v e

uR(a)

= u(a)

if

G

k

point and

of

E

or a r e g u l a r

(kb)~(a)

is the c o n n e c t e d

seen in T h e o r e m

: f u(z)d~G(z):

([CC],

(kb)ER(a)d~(b)"

fA(f

Ch.

of

44-45)

boundary

= kb(a) , so that

component

6E-(a),

pp.

R \ E

point

(7) holds. containing

I,

kb(Z)d~aG(Z)] d~(b)

A Finally, Theorem E.

let 5B,

a E E Ch.

be an i r r e g u l a r

I, the

singleton

{a]

boundary forms

point

of

a connected

R \ E.

Then,

component

So we can find a s e q u e n c e

by

of

{G : n = i, 2,...} of J o r d a n d o m a i n s in n N n=l ~ C R \ E, CI(Gn+I) = C G n and CI(G n ) = R such that a E Gn, ~Gn = {a]. Let ~ be the h a r m o n i c m e a s u r e of G at the p o i n t a. Then n n we h a v e

(kb)R(a)

= lim I ( k b ) R ( z ) d ~ n ( Z ) n -~

and t h e r e f o r e

the f u n c t i o n

b ÷ (kb)R(a)s

= nlim f -~

is m e a s u r a b l e .

Hence,

47

n÷ ~ ~ A

[lim I ( k b ) ~ ( z ) d ~ n ( Z ) ] dB (b)

: IA "n÷~ : I

(kb)~(a)dz(b)' A

as was to be proved.

3D.

[]

We are n o w in a p o s i t i o n

to p r o v e

the m a i n r e s u l t

of this

section. Theorem. cation

boundary of

R

The M a r t i n

in the A

Let

f

The h a r m o n i c

is g i v e n by

be a n y r e a l - v a l u e d

it to a c o n t i n u o u s f.

a

We m a y a s s u m e

function that

R*

is a r e s o l u t i v e

every real-valued

is r e s o l u t i v e .

at the p o i n t

Proof.

compactifieation

sense t h a t

on

measure,

kb(a)dx(b). continuous

R*, w h i c h

0 ~ f ~ 1

continuous

on

with

function

on the

support

([CC],

is d e n o t e d

R*.

eompactifi-

function

p.

on

in

AI,

140) A.

We e x t e n d

by the same l e t t e r

For e v e r y

n = i, 2,...

we

set

A i = {b e Al: E9 : {a e R*: 1

( i - ~) =< f(b) f(a)


i + i],= n

E. : E g A R , l l and

ui = I

kb dx(b) A. 1

for

i = 0, i,...,

by C o r o l l a r y Theorem

n.

3B

For e v e r y

( k b ) ~. 1 3C we see t h a t

is

(ui)R Ei is a l s o a p o t e n t i a l . f < (i + l)/n

on

Since

fixed

i

a potential

: I

A. 1

(kb)~

we see

for

.i

(ui) ~. = u i i R \ E~, we h a v e

any

A iNE~

b E Ai .

= ~

and so

By u s e

of

dx(b) q.e.

on

Ei

and

(i - l)/n


0,

function

on

function

majorant

f. f

on

and a potential

that

sequence

U

is a W i e n e r

functions one

of

for a n y

f

a superharmonic

function

a potential

(i)

clear

has

quasicontinuity

is a W i e n e r

Proof.

say t h a t

Ifl

set o f W i e n e r

such that,

u E HP'(R) then

the

differs

For any Wiener

exists

R

We

and

= U 1 + U2,

for a n y we can

s > 0,

R

CI(R2)

and

CI(Rm+I).

define

then

U'

Ifl =
i + l},= n

and

ui for

i : 0, i,...,

is a p o t e n t i a l .

n.

If

Using

: I

As

kb~(b)dx(b)

b E Ai, t h e n

Theorem

(ui)R = Ei is also a p o t e n t i a l .

A.

R \ E i 6 G(b)

so t h a t

3C, we see t h a t

A. i

1

R ~(b)dx(b ) (kb)E. l

in 3D we h a v e

n i - l(u " _ ( u i ) R ) n 1 i:0 l .

< fu < = =

n n [ i+ 1 • + [ n Ul i:0 i:0

(kb)~.

( u i ) ER

i

56

q.e. R

on

R.

So t h e r e

exists

a positive

superharmonic

function

s'

on

such that n

holds

on

class

R

~

=

i=O

for any

W(f)

n

fu
0.

i

[ i:0

(u i R )E.

I

The r i g h t - h a n d

+

ss'

member

thus b e l o n g s

to the

and so n

~[fu]


0 a n d v > 0. By T h e o r e m i = q = on R and the singularities of

vi,

if any,

be a s i n g u l a r i t y

vi

has

we h a v e where

vi(z) h(z)

tive,

we

h

v i.

= c log Izl + h(z)

see t h a t of

c ~ 0 a.

by R i e s z ' s

As

is i n n e r ,

Ch.

a

II,

therefore

suppose

inner

and

that quasi-

v ~ 0, we h a v e

v

is e v e r y w h e r e h a r m o n i c q are isolated. Let a E R

only

The

its

Since

logarithmic

in a p a r a m e t r i c

function. and

disk

singularities,

centered

function vi

is

v. being m superharmonic

at

a,

nonnegain a

v. is s u p e r h a r m o n i e o n R. l I) we h a v e v. = h + h , w h e r e l s p harmonic function and h is a p o t e n t i a l . Since p h . C o m b i n e d w i t h L e m m a 5C a n d T h e o r e m 5D, Ch. III,

theorem

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

s v.

5B,

Since

is a h a r m o n i c

neighborhood Thus,

of

5A, we m a y

so is

l

is a r b i t r a r y ,

(Theorem

6F,

Ch.

S

these

observations

AI •

So

a.e.

again

to

vi

Qq

exists

Corollary.

on

5C,

AI .

If

u

a.e.

on

that

Ch.

AI,

where

Proof•

Set

preceding ~q

a.e.

is a n o n z e r o AI

v : log

theorem with

Vq

The

We w i l l

purpose

convex complex

v

vanish

Moreover,

AI .

Hence,

and

exists

a.e.

a.e.

Vq

on

exists

and

is e q u a l

l.m.m,

of bounded

characteristic,

then

= i

and

u : UQ

u.

shows

Since that

and

UQ

= exp(prQ(lOgu)).

v E SP'(R)

Q

uI = i

a.e.

exists

by the

a.e.

Hence

we h a v e

a.e.

[]

on

assumption,

AI

and

~ = expQ

the

is e q u a l

: exp @ q

to

= ~Q.

It

B-Topology

5A. this

exist

p

and

= prQ(v).^

that

h

a.e • on

III.

u I : exp(Prl(lOgu))

is n o w t r i v i a l

5.

and

s

[]

uI on

h

and vanishes

by L e m m a

a.e.

exists

imply

we

linear field.

describe

first

spaces.

the

recall Let

Precisely

B-topology

some E

basic

f o r the

facts

be a l o c a l l y

speaking,

this

space

in the

convex

is d e f i n e d

h~(R).

theory

linear b y the

of

space

For locally over

following

the

78

conditions: (AI)

E

(A2)

is a l i n e a r

E

plication

is a H a u s d o r f f

are

the o r i g i n of

space

space

each continuous

0

has

a basis

over

the

complex

such that

in b o t h

field;

addition

variables

of n e i g h b o r h o o d s

and

jointly

consisting

scalar

and

multi-

such that

of c o n v e x

subsets

E. Let

E'

denote

space

of a l l

each

x' E E'

spondence

continuous

E

meaning

(resp.

of

E')

we

{x E E:

A

in

E'

E')

satisfying

is c a l l e d < ~

every

x E E).

and then of

only

the

totality

of p o l a r s

of

E

a basis

by

E'.

s(E',E)

The

dual

E

and

A°

of the

bounded)

that

a subset

The

strong

convex

A

of

over

0.

strong

dual

topological

linear

space

The of

w(E,E') topology (resp.

for

is subsets

for

all

if

E'

such

bounded

space and

E'

subwith

is d e n o t e d

is d e n o t e d

We c a l l

by

Identification

x' ÷ < x , x ' > E

on

semireflexive

the w e a k - ( E , E ' )

(resp.

w(E',E)),

for

(resp.

E

E".

is c a l l e d

(resp.

E')

if

gives

the

x E E

us a n a t u r a l

inclu-

E = E".

weak-(E',E))

is d e f i n e d

for

E'

of each

topology,

as t h e w e a k e s t

that makes

every

locally

functional

written

as

convex x'

E E'

x E E) c o n t i n u o u s .

Theorem. if e v e r y

A locally bounded

harmonic first

We n o w

linear

of

[37],

E

p.

on

R.

is the u s u a l

space

E

is s e m i r e f l e x i v e

is r e l a t i v e l y

if a n d o n l y

w(E,E')-compact.

(cf.

190)

l o o k at the

functions one

convex

subset

and Namioka

5B.

The

s(E',E)

E E'

(resp.

< ~

on b o u n d e d

topology

ranging

number E

is b o u n d e d

topology

the

functional

E =C E". Finally,

Kelley

of

sup{i<x,x,>I:

E

convergence

locally with

of

polar

bounded

S

of

a linear

sion

A

if

of

x E A}

a positive

a subset

is

= {0}

A

it t h e

is c a l l e d

S

bidual with

which

s u p { l < x , x ' > I : x' @ A}

of n e i g h b o r h o o d s

is c a l l e d

E

and

x' E E'}

call

corresponds hand,

E

corre-

E × E'

for a l l

and

of

weakly*

of uniform

it is the

sets

topology

E

(resp.

bounded.

that

the

(resp.

topology

words,

forms

in

in p a s s i n g

if it is w e a k l y as the

0

On the o t h e r

x' E E'

We n o t e

In o t h e r

of

bounded

for every

defined E.

U

A

x E the

For a s u b s e t

~ 1

x' E A})

linear

each Then

for all

= {0}.

I<x,x'>I

A subset

For

f o r m on

: 0

x E E}

for all

E).

E.

as t h e

x'(x).

a bilinear

A ° = {x' E E':

in

on of

{x E E: < x , x ' >

A ~ eU.

weakly

x ~ A}

in p l a c e

for all

I<x,x'> I ~ I (resp.

is d e f i n e d

functionals

defines

that

if to e a c h n e i g h b o r h o o d > 0

E, w h i c h

<x,x'>

= 0

set

of

linear

(x x') + < x , x ' >

{x' E E': < x , x ' >

(resp.

dual

we w r i t e

nondegenerate, and

the

space

h~(R)

Let us d e f i n e norm

topology,

of c o m p l e x - v a l u e d in it two k i n d s which

is g i v e n

bounded

of t o p o l o g y . by the

sup-norm

77

Ilhll

(6) for

h E h~(R).

: sup{ih(z)I:

z • R}

The second one is the B-topology

(or the strict topol-

ogy), which is the main objective of this section. this,

let

tions

f

C0(R) on

In order to define

be the space of all c o m p l e x - v a l u e d continuous

m

compact for any

i.e.

e > 0.

forms a Banaeh space with

Clearly,

C0(R)

{z • R:

func-

that vanish at infinity,

If(z)l ~ e}

is

respect to the usual a d d i t i o n and scalar m u l t i p l i c a t i o n of functions and the s u p - n o r m of the form (6). Now, for every setting

Nf(h)

f • C0(R)

= llfhll .

we define a seminorm

The totality

{Nf: f • C0(R)}

determines a locally convex t o p o l o g y for topology.

In other words,

f

ranging over

for this topology.

in

h~(R)

by

of seminorms

h~(R), which we call the B--

the c o l l e c t i o n of sets of the form

Vf = {h • h ~ ( R ) : with

Nf

llfhrl < i},

C0(R) , makes up a basis of n e i g h b o r h o o d s of

The space

h~(R)

0

e q u i p p e d with the B-topology is

denoted by h ~ ( R ) .

5C. dual of

We want to determine the dual of h~(R).

Let

Borel m e a s u r e s on

Mb(R)

R.

hB(R) , which we call the B-

be the space of all c o m p l e x - v a l u e d bounded

This forms a Banach space with respect to the

usual a d d i t i o n and scalar m u l t i p l i c a t i o n of m e a s u r e s and the total variation norm tional on

IJ~II : fR C0(R)

Id~I"

(7)

for

Each

~ e Mb(R)

determines a linear func-

by means of the formula

: I hd~ JR As we see easily by means of Riesz's r e p r e s e n t a t i o n h E C (R). 0 is identified via (7) with the dual C0(R)' the space Mb(R)

theorem,

of the Banach space N(R) and denote by

C0(R).

= {~ E Mb(R): M{(R)

the

On the other hand, we set = 0

(algebraic)

for all

h E h~(R)]

quotient space

Mb(R)/N(R).

Then

we have the following: Theorem.

The space

(algebraically)

h~(R)

is s e m i r e f l e x i v e and its dual

identified with

h~(R)'

is

M{(R).

Proof. The proof is divided into three parts. (a) First we show that the dual h~(R)'

is a l g e b r a i c a l l y equal

78 to

M{(R).

Let

is c o n t i n u o u s i, 2, ... } Un=l ~ Kn

hB(R).

of c o m p a c t and

Urysohn's R

~ E Mb(R) ; t h e n the l i n e a r

on

To see this,

subsets

such that

fn : 0

on

R.

to

on

a measure

R\ Int(Kn+l) ,

K0

=

[ n=0

= i

fn' on

defined

d~'

{Kn: n = R =

We t h e n use

n = i, 2,...,

Kn,

and

on

0 =< fn =< i

by

2-kf k

2 -n-I £ f £ 2 -n+l

denotes

by s e t t i n g

IB'J(R)

[ k=l

and in fact

where

~'

n : i, 2, . . . .

functions

T h e n the f u n c t i o n

C0(R)

n = D, i,...,

for

F : h ÷

a sequence

K n =C Int( Kn+l) ,

such that

continuous

f = belongs

R

I ~ I ( R \ I n t ( K n )) =< 4-n

l e m m a to d e f i n e

everywhere

of

functional

we c h o o s e

the e m p t y

= f-ldu.

i~'I(Kn+ I \ K n) ~

on

subset.

Then

[ n:0

Kn+ I \ K n

Finally,

for

we d e f i n e

~' E Mb(R) , b e c a u s e

2n+li~l(Kn+ I \ K ) n

co

< 2 1 ~ I ( K I) +

It f o l l o w s

that,

for every

[ n:l

2n+1-4 -n < ~.

h E h~(R),

iF (h) I : i I = l el.

6A of the G r e e n

functions

first

a • R

: {z • R: g(a,z)

I,

we h a v e

~ HI(R)

every

relation groups

of the r e g i o n

HI(R(~,a)).

So,

It then is h o m o l o R(e,a).

In

HI(R(~,a))

(Theorem R(e,a), when

2C,

Ch. I).

i.e.

a point

the a @ R

84

is h e l d

fixed,

for all

sufficiently

isomorphic

B(e,a)

with

Definition.

is a n o n i n c r e a s i n g

large

an o p e n

disk

A hyperbolic

Parreau-Widom

type

~,

function

for the r e g i o n

for all

Riemann

(abbreviated

S

large

surface

e

and

vanishes

is c o n f o r m a l l y

e. R

is c a l l e d

to a P W - s u r f a c e

B(~,a)d~

~

in

R(e,a)

a surface

or PWS)

of

if


0.

This

w C R

z = w.

we r e p e a t

property

Z(a;R)

according

= ~{g(a,w):

is a PWS

w e Z(a~R)}

if and

only

if

to

zero

region

is a c r i t i c a l Let

of p o t e n -

tends

the

then

a surface

d~

sense

ga(Z)

equivalently,

A point at

f0 B ( ~ a ' )

to be proved.

regular

= 0

ga'

and

is c a l l e d

a.

= ~ga/~ of

as was

infinity

is c o m p a c t choice

f0 B(~,a) d~

time,

a E R

point

points

(i)

for e v e r y

some

~ga/~Z

critical

same

surface

~ ~}

of the if

integrals

at the

to the

g(a,z)

ga(Z)

of all

two

if for

tends

independent of

that

A Riemann

theory z

Since and

~0

or d i v e r g e

IC.

respec-

that

B(~,a')d~

therefore

a

same

and

for any

~0

and

a',

A-ig(a',z) the

= H[ga;G]

by

and

contain

of fact,

~G, t h e n

~ R ( A - I ~ , a ') ~ R ( A - 2 ~ , a )

B(A-2~,a).

ga

a

of the

a, a' E R.

G : R \ (CI(V U V ' ) ) .

does

A > 0

As a m a t t e r

points

centers

set

and

is i n d e p e n d e n t

distinct

with

and

is c o m p a c t

definition

two

disks

constant

z @

z E G,

Ch.

above

take

closures,

G

a positive

g(a',z) R(~,a)

disjoint

for any

for

the

V'

having

Ag(a',z)

that

In fact,

and

tively,

note

a ~ R.

is a g a i n point

be the

set

to m u l t i p l i c i t y .

88

[{g(a,w):

(2) for some

(and h e n c e

Proof.

Since

2,...}

of p o s i t i v e

level

h

bordered to

Z(a;R)

curves

We set

all)

is d i s c r e t e ,

numbers

= Cl(R(an'a))"

g ( a , z ) - a n.

gn(a,z).

extended

Let

~n(a,z)

gn'

denotes

points

in

Rn, then

deg(T)

=

readily

I~a

denote

2.

(9) in Ch.

= gn' = N n

gral

B(~,a)da

:

_anB(a n

,a) -

if

~ g(a,w) foB(~,a)da

e ~, t h e n < ~,

n ÷ ~, and thus Parreau

inequality on G r e e n

of

single-valued

R'n deg(T)

has

but

and can be

Nn

in v i e w of Ch. 2, . . . .

of

Rn . : 2 g ~ - 2,

critical

zeros and two poles, n = i,

is e q u a l

so that I, 2B

It f o l l o w s

+ [{ga(W):

f0 B ( ~ , a ) d a

inition

clature

type.

w e Z(a;R), e ~

ga(W)

Conversely,

This

B(a,a)da

< ~.

> an}. if the inte-

÷ 0

H e n c e we get the e q u a l i t y

regular

in sueh surfaces.

for the s u r f a c e s

Riemann

surfaces

to d i s c u s s As we shall

e n o u g h to c o v e r is p r o b a b l y

we are d e a l i n g

[]

for w h i c h the problems

see in 3B below,

in e s s e n c e

enough

R

Diriehlet

(i).

all s u r f a c e s

to l e g i t i m a t e

based

his defof

our n o m e n -

with.

Characterization

A remarkable

a hyperbolic

~

His a i m was

is i n d e e d g e n e r a l

Widom's 2A.

[ g(a,w)

[52] c o n s i d e r e d

lines

~dB(a,a) n

then

(2) holds.

Parreau-Widom

2.

I ~a

n

anB(an,a) as

a

conjugate

and h a v e

is c l e a r

Z(a;R).

is a c o m p a c t

on the d o u b l e

gn(a,z)

for

n = i,

that

= -anB(an,a) So,

If 2N n

gn' = B(an,a)

B(an,a)

that

R'.n

h

is s i n g l e - v a l u e d T

in

w i t h pole

the h a r m o n i c

I, 10D,

s h o u l d have

Since

every

is not n e c e s s a r i l y

the genus of T

any p o i n t

gn(a,z)

differential

{an:

to zero such that the

is r e g u l a r ,

d(gn(a,z) + i~n(a,z))

where

we c o n c l u d e

R

function

~n(a,z)

to a m e r o m o r p h i c

-

decreasing

do not c o n t a i n

Since

So we can use the f o r m u l a

2N n

= a n}

its G r e e n

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

we can find a s e q u e n c e

strictly

and

The f u n c t i o n


0 f o r i i a set of s e l e m e n t s , say B = {b(1),..., b(s)},

we assume

i = s+l,...,

of

a eonformal

to a h o m e o m o r p h i s m

is a c o m p a c t

for all assume

~V i

exists

{w e ~: r i
k.

of c r i t i c a l

of e x c e p t i o n s .

IC and 4C, and the L e b e s g u e m o n o t o n e

we f i n a l l y h a v e

: l i m Mn(a)

from

that

in one of the r e g i o n s in

number

each

< gn+l(a,z)

in t u r n i m p l i e s

HI(R(~,a)).

gn(a,z)

theorem,

M(a)

Ck

M(a)

can be d r a w n

is seen to be a r e g u l a r

converge

B(e,a)

1-cycles

Thus

=< B n + l ( e , a ) .

Bn(e,a)

gn(a,z)

that

÷ Hl(Rn+l(e,a)),

Rn(~,a)

that

(9)

In fact,

gn(a,.).

R, we see s i m i l a r l y

~ Rn (~,a), = U n=l

pendent

conclusion

~ R n + l ( ~ , a ) , and t h e r e f o r e

is i n j e c t i v e

R(~,a)


0

6B gives

to

posi-

bounded

*dg(.,~)

Bj(P)

of T h e o r e m

that

is

and set

an a p p l i c a t i o n

shows

RI, R

duj e F h 0 ( C I ( R j ) ) ,

so that

each

a parti-

that

Since

?B we take

with and

Then the f o l l o w i n g

B(G)

for

in any c o m p a c t

suppose

are not.

By use of Lemma

in v i e w of the facts

CI(R I)

Since

with

~, where

lying

We may

R6,+I,... , R 6

on

fBl(p)

g0(-,~)

satisfied

on

R

j = ~' + i,..., ~; and (iii) a3. = 0 for i a.] = 27. for a moment, we p r o c e e d further. If ~' = !, then we

Then we define = uI

fixed

region

a. = a.(~) ] 3

..., ~',• (ii) A s s u m i n g these

z

j = i,...,

on

B(R);

= {RI,... , R6}.

uj

We take any

j = i,...,

boundary

for

6' ~ i.

functions

(13) for

for

function

z.

be any r e g u l a r

tion of the form ..., R6,

(mod 27)

ideal

is b o u n d e d

set not c o n t a i n i n g Proof.

of the

harmonic

L, we

107

For

j : Z' + I,...,

~

We now define

go

(15)

we set

pj ~ 0

on

R.

by the formula

g0(.,~)

=

[ aj(~)pj j=2

+ g(-, i.

j = 2,...,

for a

~, can be used and there-

of the properties (i)-(iii) of aj, g0(-, 0, it f o l l o w s ]

is h y p e r b o l i c .

I

of

*d~ ~ 0 8j(P)

from

along

if and o n l y

if

(16) that

< 0

*dg(.,{) Bj(P)

if and o n l y if follows

at once

in T h e o r e m

7E. Lemma.

R. is h y p e r b o l i c . This p r o v e s (i) and (ii). (iii) 3 f r o m the l o c a l e x p r e s s i o n of the G r e e n f u n c t i o n g i v e n

6A, Ch.

We n e e d Let

I.

some f u r t h e r

{V, z}

and its p a r a m e t r i c V

such t h a t

to

~0'

coordinate, I~I

of

{Izl

g.

disk,

< r I}.

in w h i c h we i n d e n t i f y

~0 , %0'

and let

< r I < r 2 < I.

the d i s k

with

property

be a p a r a m e t r i c

I~ol,

within

tersecting

[]

Let

Then

Proof.

*d(g(.,~)

- g ( . , % 0 )) =

differential

We use the n o t a t i o n

arg{(z-

joining ¥

in ~0

not in-

f ]

~ 2 ~

o(y), c

is a r e a l

single-valued

points

be an are

for any 1 - c y c l e

y

o(y)

be d i s t i n c t

c

c

I where

a point

branch

in Ch.

in

Fh0

I, 9.

of the f u n c t i o n

given

Let

in T h e o r e m

v(z)

9C, Ch.

(resp.

s(z))

~)}

(resp.

log{(z - %0)/(z-

I.

be a

~0)/(z- ~)}) in the a n n u l u s r I S Izl < i a n d let e(z) be 2 C - f u n c t i o n on R such that e ~ I on {Izl < r I} and

a real-valued 0

on

R \ {Izl

{Izl £ i}

and

< r2}.

k(z)

We also

= g(z,~)

of the G r e e n

Izl < i, w h e r e

is a h a r m o n i c

*dk = d { a r g ( z for

Izl < i, w h e r e We f i r s t

(17)

h2

function

= logl(z-

Then

~0 ) - a r g ( z -

is a h a r m o n i c

on

d{(l - e)k} • Fe0(R).

follows

function.

{0)/(z- ~)I

that

k = w+ h I

for

Thus, ~)}

+ dh 2

conjugate

of

hI

in

{Iz

< i}.

see that d ( k - ew)

: d{(l - e)k} + d { e ( k - w)} :

and so

w(z)

- g(z,~0).

F r o m the d e f i n i t i o n hI

set

* d ( k - ew) @ FeO ~{.

d{(l - e)k} + d ( e h I) E Feo(R) On the o t h e r hand,

dk + i * d k - d(ev)

E F e.

109

In fact,

on

V

we h a v e dk + i * d k -

which

is closed.

Next,

d(ev) on

: d{(l-

R \ {Izl

dk + i * d k which

is a n a l y t i c Let

a real

y

and

d(ev)

contained

differential

for any c l o s e d

S r 2}

we h a v e

= dk + i*dk,

so is closed.

be a 1 - c y c l e

harmonic

e)v] + d(h I + ih2) ,

in

o = o(y)

differential

~ E F

R \ {Izl E Fh0

< rl].

such

(Theorem

9B,

Then

that Ch.

there

is

fy ~ = (m,o*)

I).

Since

dk-

C

d(ev)

is exact

f

*dk

on

R \ {Iz[

= -i f

¥

< rl},

dk+ i*dk-

we have

d(ev)

= -i(dk+ = A,

Now,

from

(17)

follows d(k-

This

i*dk-

differential

say.

that ew) + i * d ( k -

is e q u a l

d k - d(ew) + i ( * d k -

ew) @ Fe0 + F e0 * "

to

e*dw-

w *de)

: d k - d(ew) + i ( * d k = dk + i * d k -

Since

Fho" ±

(Fe0 + Fe0*)

by

(4)

in Ch.

( d ( k - ew) + i * d ( k and

d(ev),o*)

y

I,

eds - w ' d e )

d(ev) + i(sde - w ' d e ) .

8E, we h a v e

ew),o*)

= 0

thus A : -i(i(w*de

- sde, o*)

: ( w ' d e - sde, o*)

rl< I z l< r 2 sded wde°*

d(ew)(]*

+ ffrl < zl 0

is given

with

equal

we have

~

to

~ m. by a character,

point

0, where

on the c o m m u t a t o r by its values

generate

at

~i'

the h o m o l o g y

HI(G0,%) = F 0 ( G 0 ) / [ F 0 ( [ 0 ) ] . Here, the paths from 0 by p e r f o r m i n g obvious d e f o r m a t i o n s .

exp(-2~iaj)

for

..., N, where the region there

a E GO.

for any line b u n d l e

subgroup

belongs

~k ) - b

same n o t a t i o n s

As n o t e d

an N-cube

U.

(mod 27)

are the points

'

the

t 0 = CI(G0).

group issue

*dg0(" Yn

8C.

Proof.

over

(Xl,...,x N) • U

- bn

~k ! s

[]

and an

say

~MN

then

N.

Suppose

a point

x k)

n = i,...,

Theorem. R

exists

side covers

ranges

r

I

that

(Xl,...,x N)

~i'''''

times,

k=l

Using

- 2~M%(Xl,...,XN).

M, the r i g h t - h a n d

*dg0(.,~+

MN

for

[ *dg0(-,~)) k:l YN

Yn

N.

N, each r e p e a t e d

Nf

j = i,...,

aj

and

bn

GO, we should

exists

~

and

0(yn)

are s u p p o s e d to We set 8(e.) = ] : e x p ( - 2 w i b n ) z for n = i,

are real numbers. have

[j=l a. - 0 ] harmonic function

a nonnegative I

*du ~

(mod

a.

Since (mod u

[j=l ~j

2w). on

bounds

By T h e o r e m

R

7C

such that

2~)

] for every

j, where

u(a)

aj.

Next we use Lemma

~MN

from a fixed

k=l

compact

Yn

~ C < ~,

C

8B to choose set

being

a finite

K

lying

*dg0(.,~ k)

~ bn -

in

f

Yn

a constant number G O \ {a}

*du

independent

of points

(mod

such that

27)

of

~i'''''

114

for

n = i,...,

where

~

N.

denotes

is h o l o m o r p h i c

We set

v = u+ [~i

the h a r m o n i c

on

R.

g0(''~k )

conjugate

of

and

v.

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

f = exp(-v-

We see first

of

go

(Theorem

i~),

that

f

7D) shows

that

I for any

*dg0(.,%)

and any

6 • GO

-

~ O

(mod 2~)

] j

= 1,...,

rI J

*dv

~,

~

we

see

that

(mod 2~).

-a.

]

We also h a v e MN -

*dv

:

-

*du

Yn Thus, ~

f

-

[

Yn

is a s e c t i o n

of

~

,~k )

-b n

-

Yn

of a line b u n d l e

is the r e s t r i c t i o n

If(a)l

*dg0(-

k=l

to

GO .

= exp(-v(a) -

~

over

R, so t h a t the b u n d l e

Moreover,

MN ~ g0(a,~k)) k=l

> m > 0,

where m : e x p ( - C - MN sup g 0 ( a , ~ ) ) %EK is i n d e p e n d e n t

8D.

of the c h o i c e

of

~.

[]

For each line b u n d l e

T

striction

of

GO .

Theorem.

Suppose

T

to the r e g i o n m(a)

= 0.

over

R

we d e n o t e

by

p0(~)

the re-

Then

Then,

for any line b u n d l e

T0

over

[0'

we h a v e inf{m(~,a): Proof.

Since

bundle

T'

the p r e c e d i n g a section where

m

m(a)

over

= 0, we see that

R

with

''-I)

is a c o n s t a n t

belongs

l(f0h)(a)l

m(T',a)

: T0}

for any

< ~.

to

~(R,T~

~ m(~',a)

such that

independent If

of

< E.

Since

e > 0

say

llf011~ ~ i ~.

h • ~(R,6)

''-I) : ~ / ( R , < ' )

= 0.

We set

t h e o r e m we f i n d an e x t e n s i o n ,

f0 e ~ ( R , ~

P0 (T) = P0 ( T )'P 0 (6'') = ~0" foh

p0(~)

and

Set with

there

exists

a line

T~ : P 0 ( T ' ) - I T 0 • T", of and

T~

to

R

By and

If0(a) I ~ m > 0,

~ = ~'T".

Then,

llhl11,a =< i, t h e n

llf0hlll,a =< I , so that

If0(a) I £ m, we have

lh(a)l ~ m-ls,

115

which means that sired result. 8E. that

~ m-it.

= 0

for some

s

is arbitrary,

open subset

V

(and hence all) of

We then choose a regular nonical regions

(Ch.

(resp.

Pjk

with

bundles

over

R

R

a @ R.

and a countable

2B, we suppose

We take a relatively

dense

subset

S

of

V.

exhaustion

j ~ k) the restriction Gk).

~n

over

We fix a line bundle

~j

mn = inf{mn(~n'b):

to

Gn

(resp.

Gj) of line

Finally we set

: sup{If(a)l:

for any line bundle

n ~ j

we get the de-

{G : n = i, 2,...} of R by can I, IC) such that CI(V) ~ G I. We denote by On

(resp.

mn(~n,a)

for

As

To finish the proof of (c) ~ (a) in Theorem

m(a)

compact

m(6,a)

[]

f • ~ ( G n , ~n ), llflll,a =< i]

Gn

and any

over

Gj

a • G n-

and take any

~n e HI(Gn;~),

b E Gj.

We set

Pjn(~n ) : ~j}

and = inf{m(~,b):

~ • HI(R;T),

pj(~)

= ~j}.

We claim (18)

lim m

n

: m.

This can be shown in the same way as in the proof of (8) (see 5A). fact,

if

belongs

~n+l E HI(Gn+I;T) to

HI(G

;~)

with

Pj,n+l(~n+l ) = ~j, then

In

Pn,n+l(~n+l )

and therefore

n

mn(Pn,n+l(~n+l)'b)

~ mn+l(~n+l'b)

~ mn+l"

Since every line bundle over see that

~ mn+ 1 ~ m

G. is the restriction ] and hence limn÷~ {n =

In order to show the reverse We take

~ E HI(R;T)

before that


0,

0

where the

~

ranges

common

over

value

all

line

is e q u a l

The

last

statement

for any

Corollary. common

point

If

zeros

R

line

Theorem.

Let

(a)

For each

pair

with

(b)

For

a simple

zero

Proof.

comes

is a PWS,

9B.

f e H~(R)

{g(a,w):

f(a)

every at

from

Theorem

R

tion.

Take

because of

let

then

the

bundle

an

R

surface

R.

statement

h(z)exp[-g(a,z) H~(R).

Since

this

the in

theorem

is

following

~(R,~)

have

no

Then

points

the

following

a, b E R

hold:

there

exists

an

a E R

there

exists

an

f E H~(R)

which

has

bounded

be the

line

bundle

-I)

such

- i~(a,z)]

is e a s i l y

and

seen,

the p r o p e r t y :

over

R

extend

choose

by this

This

function,

and

f(b)

-I)

with

~ 0.

func-

is p o s s i b l e

a function

it a n a l y t i c a l l y

resulting = 0

determined

lh(b) l ~ 0.

We t h e n

the

f(a)

function

ig(a,z))

that

Corollary.

analytic

element

to the w h o l e say

f, b e l o n g s

This

proves

~ 0.

Then,

the

(a).

To p r o v e

in

then

~ f(b).

point

h E ~(R,~

As

and has

IC.

functions

be a PWS.

of d i s t i n c t

preceding

h(z)exp[-g(a,z)

H~(R)

is r e g u l a r ,

~ E HI(R;~).

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

~

of the

R

a.

Consider

and

If

in p a r t i c u l a r

z ~ exp[-g(a,z)on

R.

w e Z(a;R)}].

a ~ R, we h a v e

for any

over

to

exp[-[

valid

bundles

(b),

we

choose

- ig(a,z)]

Since

h(a)

h E ~(R,~

gives

~ 0,

f

rise has

h(a)

to a s i n g l e - v a l u e d a simple

zero

at the

function point

f a.

[]

118

NOTES For the fundamental brilliant

Parreau's

[70],

Most

two

in 1976, paper

although

Subsection

definitions

except

some

those

was

was

chapter

shown

out

coined

in S u b s e c t i o n s

from Hasumi

are

have

[18].

contains

taken

[52].

when

from

[18].

[18]

[19].

his

this

Essential

was

As a written.

to me the r e l e v a n c e

with my work

then

which

by H a s u m i

results

pointed

small m o d i f i c a t i o n s

3 are a d a p t e d

[70],

is in P a r r e a u

Parreau's

in c o n n e c t i o n type"

Widom

in this

definition

Z. K u r a m o e h i

[52]

results

follow

results

I did not k n o w

of P a r r e a u - W i d o m All

of PWS we

Parreau's

of t h e s e

of fact,

Afterwards,

faces

theorem.

paper.

equivalence matter

definition

The n a m e

of "sur-

by M. H a y a s h i . 3 and

6 are due

b e e n made.

to W i d o m

The r e s u l t s

in

CHAPTER

Every bounded almost means

harmonic

of the

Poisson

function

integral.

has a striking type.

discovered

Parreau

by M.

paper of Parreau time.

of Parreau the case

that

problem asking

of surfaces

positive

harmonic

issuing of Green

between

show that

harmo n i c

faces

i.

limits

V), results

Green

Green

in

sections

boundary. lines

on

are devoted

We show in 52 that

every

almost

line

every Green problem

data m e a s u r a b l e

on the

lines)

of functions

space

problem

and the fine

including

to

with re-

in §3 the B r e l o t - C h o q u e t

(along

for a class

objective type,

Definition Let

of Green R

g(a,z).

= r(a;z)

PROBLEM

and

8(z)

hyperbolic

is to i n v e s t i g a t e

3B, Ch.

limits

bounded

Green

assumption

Riemann

sur-

lines

on sur-

is e v i d e n t l y

V.

ON THE SPACE

OF GREEN

LINES

Lines

be a r e g u l a r Let

only r e g u l a r

this r e g u l a r i t y

in view of T h e o r e m

THE D I R I C H L E T

function

for any bounded We solve

for the

answer,

and the M a r t i n

and that the D i r i e h l e t

we c o n s i d e r

of P a r r e a u - W i d o m

IA.

r(z)

chapter

As our u l t i m a t e

legitimate

§i.

along

in that

appeared

functions.

In this faces.

same

type.

were

Brelot-Choquet

The other

has a limit

surfaces

direction

2B, Ch.

surfaces.

is r e s o l u t i v e

the radial

lines

by

as the Fatou

affirmative

to the

along

function

It is indeed type

is given about

of P a r r e a u - W i d o m

function

the

in this

(Theorem

type,

Green

known

discussion

spect to the Green measure.

are e s s e n t i a l l y

theorem

limits

limit

to all Riemann

in 1958.

of P a r r e a u - W i d o m

from any fixed point lines

result,

us to give a c o m p l e t e l y

Riemann

the case

famous

of P a r r e a u - W i d o m

with Widom's

relations

hyperbolic

from its radial

[52] already

In §i an i n t r o d u c t o r y general

in the unit disk has

generalization

surfaces

Combined

LINES

The first main results

then permit

of surfaces

This

direct

of P a r r e a u - W i d o m

and

GREEN

all radii and can be r e s t o r e d

theorem,

first

VI.

a E R

hyperbolic be held

= e(a;z)

Riemann

surface

fixed and define

by equations

with

Green

two functions

120

dr(z)/r(z) respectively. r(z) the

In the

= exp(-g(a,z)) point

which

a

large

the

following for the

~ 0

and

8(z)

set of G r e e n

~ > 0

conformally

the

de(z)

we d e n o t e

by

any m a x i m a l

issuing

region

with

the

: -*dg(a,z), r(z)

equation.

arc

a.

CI(R(a,a))

special

line

from

(= {(a),

for

from a

on

short)

For a s u f f i c i e n t l y

= {z e R: g(a,z)

unit

solution

issuing

issuing

$(R,a)

from

closed

the

A Green open

is c o n s t a n t .

lines

closed

isomorphic

and

first

is by d e f i n i t i o n

de(z)

denotes

= -dg(a,z)

disk

under

~ ~}

is

the m a p

z ÷ e~r(z)exp(i0(z)). Thus

each

= ~8 Using

Green

in such the

R

a way

that

and

the

set of G r e e n in

~

longs

to

Green lines

~0(a)

e

{(a)

in

So

on

only

called

regular

zero.

We d e n o t e

by

{(a).

Since

is r e g u l a r ,

R

infimum, L(R,a)

The u n i o n Green

$'(a)). the

The

region

function

E L(a)

lB.

by

point

a

~'(a)

the

S(f;~(a))

all a

set.

the

point

of be-

of the

A Green ~

unit

which

~ E {(a)

over

line

~

is e q u a l

We

s

(B2)

lim

set

is b o u n d e d

Green

the p o i n t

problem

real-valued

lines

in

is d e n o t e d

connected

as a g l o b a l

S(f;{(a)) pointwise

$(a)

is c a l l e d

z

on

~

on the

function

by

{'(R,a)

and

(resp.

in

on it.

For

for w h i c h

set f

~(a)

on the

functions

(or

and we can use every

g(a,z)

~ f(~)

let

for

Suppose G[f;{(a)]

supremum)

on

R

dma-a.e, that

both

(resp.

on

s

L(a), on

R

L(a).

[(f;{(a)) G[f~{(a)])

of f u n c t i o n s

in

=

of G r e e n set

below;

= -S(-f;~(a)).

are n o n v o i d infimum

to

lines

region

coordinate

set of s u p e r h a r m o n i c

inf ÷ 0 s ( z ( ~ ; ~ ) )

S(f;~(a))

is

Green

that (BI)

as

be the

CI(~)

set of r e g u l a r

and

is a s i m p l y

Dirichlet

the

= ~0(a)

a line

g(a,z)

the

with

= d8/2~,

closure

in a c r i t i c a l

2w)

= ~ N ~R(~,a).

{(a)

~0(R,a) the

see that

z, of

and at

denote

an e x t e n d e d

{z)

[0,

= L(a) UE0(a).

centered

z(~;~)

We d e f i n e Given

in

8 6

we h a v e

z ÷ r ( z ) e i0(z)

let

lines. denote

of the

star r e g i o n

to

= L(a)

with

identify

is a c o u n t a b l e

~(a)

the

that

if it ends

E0(a)

if the

Let

such

It is easy

2~)

with

d m a ( ~ 0) = dma(8)

{(a).

{(a)

set.

if and

g(a,z).

a measure

measure ~

is p a r a m e t r i z e d

8(z) ~ 8 (mod i0 ÷ £8' we can

introduce

is a c o m p a c t

function

in

correspondence

circumference is e a l l e d

line

and be the

S(f;@(a))

we such

121

(resp. and

S(f;{(a))).

s" E S ( f ; @ ( a ) ) , so that

lies,

G[f;@(a)] if

function

G[f;@(a)]

problem

G[f;@(a)]

5A,

Ch.

Dirichlet

f

on

R

are

nonvoid

An

only

is c a l l e d of G r e e n

with

the

following,

real-valued

which

if t h e r e

function

exist

s

on

function

a harmonic R

such

solution

lines.

and

common of the

It is c l e a r

that

is an a n a l o g u e

of T h e o r e m

for any

E > 0.

Proof.

The

If this

proof

sequence

{s~:

with

n : i,

function

u

to that

in

then

a*

superharmonic

super-

u : G[f;~(a)].

of T h e o r e m

5A,

~(f;{(a))

such

Ch. then

IIl.

In fact,

there

exists

a

that

< ~,

n

is any p r e s c r i b e d

is a p o s i t i v e

is r e s o l u t i v e

and a p o s i t i v e

u = G[f;@(a)],

(s'(a*)-u(a*))

n=l

L(a)

u - Es E S ( f ; @ ( a ) )

case,

solution

2,...}

on

and

is the

is s i m i l a r

is r e s o l u t i v e

f

that

u + ss e S ( f ; { ( a ) )

m > i

the

The

HP'(R).

extended

(i)

where

resolutive.

and

Problem

We b e g i n

harmonic

f

fami-

S(f;@(a))

is c a l l e d

{(a)

s' e S ( f ; { ( a ) ) are P e r r o n

are h a r m o n i c

and

and

space

for any g(f;~(a))

~[f;@(a)]

G[f;~(a)]

to

and

IIl.

Theorem.

if

by

s' ~ s"

S(f;{(a)) then

for the

belongs

2A.

if and

and If

= G[f;@(a)],

is d e n o t e d

Dirichlet

The

we h a v e ~(f;{(a))

~ G[f;~(a)].

G[f;{(a)]

2.

Since both

point

in

function

R. on

So the R.

Then

sum

s'

for any

= ~n=l (s~ - u) fixed

integer

we h a v e

=

u(z) + m - l s ' ( z )

~ u(z)+

m -I

(s n'- u)(z) n=l

=

for e v e r y ilarly, that

z @ R.

there

u - m - l s ''

superharmonic Suppose, positive

It f o l l o w s

exists

belongs

to

S(f;@(a))

that

function

s n'(z) that

u + m-ls ' E S(f;@(a)).

superharmonic

s = s' + s"

conversely,

superharmonic

-i ~ n=l

readily

a positive

function

m

for any satisfies

we h a v e s

function integer the

a harmonic

satisfying

s"

on

m $ I.

conditions function

(I).

Then

in u

SimR

such

So the (i). and

a

122

u : lim c÷O wherever

s

conclude

that

Using usual

(u + es) ~ [ [ f ; { ( a ) ]

is finite. f

Corollary.

s

is r e s o l u t i v e

this,

Dirichlet

Since

we can e a s i l y

~ G[f;{(a)]

~ lim s÷O

is i n f i n i t e

and

( u - ss) : u,

o n l y on a p o l a r

u = G[f;{(a)].

show the f o l l o w i n g

set, we

[]

as in the case of the

problem.

(a)

If

fl'

f2

are r e s o l u t i v e

functions

on

L(a)

and

el'

~2 are r e a l n u m b e r s , then ~ifl + a2f2, m a x { f l , f 2} and m i n { f l , f 2} are r e s o l u t i v e and s a t i s f y the f o l l o w i n g , in w h i c h we w r i t e G[.] in place

of

G[-;{(a)]: G [ ~ i f I + e2f2 ] : ~ i G [ f l ] + e 2 G [ f 2 ],

G[max{fl,f2}] (b)

Let

functions

= G[f I] V G [ f 2 ] ,

{fn:

on

with

Let

limit

sequence

G[f]

is the case,

2B.

n = i, 2,...]

~(a)

if a n d o n l y if the

G[min{fl,f2}]

= lim

f

{G[f

G[f

n÷~

n

n ].

= G[f I] A G[f2].

be a m o n o t o n e

sequence

of r e s o l u t i v e

= limn÷~

Then

is

fn"

]: n = I, 2,...}

f

: n = i, 2,...} be a s e q u e n c e of p o s i t i v e n u m b e r s n d e c r e a s i n g to zero, such that DR(en,a) N Z ( a ; R ) = ~. We set

strictly

n = I, 2, . . . .

Since the

surface

R

is r e g u l a r ,

family

{R : n = i, 2,...} forms a r e g u l a r e x h a u s t i o n n sense of Ch. I, IA. For e v e r y n the G r e e n f u n c t i o n

with pole

a

is e q u a l to

function

length measure R

n

on

DR

u(a)

=

g(a,z)which

n

and if

w i t h the b o u n d a r y

(2)

u

1 ; 2~ ~R

So if

is s u m m a b l e

is the

function

an.

solution

f

=

Lemma.

Let

s

the g r e a t e s t [(~

)(~) n

T h e n we h a v e

lr2

in the

gn(a,z)

is an e x t e n d e d

with respect

for

Rn

real-

to the a r c problem

for

1 I DR = -~-~

f(z) *dg(a,z) n

f(z(go;an))dO

be a s u p e r h a r m o n i c harmonic

R

the

f, t h e n

f(z) * d g n ( a , z )

-)--~-JO

of

of the D i r i c h l e t

n

u

If this

{a

R n = R(~n,a) ,

valued

resolutive

converges.

minorant

=

I ~(a)

function of

= lim inf s ( z ( £ ; ~ n ) ) , n÷~

s

on ~(Z)

f(z(g;an))dma(~)"

which R.

is b o u n d e d

b e l o w and

We set

= lim inf s ( z ( £ ; e ) ) . e÷O

123

u(a) => I]L(a) S(c~n )(£)dm a (£) => I]L(a) -s(£)dm a (£). Proof.

Let

un

be the greatest

u n = H[s;R n]. Then, of (2) we have

: lim Un(a) n÷~

u(a)

=>

2C.

harmonic

minorant

u n => Un+ I => --- => u

I]L( a )

on

of

Rn

s

and

on

Rn, i.e.

Un ÷ u.

By use

: lim IL s(z(£;~n))dma(~) n+ ~ (a)

S(en)(Z)dm

We apply the preceding

a

(£) => I]L( a )

result

s(£)dma(Z). --

to the Dirichlet

~

problem

defined

in IB. Theorem. that

Let

f(~)

~(f;{(a))

G[f;{(a)](a) ~

and

Proof.

If

I

f(£)dma(~)

s

f(~)dm

L(a)

such

(£) > G[f;~(a)](a), a

the lower and the upper

and if

2B implies

=

--

integrals

with respect

> I]L( a )

u

is the greatest

harmonic

minorant

that

s(Z)dm a --

(£) > T]L ( a )

f(£)dm

a

(£).

is arbitrary, G[f;¢(a)](a)

The remaining tegral

~ r

on

dma, respectively.

s 6 ~(f;{(a))

u(a)

function

Then we have

//u(a)

denote

s, then Lemma

Since

real-valued

are nonvoid.

JL(a)

to the Green measure

of

S(f;{(a))

> r =

where

be an extended

and

inequality

is always

inequalities.

7

> | f(~)dm (£). = JL (a) a

can be obtained

similarly.

larger than the lower integral,

As the upper in-

we @et the desired

[]

Corollary.

If an extended

resolutive,

then

f

real-valued

function

f(~)

is dm a -sunlmable and G[f;{(a)](a)

: [

~m (a)

f(Z)dm

a

(~).

on

]L(a)

is

124

§2.

THE S P A C E OF G R E E N

In w h a t Riemann

3.

follows

surface

The G r e e n 3A.

gion

ON A S U R F A C E

OF P A R R E A U - W I D O M

in this c h a p t e r we a s s u m e

of P a r r e a u - W i d o m

that

R

TYPE

is a r e g u l a r

type.

Star R e g i o n s

We c h o o s e

notations

LINES

given

a point

in IA.

D = D(a)

a • R, w h i c h

We map the G r e e n

in the open unit d i s k

is h e l d fixed, star r e g i o n

~

and use the

¢'(a)

onto a re-

in the c o m p l e x w - p l a n e

con-

W

formally

and u n i v a l e n t l y

by m e a n s w : ~(z)

where

r(z)

= r(a;z)

9 • [0, 27) ~(a) UE0(a

with )

and

e(z)

of the g l o b a l

= Q(a;z). Since

Se,

8 E ~0' d e n o t e s

the end p o i n t

the

slit

of the G r e e n

e @ ~0'

s h o u l d be a c r i t i c a l

Theorem

iC , Ch.

V, shows,

In v i e w of the fact of the slits

theorem

there

exists

in the c o m p l e x

be the set of all we h a v e

¢(a)

~ r ~ i},

z(e)

:

point

of

[{g(a,zj):

he

g(a,z),

e • ~0'

onto

$8,

D

~D

this

is finite. ~

of

D

Each

z(e),

z(e) E Z(a;R). < =

As

and therefore

< ~. that the t o t a l

By the R i e m a n n m a p p i n g the o p e n u n i t d i s k

and u n i v a l e n t l y .

is the u n i o n

The a b o v e

a.

implies

which maps

conformally

0 • ~0"

than

i.e.

zj • Z ( a ; R ) }

Se,

a function

r(z(8))

other

z]. E Z(a;R)}

= exp(-g(a,z)),

boundary

slits

e E ~0 },

r(z)

E-plane

the t o p o l o g i c a l and all the

~0

is r e g u l a r ,

{reie:

line

I{1 - e x p ( - g ( a , z j ) ) :

length

Let

R

and so D = D w \ U{S8:

where

function

= r ( z ) e i0(z)

%~ ~ E 0 ( a ) .

(3)

being

coordinate

of the c i r c l e

observation

~

Clearly, ~D

shows that

w ~D

is r e c t i f i a b l e .

3B0 ~.

We are g o i n g

First we prove

Theorem. ci(D

~

to m e n t i o n

some u s e f u l p r o p e r t i e s

of the f u n c t i o n

the f o l l o w i n g

can be e x t e n d e d

continuously

to the c l o s e d u n i t d i s k

).

Proof.

$

set of

e E [-7,

By T h e o r e m

is a b o u n d e d 7)

2A, Ch.

hoiomorphic

function

such that the r a d i a l IV, the set

~

on

limit

~%.

Let

~

l i m r ÷ I $(re i0)

has a n e g l i g i b l e

complement

be the exists. in the

125

interval in

I-z,

D~

and

a point orem,

in

~).

by ~D

let

c

C0 : i

for the

in

to

in

~D.

sequences • -. ÷ 0

curve

sake

in

r ÷ i.

be the

{@"(n)}

> 8"(2)

>

in

~

G n : {z @ ]D :

@'(n)

G'

: ~(G

converges

In o r d e r

to p r o v e

to a point

on the

in

contrary

in

such

that

We take < arg

~D.

n

~. ZD.

Set

that

c'

points

We then

@'(i)

any

take

< @'(2)

and

to

the the-

S0 E

to a p o i n t

set of a c c u m u l a t i o n

included

.-- ÷ 0.

0 £ r < i}

c$

converges Suppose

E $ ~D

{rei@:

~, then

converging

= ~(e)

is a c o n t i n u u m and

8"(i)

the r a d i u s under

as

~

e'

let

E

{@'(n)}

c@ c@

of s i m p l i c i t y .

and

Then

by

of

@ 6 9

show t h a t

D

and

image

every

be any

only

c'

the

for

We h a v e

oscillates

If we d e n o t e

c~

of two


0 on

w

the

to

as

maps

on

We @

point

e@'

to the

boundary

n + ~.

This

to a p o i n t

extension ~

of

with

tends

e@.

the Since

v = u O ~,

so

to e i t h e r

to

~D \ ~G'n

along

with

that the

~D, as d e s i r e d .

T

to

continuously

CI(D~) onto

~D.

and So

' c@.

It f o l l o w s

contradicts in

G'.n

for the

belongs

curve

along

D

in

of

w E ~ D \ E.

set

in

is c o n t a i n e d

for

if

D. If

~

converge

continuous

= 0

~.

goes

goes ÷ 0

e'

comes

on the b o u n d a r y

and

E

{el@:

eventually

image

lies

Dirichlet

then

as

~

E

c

c@,,(n ) '

fact []

by the Then

following

The

its r a d i a l

fined

c'

~),

zero as

of the

e@.(n),'

to the arc

curve

w E E

and h a r m o n i c

< (@"(n) - @ ' ( n ) ) / 2 ~

u > 0.

of the

and t h e r e f o r e

zero

the

part

if

curves

corresponding

continuum

point

(@"(n),

thus

v(0)

= i

every

problem

v(~)

and

solution

is b o u n d e d

@'(n))

the

).

convergent

Since

oscillating

f(w)

is a c o n t i n u u m ,

~D

~C.

the r e g i o n

boundary

by two

n

derivative limit

~'(6)

is a l m o s t

belongs everywhere

to the H a r d y equal

to

class

~T/~,

HI(~) which

by 8~. i@ ) : lim ~-~<e t÷0

(~(e l(@+t) ) - ~ ( e i @ ) ) / ( e i(8+t) _ ei@).

is de-

126

is of bounded Proof. Since ~D is rectifiable,, the function. ~I~I3% variation. We set d~(e it) = (ielt)-id~(e~t) on ~I3 . Then I and for

eitd~(eit)

= i-lf

d~(eit)

= 0

n : 2, 3,... I

enitdB(eit ) = i-i I ~D{

e(n-l)itdY(e it) 3D%

(n-i)[

~(eit)e(n-l)itdt

= 0,

J because ~({) is holomorphic in I]~ and continuous up to the boundary. By the F. and M. Riesz theorem (see Appendix A.3.2 and A.3.3) dz is absolutely continuous with respect to the arc-length measure and u(rei8 ) = i

0

P(r,O - t)d~(e it)

belongs to the Hardy class HI(ID ). Setting have by the Fatou theorem (Appendix A.I.2) f(e it )

= ~~(

eit

)

d~(e it) = f(eit)dt, we

a.e.

So the radial limit of u(~) is equal to ~ I ~ a.e. on ~]D . On the other hand, since du is orthogonal to e nit , n = i, 2,..., we have I~D

e -it + ~ e-l~_~d~(eit)

= f~]D~

d~(e it)

and therefore

u(~)

=

i I i [e it + ~ e -it + ~] ~ [e-~-- + d~(e it) ~ ~9~ ~ e_mt_ ~j

= 4~ = ~i =

f

•

[elt+ ~ ~D [e±t_ ~

+I

i d~(e it) = 2~

f

~]D

it

e__

el~_

I ~13 -e ~ I - % d~(eit) = 2-~ i ; ~I] (e~(eit) it- ~)2 d(eit )

~'(~),

which implies the desired result. [] Combined with the Fatou theorem, the preceding theorem shows:

127

Corollary.

If

~Y/Z%

verges to the value

exists at a point (~Y/~6)(~0)

any fixed Stolz region 3D.

in

~{

as

with vertex

in Appexdix A.I.2.

For

the connected

S(~0;~,p) {z ~ ~:

to which the point

~0 ~ ~ '

S(~O;~,O)

through

~0

of Stolz regions

0 < ~ < ~/2 component

is adherent.

con-

Y'({)

tends to

~0"

and

given

0 < 0 < i

we de-

of the set

-~ ~ a r g ( l - Z~o) ~ ~, 0 ~ ~0

then

%

We now have to modify the definition

note by

region

%0 ~ ~ '

uniformly

Izl < I},

In order to make the new Stolz

simply connected,

we always assume that

P ~ sin ~. Theorem.

(a)

every angle (b)

(~/~6)(~0)

exists and is different

80 E [0, 2~) \ ~0' where

Let

w 0 : exp(i80)

80 E [0, 2~) \ ~0 = ~(%0 ).

Then

satisfy the condition

SD

have the same tangent at the point

with

0 < ~ < ~/2

the region

Stolz region with vertex

p

depends

S(w0;a,p) ~0

{0

~0' lies in

~%

We showed

Poisson

integral of

Theorem

in Theorem

function 2A, Ch. IV.

point in

the

in

~

3C that

Stolz region

(< ~) in the disk sides which, at

%0

and ~0 ei(~+~),

Since

~-i

Stolz region

~/~ ~'(~)

Hence the statement

~6

any

except the vertex to the two adjacent

respectively.

exists a.e. and is not identically

can vanish only on a set of measure

c

which,

(a) holds

~'

is the

zero, its zero by

for almost every

~0"

~

satisfies

the condition

except the initial point

and has a tangent at

tial to the circumference tinct from

2~

80 E [0, 2~) \ ~0

Draw a smooth curve ~

Under the map

~.

Suppose that disk

Z~/$~. ~T/~

D) and

2e.

and such that the tangents ~0 ei(~-e)

of

for any

a Stolz region of the form ~.

is sent to a curvilinear

triangle with smooth

Proof.

Consequently,

(b) above we mean by a curvilinear

sides have directions

boundary

and

and angular measure

closed eurvilinear

w 0.

contains w0

and angular measure

In the statement with vertex

D on

in (a) and set

(see (3) for the definition

~w

S(w0;~,0) , where

from zero at almost

%0 = ~-l(exp(i80))"

t 0. .

Let

We suppose that ~i' ~2

We have

~(~i ) - ~(~2 ) =

T'(~)d~,

in (a).

~0' lies in the c

is not tangen-

be two points on

c

dis-

128

where

the r i g h t - h a n d

and let

%2

side is the i n t e g r a l

tend to the p o i n t

~0

along

along

c.

the c u r v e Then

c.

Fix

tl

~(~2 ) ÷ ~(%0 ), so

that

I

~(t I) - ~(t 0) = Since

~'(t)

tends

to

(3~/3~)(~0)

shown that the d e r i v a t i v e

of

is e q u a l to

(~/~t)({0).

arg{~(%0)}

u n d e r the map

angle

between

with respect boundary tangent Hence

~D to

two c u r v e s

to the c i r c u m f e r e n c e

~.

at

cI

{0

between

~D

and

at

~

÷ ~0

~(e).

~(~0)

to

tends

As

~D

Since

~(t ~ )

to

~0' we h a v e

the c u r v e e

issuing

under a rotation the c i r c l e

and

along

at

c

{0 ~0

(3~/~)(~0)

~ 0, the

is o b t a i n e d

e and

e 3~ w .

f r o m the

arg{~(t0)}.

= w 0 6 3~ w

{~n*: n = i, 2,...}

the

nontangentially

by the a n g l e

( = w n, say)

by

preserves

and the curve

~(~0)

e x i s t s and

is r o t a t e d

~(t)

from

w 0 = ~(~0 ), w h i c h

we can find a s e q u e n c e

~ ~0'

t E c

%@

and t h e r e f o r e

e2

~Dt

as

~ + ~(~) and

the a n g l e

tifiable,

at

~'(t)dt.

So the t a n g e n t

has a t a n g e n t

that b e t w e e n

~

{l

is e q u a l to SD

~C ~

is r e c -

such that

Then,

wn ÷ w 0

and

~ ( t ~ ) - ~(t O) w - wO % ~ - ~0 = lim n n÷~ ~tO

= lim n÷~

We thus h a v e a r g t0~r[3~~ " )} = a r g w 0 - arg ~0 This

shows

w 0.

The first h a l f of

in p a r t i c u l a r

c'

~D

and

a Stolz-region

be any one of its sides

issuing

arg(1-w~ for

w E c'

D t.

Set

We s h o u l d

with

w i t h the c o m m o n

for

~i 6 ci,

above,

c!l

equations c'

c

arg(l-

initial

i : i, 2.

converge

to

~ 0 ) = e.

a r g ( l - w i w 0) = ~l.

point

~0

c

from

for i n s t a n c e

to

t0

for

w0:

c

in

at

D

is a s m o o t h

and

cI

arg(l-

and

determined

lies in a Stolz r e g i o n w i t h v e r t e x

in

As s h o w n

w0, w h i c h

w.l ~ c!l' r e s p e c t i v e l y . w0

~I' c2

t i ~ 0) = ~i

c i' = ~(c.). ± at

in

at

we take any

segments

such that we set

curve

and has a t a n g e n t

To see this,

and have t a n g e n t s

lies in a Stolz r e g i o n w i t h v e r t e x

and c o n s e q u e n t l y

included

and d r a w two

And f u r t h e r w0

S(w0;a,p)

so t h a t

converges

0 < ~i < e < e2 < 7/2

~t

h a v e the same t a n g e n t

w

0) =

c = ~-l(c' \ {w0}),

show that

t0 , for w h i c h we h a v e ~2

$~

(b) is n o w clear.

Let us now c o n s i d e r let

that

(mod 2n)

satisfy

the

So the c u r v e c I'

by ~0

and

c~,

determined

129

by

cI

to

e, we h a v e

and

satisfying of the

c 2.

the

Since

shown

equation

statement

4.

Limit

X.

For a g i v e n

Let

can be t a k e n to

~0

~ 0 ) = ~.

arbitrarily

and has

This

proves

close

a tangent the

at

latter

~0

half

Lines

be a f u n c t i o n Green

u~(~)

e2

converges

[]

Green u

and c

arg(l-

(b).

along

4A.

el

that

line :

on

R

with

~ ~ ~(a)

n

we

Cl[u({z(~;B):

values

in a c o m p a c t

space

set o

~ < ~})].


o If

u~(~)

dial

consists

limit

Theorem.

of Let

Green

star

~(~)

exists

on

(i)

~ in

~

u

that,

Since

to

E

this

property.

~-io

¢

Hence

Then

(ii) {'(a)

Let

u

is s i m p l y

uniquely

under

the

in any So on

zero.

is a b o u n d e d

holomorphie every

function

~0

function

through

under

with

vertex

in

for a l m o s t

every

so that

its r a d i a l

limit

harmonic

the h a r m o n i c ~(a)

= 0.

: exp(-u(z)

function ~ E L(a).

on

for a l m o s t function

conjugate Sinc~

u

say)

the map in

E.

~ e L(a). vanishes

every

on ~

{0

possess

~80

exists

~ 0

~%

80

Stolz

we see

(= {0'

Let

~, ~0

fixed

zero,

~-l(exp(ie0))

line

on

of such

of m e a s u r e

n0).

and

point

any

totality

region

~(~)

limit

~'(a)

every

Stolz

,

on the

the

on

~(~) •

condition

function

of

Green

~(~).

dma-measure.

sets

2~) \ n0,

of the

u

be the

definition

the ra-

is dm a - m e a s u r a b l e vanishes identically

holomorphic

to

E

by

~ E L(a)

for a l m o s t

preserves

be any p o s i t i v e

the

for a l m o s t

tends

Hence

connected,

tends

Let

[0,

image

h(z)

exists

%

t 0.

80 E

v ~ 0

of m e a s u r e

as

÷ ~D

exists.

u ~ 0, t h e n

is a b o u n d e d

3A for the

is c o n t a i n e d

on a set

~

every

(see

~(Ze0 )

limit

harmonic

then

holomorphie

v(~)

vertex

~:

for a l m o s t

v

is c a l l e d

is d e n o t e d

~ ÷ ~(~)

holomorphic,

theorem

point

every

function

on a set of p o s i t i v e

Then

with

and

for a l m o s t

be a b o u n d e d

Fatou

~

the

be a p o s i t i v e

The

is b o u n d e d

then

line

u

Then

vanishes

~%

.

belongs

If

u

to a f i n i t e

region

let

{'(a).

point,

Green

is finite.

Let

by the

one

the

and

v = u o ~ - i o ~.

so that

in

and

~(Z)

Proof.

in

a E R

If

whenever

along

region

L(a).

set

u

of o n l y

{'(a). of

only

~ ~ L(a).

u

Since

exists

is p o s i t i v e ,

- i[(z))

~'(a) Since

and t h e r e f o r e ho ~-io ~

does

by

(i)

not

h(~)

vanish,

130

its r a d i a l

limit

that

h(~)

~ 0

~(Z)

= -log

4B. fact,

lh(Z) I

In the

Let

case

f(~)

on a set of m e a s u r e

and

~ E ~(a).

is f i n i t e

of a PWS,

Theorem

and

Consider

•

be a b o u n d e d the r a d i a l

for a l m o s t

Rn

with

every

in 2B.

limit

{R

is d e f i n e d

the o r d i n a r y

Then

fn

a.e.

We

Dirichlet

bounded.

It is e a s y

bounded

equioontinuous

subsequence monic

data

set

clear

see

In

that going

S~

to

S~:

+ S 0) if

solve

w

rectifiable

boundary

let can

slit

the

and

of

with

of the

can be v i e w e d

Then is equal

Rn

Dirichlet

=

problem

conformal

limit

So,

forms

of

a uniformly passing

converges

to a

to a h a r -

map

}

of the

we d e f i n e d

in

to b e l o n g w' E D

D.

For

of its

to

two

Se

satisfying

common

point

exp(-g(a,z(0))

sense

3A.

Pn = It is

as the u n i o n

only

i.e.

Green

z E R'n, w h e r e

for the r e g i o n

of

The

in an o b v i o u s

~R n,

is u n i f o r m l y

D n = (pn D) D D w .

e E ~0'

> 0).

2,...}

which

for

is d e f i n e d

as the

on

solution

R.

problems

$0,

is the

{u n}

on

D : D(a),

slit

R

the

Consequently,

= pn ¢(z)

w E $8

I m ( w ' w -I)

of

Se

+ i0).

as a J o r d a n

region

Then with

L, w h e r e

L = U{S +e U S~:

v

n Un+2,...}

).

uniformly

Dirichlet

is r e g a r d e d

D

exists

function

: n = I,

n a s s u m e that

Cn(Z)

each

.. }

measurable

{u

CI(R

the r e g i o n

a point

the r e g i o n

Now

on

we can

and

I m ( w ' w -I) < 0 (resp. + and S 8 is the v e r t e x

L(a).

= f(~)

{Un+l,

we c o n s i d e r

we r e g a r d

and

on

where

Then

that

almost

onto

.

u n = H[fn;Rn] , which

family

hand,

2,

we c o n s i d e r

be the image of Cn' i.e. D I =D D 2 ~ .. • =D D n =D . . . D D .=

Dn

purpose,

(resp.

to

u,

~'(a)

Let

We are

n ary

lh(z)1,

strengthened.

function

i,

=

n

fn'

problem.

R'n = {'(a) D R n

exp ~n"

edges

say

other

star r e g i o n

this

It f o l l o w s

= -log

[]

2C is m u c h

is a b o u n d e d set

if n e c e s s a r y ,

function, On the

then

zero.

u(z)

G[f;{(a)](~)

: n

n

For e a c h

the b o u n d a r y

~ E L(a).

which

We

a.e.

measurable

fn(Z(~;en)) for

As

~ E L(a).

an e x h a u s t i o n

R(~n,a) , as g i v e n for

only every

exists

f(~)

f is r e s o l u t i v e

Proof

vanish

we have:

Theorem.

to

can

for a l m o s t

0 E ~0}

U {e ie : 8 E

[0,

2w) \ ~0 }.

v be the r e s t r i c t i o n of u o ¢-i to the r e g i o n D. Then n n n c l e a r l y be e x t e n d e d c o n t i n u o u s l y to the b o u n d a r y L, the b o u n d -

values

being

denoted

by

f~.

If

Z = ~e E ~ ( a ) ,

then

we h a v e

131

+

f~(e 18)

point

= f(£).

If

Zn(W) 6 R

sequence

{zj:

~n(Zj) ÷ w tinuity.

D.

W E S;

(or

converges

w

of

f~(w)

C R'

=

f*(W)n

the

is

sequence

£ = £8 E l ( a ) ,

of

z 6 S + US;. e D supported

see that of

L.

d~ w

exists

there

3

÷ zO(w).

+ U(Zo(W)).

function

f*,

is

z. ÷ z (w)

a

a

and

n

determined

by c o n -

(w): n = I, 2,...} n to a p o i n t z0(w) 6 R.

: Un(Zn(W))

Thus,

Namely,

stays In

if {f~}

i.e.

: f(£)

and

f*(w) if

properties

uniquely

~-l(pnlw')

to a w e l l - d e f i n e d

there

and (ii)

{z

converges

f*(e ie) if

S8) , then

= Un(Zn(W))

with

w' ÷ w, t h e n

then

pointwise

and

(or

Zn(W)

n

the

R

and

S:),

(i)

2,...}

such

subset

w' E D

w e S;

Such a p o i n t

For e a c h

if

and

such that

j : i,

in

in a c o m p a c t

fact,

8 e ~0

: u(z0(w))

For any

given

w E D

on

Since

L

L.

is a b s o l u t e l y

let

d~

w is r e c t i f i a b l e ,

continuous

be the h a r m o n i c it is not

with

respect

: S

f~d~w'

hard

measure to

to the a r c - l e n g t h

As we have

(U n o }nl)(w)

: Vn(W)

L so we d e d u c e , vergence

by l e t t i n g

n ÷ =

and

using

the

f

(u o ¢-l)(w)

= I

f* dm

JL Namely, D

u o ¢-i

is e x a c t l y

corresponding As we h a v e

i}

with

theorem

dominated

con-

8 6 shows

to the seen

[0,

the

solution

function

in T h e o r e m

2w) \ ~0

of

r ÷ i

of the

problem

3D, ~w

almost

every

is o r t h o g o n a l

radius

for a l m o s t

every

8 E

every

£ E ~(a),

Every

we can

[0,

as was

to

~D.

So the

= f(£8 )

2z) \ ~0"

This

means

that

= f(£)

to be

shown.

[]

show:

dm a - s u m m a b l e

for

{re ie : 0 ~ r
0 = 0i(b ~)

last equality sign being true by T h e o r e m 4B, Ch. IV. bounded l.a.m, fined by

h

h = exp(-vi . - v q ) and denote by q the line bundle deThen we take any nonzero l.a.m, k < i,

(Ch. II, 2C).

whose line bundle is

q-i

(Theorem 2B, Ch. V).

We set

0

and

Vq'

This is possible in view of Widom's theorem - l o g k = v i' + v'q

~q(£)

for almost every Since

with

v'i : P r i ( - l o g k) =>

pr Q (-log k) => 0, and see as above that

=

(17)

h

and

7(£) = f(b£)

almost every

: Q'(b~)q

Z 6 A(a). k

c o r r e s p o n d to m u t u a l l y inverse line bundles,

there exists a nonzero have

a.e., the

We look at the

f 6 H=(R)

such that

for almost every

hk : IfI

£ E A(a).

on

R.

We thus

This means that for

£ 6 A(a) lim h ( z ( ~ ; ~ ) ) k ( z ( ~ ; e ) )

exists and is equal to

(18)

If(m)l

: If(bm)l =

exp(-Qq(b~))exp(-~'(b~))q

by means of Corollary 4B, Ch. IV. almost every

: ~(bm)~(bm)

From (16) and (17) follows that,

for

£ E A(a), lim e x p [ - V q ( Z ( ~ ; ~ ) ) - v'(z(~;~))] ~+0 q

exists and is equal to

exp[-vq(b£) - 0~(b£)]

~ 0.

Therefore

lim exp[-v.(z(~;e)) - v!(z(£;e))] ~+0 1 1 exists

and

E A(a). value

is

equal

On t h e

to other

If(b~)[expEOq(b~) hand,

since

+ O~(b~)]~

v.

1

and

v!

1

for are

almost

(18) is m a j o r i z e d by lim inf e x p [ - v i ( z ( ~ ; ~ ) ) ] ~+0

Combined with

every

nonnegative,

: exp[-v~(b£)] $ i.

(18), this gives

i ~ If(b~)lexp[0q(b£) + 0~(b£)] ~ exp[-v~(b~)] ~ i.

the

140

Hence, to

v~(b~)

0

: 0

for a l m o s t

a.e., every

which means ~ E A(a).

that

on

A(a).

This e s t a b l i s h e s

The g e n e r a l to the p o s i t i v e

7.

Boundary 7A.

R

Behavior

Let

R

be a r e g u l a r

with vertices

in the M a r t i n

We d e n o t e

by

our a s s e r t i o n s

parts

of A n a l y t i c

~(a,a')

and is e q u a l

= 0

case can be s h o w n by a p p l y i n g and the n e g a t i v e

exists

Thus

~'(£)z = Oi(b~) a.e.

~i(~)

of

when

log u $ 0.

the a b o v e

log u.

consideration

[]

Maps

PWS.

We first d e f i n e

boundary

the set of

41 .

e E [0,

Let

2~)

Stolz r e g i o n s

a, a' E R

satisfying

in

be fixed.

the f o l l o w i n g

conditions: (i) (ii)

~8 E A(a); (~I~¢)(~)

exists

and

is d i f f e r e n t

from

0

for

6 =

~-l(exp(ie)); (iii)

both

P ( a , a ' ; Z 8)

and

P ( a , a ' ; b 0)

exist,

are f i n i t e

and

satisfy P(a,a';~8) (iv)

both

l i m i t s at

e ie

respectively, Moreover,

: P(a,a';be)

F0 o ~-i

where

(F 0 o ¢ - l ) ( e i e )

F0(z)

and

and

have nonvanishing

(F 0 o % - i ) ( e i 8 )

= F(a,a';z)

F 0 o ~-l(w)

~

A(a)

and

and

and

Fl(Z)

with

~

were

were g i v e n

and

finite radial

(F I o % - I ) ( e i 8 )

= F(a,a';z)P(a,a';z).

F I o %-l(w)

respectively,

S(eie;~,p)

statements,

F(a,a';z)

by

(F I o ~ - l ) ( e i S ) ,

through any Stolz region

z),

FI o ¢-i

w h i c h we d e n o t e

the f u n c t i o n s

In t h e s e

and

: k(be,a')/k(bs,a) ;

tend uniformly

as

w

tends

to

to e ie

0 < ~ < 7/2. defined

in 5B.

in 3A, w h i l e

Our p r e v i o u s

P(a,a';

observations

s h o w the f o l l o w i n g Lemma.

The

7B. included

set

Let

set

D(a)

This m a y be c a l l e d be .

We c o n s i d e r (Theorem

complement

in

[0, 27).

any Stolz r e g i o n

3D), w h e r e

0 < ~ < 7/2

S(eiQ;e,p) and

0
0,

a compact

proof

subset

of T h e o r e m

5D,

- eU < F < h [ F ]

K

of

Ch.

Ill,

R.

+ sU

Since

shows

that

F

is f i n i t e

we m a y

assume

everywhere, U(a)

< ~.

153

Given

any

~ > 0, we take

inequality

n

(7) w i t h r e s p e c t

differential

so large to

-(2~i)-16g(a,z)

hEnCa) : I

d~n, to

Ks =C R n

that

which

SR . n

and i n t e g r a t e

is the r e s t r i c t i o n

the

of the

Since we have

hEFl 0, a n u m b e r

Sinee

9

m ~ m 0. < u- u

F(b)kb(a)dx(b)

=

m

5B-(b),

m = i, 2,... a.e.

on

A I.

R, w h e r e

0 < I

llI,

h[F m] =

By the fact we

m = I, 2,...

and any

¢ > 0,

such that

f

Fm(b)kb(a)dx(b)l

< s

and

Fm + 9

a.e.,

there

exists,

for

such that

< I

Fm(b)kb(a)dx(b)

I

u

m

we have

= min{u,

0 ~ F ~ u

m}.

F(z)d~n(Z) - I DR

+ ~ and t h e r e f o r e that

n U m ( Z ) d Z n (z)

r

DR

It follows

Fm(Z)d~n(Z) ~R

n

u(z)dDn(Z) -]DR n

n

< u(a) - (u A m)(a). If we take

Ch.

with

AI

On the other hand, on

m}

is s u m m a b l e m 0 = m0(s)

AI

m

(6) by let-

A1

any

for

the i d e n t i t y

By T h e o r e m

for

for any

n o = n0(m,¢)

~R n

I

case.

function

Fm = min{F,

there

f

F

we have

s ÷ 0.

m ~ m0(s)

and

n ~ n0(m,s) , then we o b t a i n

F-

154

II

F(b)kb(a)dx(b) - I AI

Since

u

implies

F ( z ) d ~ n ( Z ) [ < 2e + u(a) - ( u A m ) ( a ) . ~R n

is q u a s i b o u n d e d , the d e s i r e d

4B.

we h a v e

result.

By the d i r e c t

(u A m)(a)

+ u(a)

as

m ÷ ~, w h i c h

[]

Cauchy

theorem

of w e a k type we m e a n

the f o l l o w -

ing result: Theorem.

Let

..., z I

R

be a r e g u l a r

be any f i n i t e

subset

g0(z) If

f

is a m e r o m o r p h i c

majorant

on

R, t h e n

= exp(on

R

a.e.

on

= I

surface

and let

Zl,

We set

I ~ g(zj,z)). j=l

exists

f(a)

Riemann

Z(a;R).

function

f

(8)

hyperbolic of

s u c h that AI,

has a h a r m o n i c

Iflg o

is s u m m a b l e

and

f(b)kb(a)dx(b)" A1

Proof.

Since

Theorem

4A, Ch.

surface

R

compact

set

z~

and

that

c

R \ K.

IV,

is an l.a.m, shows that

being regular, K

in

go ~ e -i

harmonic bounded

]flg 0

u

R

on

R\ K

on

Re(f)

We then m o d i f y

functions

of T h e o r e m on

R.

(resp.

fl

on

R

Since

with

III,

-

by

X(A I \ N ( f j / u ) )

f2/u = 0

j = i,

the p r e c e d i n g

if

lemma

(fl + i f 2 ) ( z ) 4 g ( a ' z ) 8R n

Im(f))

for

and

is

it is q u a s i function K

on

so as to

By a p p l y i n g

A I \ ~(Re(f)) and

set

We see

f2 ) on the s u r f a c e

n

lim ~ n+~

(resp.

and a

Zl,... ,

R \ K.

c-lu,

on the

f(z)6g(a,z) 8R

on

Re(f)

If21 S c-lu).

fl/u

R, Our

c > 0

is thus a W i e n e r

Im(f))

on

the p o i n t s

]fl ~ c - l u

(resp.

fj = (fj/u)u,

Ifjl S e-lu,

-lim ~ n÷~

that

number

contains

Im(f))

(resp.

function

It f o l l o w s

5D, Ch.

a positive

Since

majorant

is q u a s i b o u n d e d .

in m o d u l u s

G = R \ K, we see that

are n e g l i g i b l e . uous

R \ K.

(resp.

Ifll S c-lu

III, w i t h

on

Re(f)

a harmonic

Int(K)

and is m a j o r i z e d

R \ K.

the p r o p e r t y

exists

So we have

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

on

have a real c o n t i n u o u s

Ch.

there

such that R \ K.

having

u = L H M ( I f l g 0)

R

Lemma

with 5C,

A I \ N(Im(f))

are b o u n d e d j = i,

2, are W i e n e r shows that

2.

continIn v i e w

functions

155

= I

(31+ if2)(b)kb(a)dx(b)

= S

AI

f(b)kb(a)dx(b)' AI

where

{R : n = i, 2,...} is an e x h a u s t i o n of R m e n t i o n e d in 4A. If n n is so l a r g e that R includes K, t h e n f(z)6g(a,z) is a m e r o m o r n phic d i f f e r e n t i a l in z on the c l o s e d r e g i o n C I ( R ) w i t h o n l y one n p o l e at the p o i n t a, w h o s e r e s i d u e is equal to -2~if(a). H e n c e , we get the f o r m u l a

§3.

5.

(8).

[]

APPLICATIONS

Weak-star 5A.

which

(see H e l s o n Theorem.

that

extends

[33],

p.

H~(dx ) Let

algebra

of

s* = 0.

L~(dx )

that

bounded

among weak*

generated

S (a)

is b o u n d e d

on

there

and analytic

closed

and let

by

C

H=(dx ) to

we p r o v e the

on

R.

on

R.

a

function

Since

C

closed f*.

Suppose

is any f i x e d p o i n t

in

B E H~(R)

B

is c l o s e d

in

H ~ ( d x ), we h a v e

I

B(b)u(b)(f*(b))ns*(b)dx(b)

under

such R. such

is o b t a i n e d

~(R,(~(a))-l).

sub-

to s h o w that R

function

u

L~(dx).

on

Such a function in

of

be the w e a k l y *

It is s u f f i c i e n t

a nonzero

element

subalgebras

and the f u n c t i o n

C.

R, w h e r e

exists

and a n o n z e r o

t i o n by f u n c t i o n s

theorem,

in the ease of the unit d i s k

we take any m e r o m o r p h i c

IB(z) I ~ g(a)(z)

tiplying

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

is o r t h o g o n a l

is a PWS,

Cauchy

27).

To see this,

R

of the i n v e r s e

f* ~ L=(dx ) \ H~(dx)

Iulg (a)

Since

H~

is m a x i m a l

s* E LI(dx)

that

of

As an a p p l i c a t i o n

following,

Proof.

Maximality

by m u l -

Thus

Bu

is

the m u l t i p l i c a -

: 0

AI for

n = 0, i, . . . .

each

n = 0, i,...,

hn : B s * ( f * ) n m a p of

R

by T h e o r e m

with

a.e.

By the i n v e r s e a function on

Al"

%R(0)

= a.

7B, Ch.

h

Let Then

n

Cauchy

~R:

D

on

~D. of

there

exists,

for

such that

÷ R

h (a) = 0 and n be a u n i v e r s a l c o v e r i n g

h n o %R E H I ( D ) ,

h n o %R ( O) : 0

and,

III,

(h n o ~R )^ = ( ( B s * ) o S R ) ( f * o

C'

theorem

E HI(R)

Thus

(Bs*) o SR

L~(do)

generated

is o r t h o g o n a l by

H~(do)

and

SR )n

a.e

to the w e a k l y * f* o SR"

closed

Since

subalgebra

f* ~ H ~ ( d x ),

156

it is e a s y to see, H (do)

C' = L=(do) 0.

that

B ~ 0

Ill,

closed

Common

(Bs*) o SR = 0 a.e.,

Lemma.

a.e.

we c o n c l u d e

~ H~(do).

Since

L (do), we see that This

shows that

Bs* =

s* = 0, as was to be

We b e g i n w i t h the f o l l o w i n g Let

~

Proof.

Let

~C

be a line b u n d l e

common

no n o n c o n s t a n t

inner

common

Q0

over

factors

inner

factor,

common

--~{-(R,~-I~ I)

i n n e r f a c t o r of a n y e l e m e n t ~(R,~

0)

of i n d u c t i o n

2, . . . .

T h i s means,

k : i, 2 . . . . .

constant

function,

6B.

has

Q0"

Then

of

and

~ ( R , q -I) -i ) :

~{~(R,~

common

inner

0)

should

be d i v i d e d

by

Q0"

and t h e r e f o r e ~ ( R , n 0) = QoH~(R). k = Q0 H (R) for each k = i,

that

a point

)

~ ( R , ~ k) 9A,

Ch. V, that

IQ0(a) I = i

as was to be shown.

Let us c h o o s e

has no

~ (R,~

has no n o n c o n s t a n t

in v i e w of T h e o r e m

It f o l l o w s

Then

inner factor to

~(R,~

Q0

we see that

~{~(R,~)

5D).

-i) =c S{=(R,q -I),

in

contains

By m e a n s

such t h a t II,

too.

associated

~(R,n0)~(R,n-ln0

Obviously,

R

(see Ch.

be the g r e a t e s t

be the line b u n d l e

Since Q0 ~ (R,n -i ~i)f a c t o r s and since

for

f* O S R of

Inner Factors

nonconstant

the

7D, that

subalgebra

[]

6A.

let

weakly*

and t h e r e f o r e

As we k n o w

proved.

6.

in v i e w of Ch.

is a m a x i m a l

and so

Q0

is a

[]

a E R, w h i c h

is h e l d

fixed,

and set

Jn (a) = Jn = s ( a ) ~ ( R ' ( n ~n(a))-l)' S

n

= S

(a) n

and

~n = ~n~(a) for

n = i, 2, .

. (see . . IA)

T h e n we h a v e

oo

Lemma.

Un=l Jn

Proof.

First we look at e a c h

common

has no n o n c o n s t a n t

i n n e r f a c t o r of

c o m m o n i n n e r factors.

~=(R,~n) ,

~{~(R,~n) , then

were nonconstant,

t h e n it s h o u l d

n.

w o u l d have a c o m m o n

So

~(R,~

)

n = i, 2, . . . .

Q

should divide

v a n i s h at some zero,

If S n.

Q

is a If

Q

z. E Z(a;R) with j > ] c o n t r a d i c t i n g to C o r o l l a r y

157

9A, Ch. V.

This means that

~(R,~

)

has no nonconstant

common

inner

-n

factors.

By the preceding

lemma

inner factors.

Thus the function

factor of

Since the eommon

Jn"

n = i, 2,...}

are only constant

~ ( R , [ [ I)

has no noneonstant

S (a) ~is the greatest n

common inner ~(a)

inner factors of the sequence functions,

we are done.

common i~ n

:

[]

oo

7.

The Orthocomplement 7A.

space

of

H (dX)

We want to determine

Ll(dXa ).

the orthocomplement

For this purpose we fix a point

following notations

besides

those given in IA.

S n'(z) = s(a)(z)s(a)(z)-lexp(-g(a'Z)n for

n : I, 2,...

of

H~(dXa )

a E R

in the

and use the

We set - ig(a,z))

and

S'(z) whose line bundles

: S (a)(z)-lexp(-g(a~z)

are denoted by

~n

and

- i~(a,z)), ~', respectively.

We also

set

for

Let in

n : i, 2,...

Jn Jn

(resp. (resp.

III (resp. in

n

K'(a)

= K' = S'~I(R,~'-I).

and

Kn )

Lemma

lowing result

= Kn

be the set of fine boundary

functions

for elements

Kn) , whose existence has been shown in Theorem

the weak* closure Un=l = Kn

Kn(a)

3B). of

Ll(dXa)),

We further denote by U n=l ~ Jn where

concerning

in

L~(dXa )

J(a)

dXa = kb(a)dx(b).

orthocomplements,

= J

(resp.

5E, Ch.

(resp.

K(a)

the closure

= K)

of

Then we have the fol-

which will be used later and

has its own interest. Theorem.

For any PWS

R

(a)

K : H~(dXa )-L,

(b)

~ : (~')±,

we have the following:

where the o r t h o c o m p l e m e n t a t i o n (Ll(dXa), L~(dXa)). Proof.

is taken with respect to the dual pair

(a) If f E Kn, then f is meromorphie on n If(z)iexp(- [ j=l g(zj,z)) has a harmonic majorant.

R,

f(a)

= 0

By Theorem

and

4B we

158

have

I

f(b)h(b)dXa(b)

: f(a)h(a)

: 0

A1

for any

h E H~(R) •

So ,

Kn C = H ~ ( d X a )±

(9)

for e v e r y

and t h e r e f o r e

n

K ~ H~(dXa)1. N e x t we take any

i, 2~...,

f~ E K±

= L ~ ( d X a )) . (C

K'Jn C = Kn

Since

for

n =

we have

f

^^

f~hk dXa

: 0

AI for any

h E K'

theorem

(Theorem

f~k.

The

bounded u/k

and

IB) t h e r e

function

and thus

a.e.

analytic

k 6 Jn"

f*k

u 6 H~(R).

If

on

u E HI(R)

k ~ 0, t h e n

k ~ 0

f*

can be v i e w e d

as the b o u n d a r y

f, of b o u n d e d

characteristic.

R

7B, Ch.

on the disk,

are c o m p l e t e l y

is d e t e r m i n e d

f2 E H~(R)

determined

uniquely

by use of L e m m a

III, w i t h analytic

3A.

by

R' = •

f*.

a n d so

function

f* =

for an

As seen for in-

of b o u n d e d

boundary

We w r i t e

So we h a v e

u =

is also

and p r o p e r t i e s

functions

by t h e i r

u

a.e.

say

Cauchy

such that

AI, we see that

Namely,

functions

f

a function

By the i n v e r s e

function,

istic on that

exists

f*k ± K'.

being bounded

stance by u s i n g T h e o r e m analytic

Thus,

functions,

f = fl/f2

seen that

of

character-

with

so fl'

k f l / f 2 E H~(R)

~ ~ J n has, by L e m m a 6B, no n o n c o n s t a n t for any k C U n=l Jn . Since Un=l c o m m o n i n n e r f a c t o r s , the i n n e r f a c t o r of f2 m u s t be an i n n e r f a c t o r

of

fl"

It f o l l o w s

belongs closed

to

H~(dXa )

in the

space

that

f = fl/f2

belongs

and c o n s e q u e n t l y

to

H~(R).

K± ~ H ~ ( d X a ).

L l ( d X a ), we c o n c l u d e

Hence,

Since

K

f* = is

that

: ~±± ~ H~(dXa )Combining (b) and

this w i t h

(9), we get the d e s i r e d

T a k e any

f E K',

Iflg (a)

meromorphic a harmonic

has a h a r m o n i c on

R,

majorant.

(fu)(a)

so that majorant = 0

By T h e o r e m

f

and

f on

R.

If

on

u E Jn'

l(fu)(z)lexp(-[j~l

R,

f(a)

then

fu

g(zj,z))

= 0 is has

4B we h a v e

^^ AI fu dXa = (fu)(a)

So

identity•

is m e r o m o r p h i e

= 0.

f 6 J± and as n is a r b i t r a r y f 6 ~ Thus n ' ' " N e x t we take any f~ e 9 ± (~ Ll(dXa)). Since

~v C J

n

J±.

is an ideal of

159

H~(R),

h E Jn

and

u E H=(R)

imply

I

hu E Jn

and so

^ ^

A1 f*hu dXa = 0.

f*h E H ~ ( d X a )± = K

Namely,

n : I, 2 .... So there

and since

exists

the a r g u m e n t belongs

is closed

a function

used

(a) .

in view of

K'

in

u E ~/(R,~ '-I)

in (a), we deduce

such that

from this

7B.

Theorem.

Let

R

be a PWS

for every

to

K _C K,.

f*h : S'u.

observation

to ~ ( R , ~ '-I) and thus f* b e l o n g s ^± ^ ± J = K' or, e q u i v a l e n t l y , J : K'

Hence,

K n =C K'

Since

Ll(dXa ), we have

that

By

u/h

K'; namely,

5± C K,.

[]

for w h i c h

( D C T )a

holds.

Then

~'(a) : ~(a) : H~(d×a )±. Proof.

We have

is i n c l u d e d any

in

h E H~(R)

which

is equal

shown

in the p r o o f

K'(a).

and thus, to

K.

of the p r e c e d i n g

Now we take any by

(DCTa) ,

Hence,

f E K'

theorem

Then,

f£1 fh ^^ dXa

: 0"

K' ~ K, as desired.

that

K(a)

fh E K' So

for

f E H ~ ( d X a )_L '

[]

NOTES The c l a s s i c a l IV of Heins Neville tion.

[31].

theorem Cauchy

inverse

[47] has a v e r s i o n It is p o s s i b l e

pactification, false

as was

of this

to f o r m u l a t e indicated

in its full g e n e r a l i t y .

strong

restriction.

The w e a k - s t a r lished

[27].

Cauchy-Read The

note

[20].

A weaker maximality The r e s u l t s

is d i s c u s s e d theorem

depending

in [18].

version

in

in C h a p t e r

in H a s u m i

6 and

com-

theorem

[17] under

4B) is in H a s u m i L~(dx)

[18].

compactificaof W i e n e r

Cauchy

in H a s u m i

(Theorem

H~(dx)

in terms

The direct

in S u b s e c t i o n s

in detail

shown

on a n o t h e r

our r e s u l t s

It was p r o v e d of

was

is

a

[17].

is in an unpub7 are due to H a y a s h i

CHAPTER

The themes

ence of A.

theory

Beurling's

began

the

such modules and simply is strong simply

role.

tion that

every

plicative)

As a m a t t e r

function.

covering

as a preliminary. doubly

Cauchy

theorem,

Beurling a E R

subspaces

chapter,

will

and their

is not

all

simply

be given

(multi-

in Ch.

here

in §2.

the direct

subspaces

that the statements

X.

to the

is r e c a l l e d

Assuming

invariant

plays

to the condi-

over a PWS are d i s c u s s e d

show that

so

to sur-

(DCT)

by some

liftings disk

theorem

As for

extends

is equivalent

of the unit

subspaees

Cauchy

theorem

is g e n e r a t e d

on the

As usual,

subspaees.

theorem

are determined.

it is proved

invariant

the p r o b l e m

subspaces

in Ch.

PRELIMINARY

let

measure III.

sense of p o t e n t i a l

i.

Invariant

Finally,

the h a r m o n i c

as defined

51.

The case

type.

inverse

Cauchy

discussion

subspaces

surface.

we next

interesting.

are of

(DCT a ) with

are all equivalent. In this

dX

(DCT)

H~(R)

Further

invariant

invariant

type.

of

sur-

defined

invariant hand,

Gamelin Riemann

LP(dx)

doubly

the direct

of fact,

ideal

over

influ-

it very

out that the

on the other

Here

of Helson,

of P a r r e a u - W i d o m

all doubly

type.

6-closed

inner

First,

It turns

of

to see when and how Beurling's

In §i we define universal

R

the decisive

subspaces

found

into two classes:

ones.

of P a r r e a u - W i d o m

a crucial

in books

already

surface

subspaces,

We have

under

H~(dx)-modules

to d e t e r m i n e

invariant

simple. faces

are divided

enough

we have

closed

of a given

invariant

[2], as seen

SUBSPACES

has been one of the central

classes

study of invariant

only recently,

boundary

subspaces

of Hardy

paper

Although

Our aim is to c l a s s i f y Martin

SHIFT-INVARIANT

study of s h i f t - i n v a r i a n t

in the m o d e r n

and others. faces

VIII.

R on

denote AI

a hyperbolic

for the point

We assume

as before

Riemann

0--the

that

R

surface

origin

of

is r e g u l a r

and R--

in the

theory.

OBSERVATIONS

Generalities IA.

By T h e o r e m

We are going 3A, Ch.

to c o n s i d e r

IV, the map

LP-spaces

f ÷ f

gives

LP(dx)

in what

an isometric

follows.

linear

injec-

161

tion of space

HP(R) HP(R)

into

LP(dx)

for every

p

is equipped with the norm

with

((LHM(IflP))(0)) I/p ]IflIp : {sup{if(z)l: The space

HP(R)

is denoted by

A subspace is closed

M

Moreover, we set

of

if

for

LP(dx)

(weakly* closed,

invariant subspace variant)

z • R}

if

0 < p
n.

Then,

if

that

i =< Ifl =< n

[fnleXp(-Un)

and

I undo =< I [l°g+Ifni - l ° g + i f I l d °

= I I + I2,

of

to a s u b s e q u e n c e

Un = m a x { l ° g + I f n l

if

Hence

so that t h e r e

iif- fnliq ÷ 0.

with

by p a s s i n g

We set

= log+Ifn I

subspaee

f E N A LP(do), M

j ÷ ~.

dominated

LP(do).

is an i n v a r i a n t

we take any

{fn : n : i, 2,...} sure,

in

By L e b e s g u e ' s

+ I

ifi>n

l°g+ifld°

say. q ~ i

the case

and

~ q-lilf n - fit2 ~

p < ~.

for

0 < q < i.

T h e n we see that

JflPdo. Ifl>n Thus,

I I + 12 + 0

be the h a r m o n i c

as

n + ~

conjugate

of

and c o n s e q u e n t l y u

for

{Un( j) + i [ n ( j ) } , w h i c h

0 < s < i. converges

T h e n by

Let

with

~ (0) = 0.

As in

(a), we f i n d a s u b s e q u e n c e

n

llUn~is ~ Cs]lUnllI

f U n d o ÷ 0.

(i) we h a v e

n

to

0

a.e.

on

~.

We set

~n

h 3• :

exp(-Un(j) - i~n(j)) a.e. and

and see that

hjfn(j)

[hjfn(j) - fl p ~ 21fl p + i e Ll(da).

c o n v e r g e n c e theorem, LP(da), we have

hjfn(j) + f

f E M

in

and t h e r e f o r e

We finally suppose enough, then

12 = 0

p = ~.

So

and thus

we see that

w(L~(da),Ll(da)).

f E M

and so

NNL~(da)

and we are done.

2C.

is bounded.

f Unda + 0

M

C M.

as

If

n + ~.

= M,

n

M.

is large

P r o c e e d i n g as

hjfn(j) + f

~ f

in

L~(da)

a.e. and

in the weak* topol-

is weakly* closed in

L~(da), we have

The converse inclusion is true as before

[]

As is easily seen, an invariant

doubly invariant

is closed in

A g a i n by the d o m i n a t e d c o n v e r g e n c e

hjfn(j)

Since

M

is included in

we have got the desired result.

hjfn(j) e M A L ~ ( d a )

theorem, we conclude that ogy

Since

N N LP(da)

f

[hjfn(j) - fl £ 21fl + i ~ 211fll + i.

hjfn(j ) ÷ f

By Lebesgue's d o m i n a t e d

LP(da).

The converse inclusion being trivial,

above,

e M AL~(da),

(resp.

simply invariant)

subspace if

M

of

ei0M : M

LP(da)

(resp.

is

eiSM

M).

Theorem. LP(da),

Let

M

be a closed

(weakly* closed,

(a)

M

S

of

T

such that

a c t e r i s t i c function of

S.

(b)

M

q E L~(da)

M = CsLP(da),

The set

up to a set of Lebesgue m e a s u r e

S

where

CS

is the char-

is d e t e r m i n e d u n i q u e l y by

with

(i)

subspace of

zero.

lql = 1

a.e.

such that

M : qHP(da).

We begin with the case L2(da).

p = 2.

Suppose first that

The function

factor of modulus one. Let

M

be an invariant

ei0M = M.

Then let

orthogonal p r o j e c t i o n of the constant function i E L2(da) in0 space M. So i- q w e q for all integers n, i.e. I T for all integers Hilbert space

n.

Since

L2(da),

(i

I

P(T) + P(~). hq E M we have

e-i0M = M Since

for any

M

{e ine}

be the

forms an o r t h o n o r m a l basis of the a.e.,

subset,

is closed in

q

to the sub-

0

say

so that S, of

and therefore that

h ~ L~(da)

qL2(da) ~ M.

q)qeineda

( I - q)q = 0

istic function of a m e a s u r a b l e implies that

M

is simply invariant if and only if there exists a function

is d e t e r m i n e d u n i q u e l y up to a constant

Proof.

p = ~) subspaee of

is doubly invariant if and only if there exists a measur-

able subset

q

if

0 < p ~ ~.

q T.

hq E M

is the c h a r a c t e r Our a s s u m p t i o n for any

L2(da), A p p e n d i x A.I.4

and c o n s e q u e n t l y

for any

To see the reverse inclusion,

h

in

shows that

h ~ L2(da). take any

f E M

So

166

which

is o r t h o g o n a l

to

qL2(da).

i- q ± ein0f

for all i n t e g e r s

Hence

a.e.,

f = 0

a null

set,

M, there

ei@M ~ M.

a function

Then

q ± qe in@

is equal to a c o n s t a n t Jql = i

a.e.

qH2(da) space for n

Since

~ M.

Then

a.e.

that

qlq2

to a c o n s t a n t Finally,

fq

a.e.

qlq2

a doubly

L2(da).

This p r o v e s

the case

Suppose

tersection of

CsL2(da) suppose (i),

N = CsL2(do) It f o l l o w s

Finally doubly (resp.

for

It f o l l o w s

that q ± fe in0

w i t h all f = 0

integral

a.e.

= [q2J

that

to the

and also

JqlJ

Hence

= i

of

a.e.

qlq2

is equal

(resp.

qH2(do)

invariant

subspace

By

(i)

subspace

or to

by L e m m a

and

M

with of

qH2(da)

2B that

of

in the s e n s e d e s c r i b e d qHP(da)

of

M.

: qH2(da)

M : N nLP(da)

CsLP(da)

subspace

or

to

lqJ : i

= qHP(da).

be the L 2 - c l o s u r e S ~ T

either

with or

subspace

2B, the in-

is the c l o s u r e

is e q u a l

M : CsLP(da) N

As for

be an i n v a r i a n t Then,

MnL2(da)

S ~ T

invariant

M

0 < p < 2.

and let

CsLP(da).

2B and the case

Jql 2

p = 2.

p ~ 2, and let

from Lemma

q) is u n i q u e

is o b v i o u s

simply)

with a measurable

simply)

that

then implies

and h e n c e

CsL2(da)

(resp.

we h a v e to show that

(resp.

with

of

as desired.

we find that

2 < p S ~

subspace

To see the u n i q u e n e s s

H2(da)

is an i n v a r i a n t

LP(da).

LP-closure,

that

i a.e. qHP(da).

Lemma

in

with a measurable

By t a k i n g

to

one,

first that

M nL2(da)

MNL2(da)

a.e.

S, up to

is o r t h o g o n a l

a.e.,

M = qH2(da). = q2H2(da)

gives

LP(da).

lql : i

belong

a.e.)

of

~ M.

which

for

Since

of m o d u l u s

0 < p S ~,

qP(~)

f E M

it is easy to see that

Let

we have

is o r t h o g o n a l

This m e a n s

lJqil2 = i

n : 0, i,... in0 is o r t h o g o n a l to e

Jq] : i

(ii)

n = i, 2, . . . .

take any

qlH2(da)

and

of

is a c l o s e d

JfqJJ2 = i, w h i c h

Our a s s u m p t i o n

and c o n s e q u e n t l y

q, we s u p p o s e Then b o t h

for

f ± qe in0 So

vanishes

M ~ qH2(da)

ei@M

Since

eieM ~ M, we have

n = i, 2, . . . . and h e n c e

Moreover, (i - q)f = 0

The u n i q u e n e s s

q @ M, w i t h

a.e.

Conversely,

qH2(da).

a.e.

obvious.

next that

exists

ei@M.

to

fq = 0

and c o n s e q u e n t l y

w h i c h we had to show.

is a l m o s t

Suppose

Then n

with

(resp.

qHP(da))

LP(da)

and a l s o that

the r e s u l t

p = 2, w h i c h has a l r e a d y

Next

Then,

= CsLP(da )

in the t h e o r e m .

a.e. by

JqJ = or

:

is a CS

The r e s u l t

follows

from

been established.

[]

167

§2.

3.

INVARIANT

Doubly 3A.

SUBSPACES

Invariant

We now c o n s i d e r

tions m e n t i o n e d Theorem.

Let

in IA. E

variant

subspace

Proof.

Suppose

H~(dx).M

the case of R i e m a n n

function

of

of

LP(dx )

that

subset

Z.

for

of

Then

AI

0 < p < ~.

Since

then

its b o u n d a r y

on a n e g l i g i b l e

subset

£ I.

So t h e r e

h(b) -I

if

b e E

for any g i v e n

such that

with

f • M

f

n

fn • H ~ ( d a ) . M .

I f - fnl ~

and

If[

t h e o r e m we h a v e M.

The case

3B.

Here

Theorem.

that

i ~ p ~ ~;

(b)

0 < p < i

Since

L e m m a IC.

3C.

let

X(Z)

M ~ CzLP(dx )

S

is r e g a r d e d

Suppose

B • S(0)~(R,(~(0))

IB

which

on

=

A I.

And

is b o u n d e d ,

n

un

implies

dominated H0(dx).M

that convergence

is d e n s e

is omitted.

in

[]

section.

Let

M

be a d o u b l y

support

of

the p r o b l e m

is also d o u b l y

M.

invar-

Then

M =

is to p r o v e invariant

for some m e a s u r a b l e

as the s u p p o r t

f i r s t that -I)

true,

{M]p

is n o w d i v i d e d

be any e l e m e n t

with

Un(b)

< i.

{M}p : C s L P ( d a )

The p r o o f

of

in this PWS.

M. only

= M

Hence

be the

in

holds:

is a l w a y s

By L e m m a

2C-(a),

Z

E ~ AI Set

and the p r o o f

is a r e g u l a r and

and

inclusion.

by T h e o r e m

p / ( p - i)

R

LP(dx)

u

By L e b e s g u e ' s n ÷ ~.

similar

if one of the f o l l o w i n g

Since

T.

of

(a)

reverse

a.e. as

is a l m o s t

in-

is c l o s e d

vanishes

otherwise

Since

~ CELP(do)

is our first m a i n r e s u l t

Suppose

iant s u b s p a c e cELP(dx )

f - fn ÷ 0

E.

: 0

The d e f i n i t i o n

llf- fnlIp ÷ 0

p = =

and

h

a set

on

M

is d e n s e

function

exists

: hu f. n

Unf • UnCELP(do) and t h e r e f o r e

denote

is a d o u b l y

H0(dx)'M

0 < lh[ < ~

[h(b) I ~ i/n

we set

CE

it is c l e a r that

h • H~(R); of

and let

M = c z L P ( d X)

~ M, we h a v e only to show that

complement

and use the n o t a -

0 < p ~ ~.

Take a n o n z e r o

negligible

surfaces

First we show the f o l l o w i n g

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

the c h a r a c t e r i s t i c

and

Subspaces

of

M ° $,

i ~ p ~ ~.

Let

is o r t h o g o n a l

in the n o t a t i o n

subset

CE o $ = CS

into two cases

(a) and

described

M.

S

of

a.e.

with

p'

Take any n o n z e r o

in Ch.

by

(b).

s* E LP'(d×) to

the

and so,

VII,

IA.

If

=

168

u

is any m e r o m o r p h i c

then

Bu E H~(R)

function

and so

on

R

Buf* @ M

(2)

[ i

such that for any

lulg (0)

f* E M.

is b o u n d e d ,

Thus we h a v e

B u f * s * d X = 0. A1

By the i n v e r s e tion

Cauchy

k E HI(R)

theorem

such that

(Theorem

IB,

k = Bf*s*

a.e.

Ch.

VII)

on

A I.

there

exists

Setting

a func-

u = 1

in

(2), we h a v e

(3)

t : ~ J

k(0

On the o t h e r hand,

the f o r m u l a

~ d X : 0. AI

(12)

in Ch.

III shows that

H [ ( B f * s * ) o $] = H [ ~ f * s * ] o ~ = H[~] o ~ = k O % 6 H I ( ~ ) . For any

v E H~(D)

we thus have,

by

(3) and T h e o r e m

I~ v ( e i S ) ( ( B f * s * ) o $ ) ( e i S ) d o ( 8 )

:

By t a k i n g

LP-limits

in

I

for any B

is n o n z e r o ,

Hence

s* = 0

have

S

{M}p = C s L P ( d g ) , we c o n c l u d e

on

Z.

Lemma.

This

implies

ment Proof.

f* 6 LP(dx ),

~ e H~(dx) (Lemma)

0 < p < ~.

such that Let

a.e.

~f*

that

that on

(~s ~) o

E.

zero

s* I C z L P ( d x ) .

Since in

A I.

We thus

(a) is v e r i f i e d .

In o r d e r to deal w i t h the case Let

Bs* = 0

o n l y on a set of x - m e a s u r e

C E L P ( d X) ~ M, so that the case 3D.

0.

:

and c o n s e q u e n t l y

can v a n i s h

a.e.

~)(o)

= v(0)(kO

v - ( f * o $), we see

Since

on B

III,

((as*) o ~ ) ( e i S ) f l ( e i S ) d g ( @ ) = 0

fl E {M}p.

m u s t v a n i s h a.e.

v(0)k(0)

7B, Ch.

(b), we p r o v e

Then there

exists

a nonzero

ele-

is b o u n d e d .

u* : l o g + I f * [ .

Then

u*

is p o s i t i v e

and

summable,

for

0 ~ u~ : log If~l : p-flog If~l p ~ p-ilf~IP on the

set

R

u*

of

{If*I

> i}.

and let

v

Let

u : H[u*]

be the h a r m o n i c

be the h a r m o n i c conjugate

of

u.

extension Then

u

to is a

169

positive m. on

harmonic

R.

If

function

6

denotes

on the

R

and t h e r e f o r e

line b u n d l e

e -u

is a b o u n d e d

associated

to

l.a.

e -u, t h e n t h e r e

exists

an e l e m e n t h I ~ ~ (R,6 ) with llhl~l~ = i, for R is a PWS. -u-iv Then h l e d e t e r m i n e s a s i n g l e - v a l u e d f u n c t i o n , say h, b e l o n g i n g

to

H (R)

and we h a v e

Let whose

M

be a d o u b l y

support

an i n v a r i a n t vanishes and

N

E

subspaee

of

2C, that

The a r g u m e n t

{N} 2

zero,

namely

S

given

Since

o(S)

m u s t be d o u b l y

in 3C t h e n

shows t h a t

Since

LP(dx)

3E. LP(dx)

= [L2(dX)]p,

(b) is proved. Problem. with

Can one show that

4.

Simply 4A.

point

In w h a t

consider lowing

then

element

lemma

= X(Z)

is H~(dX)

implies

that

M

N ~ C z L 2 ( d x ).

in 3B is the s u p p o r t < I, we see,

invariant.

N

of

Hence

of

{N}2,

so

in v i e w of T h e {N} 2 = C s L 2 ( d ~ ) .

and t h e r e f o r e

) ~ M ~ czLP(dx). result

M = CzLP(dx).

So

that

M

is a d o u b l y

and that the s u p p o r t

invariant

of

M

subspace

coincides

of

with

A I.

?

Subspaces

is h e l d

R, c o n s i s t i n g a function

0 < p < i,

So we h a v e

f o l l o w s we w o r k on a r e g u l a r

a @ R, w h i c h

group of

the a b o v e Z.

we get the d e s i r e d

M = LP(dx)

Invariant

LP(dx),

[]

Suppose

0 < p < i

of

[]

N = MNL2(dx);

N = CEL2(dx)

C E L 2 ( d X) : M n L 2 ( d x

the case

Set

S i n c e any n o n z e r o

o n l y on a set of m e a s u r e

the set

~ i, as desired.

subspace

< i.

L2(dx).

have the same support,

{N} 2 ~ C s L 2 ( d o ) .

orem

invariant

has x - m e a s u r e

On the o t h e r hand, that

lhf* I ~ e-U*If*l

Q

fixed.

of c l o s e d

We d e n o t e curves

on the p r o d u c t

PWS by

Let us c h o o s e a

Fa(R)

issuing

space

R.

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

f r o m the p o i n t

A I × Fa(R)

with

a, and

the fol-

properties:

(AI)

for each

(A2)

there

y E Fa(R) ,

exists

b ÷ Q(b;y)

a line b u n d l e

~

is m e a s u r a b l e

over

Q(.;yl ) = ~(yl)~(y2)-iQ(.;y2 ) for any

YI' Y2 ~ Fa(R)"

corresponding

character

Such a f u n c t i o n tive")

on

A I x Fa(R)

scribe

the b o u n d a r y

Q

Here at the

~

is c a l l e d

of c h a r a c t e r values

say that two m - f u n c t i o n s

represents

same time

(Ch.

~

and

Q2

on

AI

a line b u n d l e II,

AI~

such that

a.e.

an m - f u n c t i o n

and the

2B). (m for

and w i l l be u s e d

of a m u l t i p l i c a t i v e QI

R

on

analytic

are e q u i v a l e n t

"multiplicain o r d e r

to de-

function.

We

if t h e y h a v e the

170 same bundle

(or character),

say

~, and there exists a

Y0 E Fa(R)

such that Ql(-;y) for every

y E Fa(R).

= $(y0)Q2(-;y)

We write

a.e. on

QI e Q2

if

QI

AI and

Q2

are equiva-

lent. To each as follows: $'(a)

f e ~P(R,~)

we associate

take a branch,

= {'(R,a)

say

and denote

it by

Fa (R).

For each

tity of the group

f0(z;y) for

z E $'(a).

Then,

most every Green line

for

b = b£,

determined

£ E A(a),

a.e.

on

AI

A I.

tions

fl(z;y)

f0

and

fl

function

where

y E F a (R)

on

A I × Fa(R)

l

denotes

the iden-

we set

= $(¥)f0(z;l) 4A, Ch. VI,

% E A(a).

We set

if the right-hand

f0(%;y)

exists

side exists.

So

for al-

f0(b;y)

is

and

: ~(y)fo(b;1)

fl(b;y)

say

as above,

= ~(y0)fl(z;y)

are equivalent.

for

4B.

and

f0(z;y)

f

on the Green star region

f0(z;l),

If we choose another branch,

such that

f

by Theorem

fo(b~y) on

an m-function

f0' of

and

fl' of

f

and define

then there exists f0(b;y)

a

Y0 E Fa(R)

= ~(yO)fl(b;y);

We call any one of

~

func-

namely,

the boundary m-

f.

In the following we consider

shall see, results

are independent

the case

a = 0.

of the choice of

In fact,

a.

as we

We first show

the following Theorem.

Let

IQ(.;y)I

= i

for some Proof.

R

with

uj(z;y)

procedure

be an m-function for each

is a PWS, there exist

and

~j(b;y),

and

~2(b;y)Q(b;y)

are independent

functions

and

in

Then

of character

~

M = {f* e LP(dx):

uI E ~(R,~)

lu2(0) I ~ 0

a = 0. of

Then

7 e F0(R)

and

(Corollary

j = i, 2, be constructed

in 4A with v~

A I × F0(R)

with f*/Q

I ~ p ~ ~, is simply invariant.

lUl(0) I ~ 0

described v~

on

y e F0(R).

h E ~P(R,~-I)},

Since

~ ( R , ~ -I) Let

Q a.e.

u2 E 9A, Ch. V).

from

Ul(b;y)/Q(b;y) and define

L~(dx ), respectively.

uj

by the and

single-valued

Ul(Z;Y)U2(z;y)

is

171

also independent

of

y

and defines

We now take any If

h(b;y)

denotes

the boundary

f~(b)/Q(b;y)

and if

corresponding

to

and defines HI(R)

h(z;y)

h(b;y),

an analytic

say

u3, in

f*/Q

~ h

for an

m-function

for

h

Ul(Z;y)h(z;y)

function,

say

hl,

H~(R).

h • ~P(R,[-I).

satisfying

is the multiplicative

then

analytic

h(b;y)

is independent

in

HP(R).

:

function of

For any

y k •

we have k(b)f*(b)v~(b)

= ~(b)(f*(b)/Q(b;y))Q(b;y)v~(b)

~(b)~(b;Y)~l(b;~) = ~(b)fil(b)

:

a.e.

a function,

f* • M, so that

on

AI

(cf. Theorem

6A, Ch. VI) and therefore

;

~(b)f*(b)v~(b)dx(b)

= I

41

k(b)hl(b)dx(b)

: k(O)hl(O)

: 0-

41

Namely,

v[ ± Ho(dx).M.

I

But, on the other hand,

v~(b)v~(b)dx(b)

= I

hl

we have

v~ E M

and

u2(b;y)Ul(b;y)dx(b) AI

= I

u3(b)d×(b)

= u3(0)

~ 0,

AI for

]u3(0) I : lUl(0)llu2(0) I f 0.

M, as was to be proved. 4C. (DCT a)

In order to proceed

(Ch. VII,

Theorem.

Suppose

space of

LP(dx),

4 l x F0(R )

Thus

H~(d X).M

cannot

be dense

in

[] further,

we need the direct

Cauchy theorem

3C), which we use here as a hypothesis. that

(DCT 0) holds.

If

M

I ~ p £ ~, then there

of some character

~

with

is a simply

exists

invariant

an m-function

IQ(-;y)I

= i

a.e.

Q

on

subon

AI

such

that M : {f* e LP(dx): The proof is divided an m-function

Q

on

f*/Q

of modulus

M ~ {f* e LP(dx):

f*/Q

character

And in the second

4D. p ~ ~.

of

Q.

Let Then

were doubly

M

for some

into two parts.

A I × F0(R) ~ h

~ h

for some

be a simply

h e ~P(R,~-I)}.

In the first part we define one a.e.

h • ~P(R,~-I)}, part we verify

invariant

subspace

of

in such a way that where

~

is the

the reverse

inclusion.

LP(dx)

i

with

{M} is a simply invariant subspace of LP(do). If it P invariant, then M would turn out to be doubly invariant

172

by our a r g u m e n t L~(do)

with

in 3C.

So, by T h e o r e m

lql = i

a.e.

on

(4)

~

{M)p

c(T)q

a.e.

complex

is T - i n v a r i a n t , on

~

number

the u n i q u e n e s s

for e v e r y

of m o d u l u s

T ÷ y~

numbers

of

T

a character T E T. to Ch.

F0(R)

~ E F0(R)*

This II,

of m o d u l u s

onto ~

it by

N(z;~),

on

~

w E ~

F0(R)

7C, Ch.

by s e t t i n g

~(yT)

as a line b u n d l e

group

isomorphism

= c(T)

over

which

star r e g i o n

where

l

is the i d e n t i t y

as in 4A.

on

and

Moreover,

= IN I o ~

= N(~(w);y

R

is h e l d

for e v e r y according

fixed. 6'(0)

of the g r o u p

we d e f i n e

for some

N1

We take and de-

F0(R).

an a n a l y t i c

Then

function

then,

= N(0;I). for any

everywhere

on

F • HP(~).

in NIF

By the

= c(T)NI(W)

f* • M.

Multiplying

same t r a n s f o r m a t i o n

function

T • T,

~.

Q, we take any n o n z e r o

o b e y the

k E HI(R).

NI(0)

) = ~(yT)N(~(w);~)

N1 o T = c ( T ) N 1

T

and

%(w) E 6'(0);

(5)

HI(~),

N1

Then on b o t h

= k o

formula

(12)

in Ch.

follows

that

~ o $ : ~i ~

a.e.

(f*/i) o $ : ( f * o $ ) / ~ ] ~

on

T

= q/Nl

f* o $ = sides,

rule w i t h r e s p e c t

so that

III we h a v e

H[k o $] = H[k] o ~ : k o ~ : NIF : H [ ~ I g ] . From this

defined

is e a s i l y

III, we can d e f i n e

N • ~(R,~),

is a T - i n v a r i a n t

for some

q o T =

into the m u l t i p l i o a t i v e

By use of the c a n o n i c a l

on the G r e e n

In o r d e r to d e f i n e

q

T ÷ c(T)

N

satisfy

and t h e r e f o r e

(6)

q G

by the c o n d i t i o n s

N 1 o T(w)

Since

T

that

is a u n i q u e l y

of

branch

INII

NIF

of

is a l s o r e g a r d e d

are d e f i n e d

qF a.e. we get

shows

c(~)

g i v e n by T h e o r e m

of

q

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

one.

Take any n o n z e r o m e m b e r

Let

a function

2B.

a single-valued

N(z;y)

of

~ E T, w h e r e

one.

seen to be a g r o u p h o m o m o r p h i s m of c o m p l e x

Nl(W)

exists

{M}p = qHP(da).

Since

note

2C, t h e r e

such t h a t

and t h e r e f o r e a.e.

to

T,

173

This m e a n s

that

the n o t a t i o n

f*/k

is i n d e p e n d e n t

introduced

Q(b;y) Then,

for e a c h

dependent

of the c h o i c e

of

f* E M.

on

AI

Using

in 4A, we set

y ~ F0(R) ,

of the c h o i c e

: f*(b)N(b;y)/k(b). Q(b;y)

of

is d e f i n e d

f* E M.

Moreover

a.e.

we h a v e

and is in-

Q(b;y)

: ~(y) ×

Q(b;~). Let implies

u = INI . that

E A(0).

Since

u

exists

a n d is e q u a l to

We thus have,

for a l m o s t recall

](£)

every

is a b o u n d e d

for e v e r y

£ E A(0)

Theorem

u(b£)

6B,

for a l m o s t

Ch. VI, every

y E F0(R) ,

and so

IQ(b;Y)I

= 1

a.e.

on

We n o w

A I.

the e q u a t i o n

(7)

f~(b)/Q(b;y)

where

k

is d e t e r m i n e d

[NIFI/INI[ longing

uniquely

~P(R,~-I).

Lemma.

Let

Q

and

by (5).

on

Since

k(z)/N(z;y)

fe/Q

To p r o v e the r e v e r s e

for some

a.e.

£i'

Ik(z)/N(z;y)I o % =

is a s e c t i o n ,

say

h, be-

Hence

M ~ {f* 6 LP(dx):

4E.

= k(b)/N(b;y)

= IFI, we see that

to

(8)

f*/Q

l.a.m.,

~

~ [

for some

inclusion,

we f i r s t n o t e

be as in 4D and let

f* E M}.

Then

J

h 6 A'P(R, o.

an

186

Thus every element f* in H0(d X) is regarded as the boundary function of a meromorphic function f, of bounded characteristic, on R, so that its value at 0 is determined. Theorem.

For every

f* E H0(dx )±

we have

f(0) = I

f*(b)dx(b)" AI

Proof. Since H~(dx )m = [10HI(dX,Qc-I)]I ~ HI(dX,Qc-I), each f* in Ho(dx )± determines an F E HI(~) such that f o ~ = Qc-IF, where f is the meromorphic extension of f* into R. We define a linear functional L on H0(dx )± by setting L(f*) = F(0). Since we have IL(f*)I = IF(°)I ~ S

IF(eiS)Id°(8)

t = J

If*(b)Idx(b)

: I T

IQc(eiS)-iF(eie)id°(e)

: IIf*IJl,

AI the Hahn-Banach theorem says that L has a norm-preserving extension to LI(dx). So there exists a function u* @ L~(dx ) such that Iiu*II = IILII < i and =

F(D) : I

f*(b)u*(b)dx(b) AI

for every f* E H0(d X) ± . belongs to H~(dx )± and that I

If f* E H ~ (dx) ± and h E Ho(R) , then ~f* (fif*) o $ = (~o $)(f* o $) = (~o $)Qc-IF, so

hf*u*d X : L(hf*) = (hO %)(O)F(0)

: 0.

AI As h E H~(R) is arbitrary, f'u* E H~(dx )± ~ HI(dX,Qc-I). Therefore, Qc-IF.(u * o $) E Qc-IHI(do), where F ranges over (I 0 o $)Hl(do,~C ). Since (I 0 o $)Hl(do,~C ) has no nonconstant common inner factors, we conclude that u* belongs to HI(dx ) and consequently to H~(dx ). It thus follows that, for any f* E H~(dx )±, (6)

F(0) = S

f*u*dx = I Al

f*(u*-u(0))dx

+ u(0) I

£i

: u(0) S

f*d X AI

f*dx. A1

Since

HI(dX,$c )

has no common inner factors,

it has no common

187

zeros. over, an

Namely, since

there

I0 o $

h e I0

with

exists

an

F 0 E Hl(d~,[C )

has no noneonstant h(0)

# 0.

Let

common

f~ e LI(dx )

(~ o ~) Qc-IF0 " Then, f~ E H0(d ~ X) ± and fl is the meromorphic extension of f[ corresponding FI(0)

~ 0

we have any

to this

f[

and hence, F = QC

u(0)

~ 0.

and so, again by (6),

f* C H0(d X)

FD(0)

be defined

by

# 0.

More-

there

is

f{o $ =

fl o ~ = (hO ~)Qc-IF0 , where into R. The function FI

is thus equal to

by (6),

with

inner factors,

(hO %)F 0.

Consequently,

In particular, QC(0)

= u(0)

for

~ 0.

f* ~ i

Hence,

for

, S

f*dx

= u(0)-iF(0)

= QC(0)-IF(0)

AI =

as was to be proved.

3.

= f(0),

[]

In the proof we have Corollary.

(f o ~)(0)

shown the following

IQc(O) I # o.

Proof of the Main Theorem 3A.

We are now in a position

Theorem. Let

Let

phic on

Let

can find an function h

satisfy

R, such that

exists

Proof.

Let

be the inner l.a.m,

u* 6 LI(dx )

there

Then hand,

qc

h*

an

Ihiq C

is bounded such that Since

F E Hl(da,~C )

such that

being bounded,

F

function

h(b)dx(b) A1

the last equality hypothesis

is arbitrary, proved.

[]

we conclude

= 0

on

R

u* = f

q c ° % : IQCI.

for any

and a.e.

h(0) on

h* o $ = Qc-IF

on

h, meromor= 0~ then

A I.

R

with

= I

a.e.

on

T.

we The

i.e.

F E H~(da,~C )D. h O # = Qc-IF on

h* : h a.e. on 2D implies h*(b)dx(b)

On the other

A I.

= O'

A1

sign being true because

in the theorem

such that

is also bounded,

is bounded on R and h* E H~(dx) l, Theorem = I

R

(c) ~ (b).

HI(dx) ± ~ H0(d X) ± ~ H I (dX,Qc-I),

h* 6 HI(dx) m.

h(O)

on

/£I h(b)u*(b)dx(b)

f E HI(R)

be a meromorphic lhlqc since

to prove the implication

h* A I.

and thus

/A I u*h*d X = 0.

that

belongs

u*

to

So As

HI(dx),

h

satisfies h* C HI(dx) ±

as was to be

the

188

3B.

We now begin the proof of the implication

assumption that H=(R) separates the points of we have to look into HP(do,

H~(d~,~C ) ~ {0}.

Suppose now that

~ IV(0)1 : IQc(0) I > 0.

HI(dx,Q~ -I)

On the other hand,

so that outer.

mP(~,0)

Hence,

mP($,0)mP'(~-I~c,0)

1-outer.

shows that

llVllp, = IIv*11p, , we have

[]

is standard and therefore

H (do,~ C)

It then follows from Lemma 3B that

~ E T~

is 1-outer.

is

~C

QC

is

Then, because of the inclu-

sion relation Hl(d~'~)H~(d°'~C ) ~ H l ( d ° ' ~ C )' So, by Lemma 3D, m ~(~-i ,0) ~ m l ( ~ c , O )m~(( 0,

~ - lim log r k ~ - log g (0)(0) k÷~ and then

m ÷ ~, we see that

the l e f t - h a n d

easily

follows

side of

from the

(19)

finishes 5C.

the proof.

We will

group

F0(R)

now study

F0(R)*

characters for each

with

with

the c o n t i n u i t y

fixed metric

space.

PWS

(DCT)

implies

(20)

Since

~ F0(R)

~ E F0(R)*

an i m m e d i a t e

and t h e r e f o r e

namely,

and any

if

~n(y)

is c o u n t a b l e , that,

{~n }

of

÷ ~(y)

F0(R)*

is a

for any r e g u l a r

= ml(~(0),0)

i S p S ~, w h e r e

of T h e o r e m

we equip

its c h a r a c t e r

a sequence

if and only

We b e g i n with r e m a r k i n g

consequence

~ ÷ mP($,0),

For this p u r p o s e

topology

mP'(~,0)mP(~-I~(0),0)

for any

).

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

F0(R)*.

topology;

to a c h a r a c t e r

y E F0(R).

compact R,

group

the d i s c r e t e

the c o m p a c t

converges

than

relation

[]

i S p ~ ~, on the c h a r a c t e r the group

tend

is not larger

inclusion

~p' (R , ~)~P(R, ~-l{(O)) =C ~(R,~(O) This

rk

our claim.

we show that But this

w • Z(0;Rn+ I) \ R n}

(0)(0).

l ~ - log g

-logrn+

exists

+ ~{g(0,w) - ~n+l:

p' = p / ( p - i).

5A and Lemma

5B.

We next

This prove

is the

following Lemma. mP($,0) Proof.

If (20) holds is c o n t i n u o u s

for any on

Take any s e q u e n c e

~ E F0(R)* , then the f u n c t i o n a l

F0(R)* {~n:

for every

n = i, 2,...}

p

C F0(R)*

to the i d e n t i t y c h a r a c t e r Id. Since m~(~n,0) implies that ml(~n-l~(0),0) ~ ml(~(0),0) for lim inf m l ( ~ n - l ~ ( 0 ) , 0 ) n~

We now c l a i m

the r e v e r s e

inequality:

with

~ ÷

i ~ p ~ =. which

converges

~ i, the e q u a t i o n (20) n = I, 2,... and thus

~ ml($(0),0).

205

lim sup ml(~n-16(0),0)

£ ml(~(0),0).

n+~

If this were not the case,

then we could assume,

quence

if necessary,

ml(~n-l O.

[fn(0)[

to a further

> ml(~ (0),0) + s

Then,

choose

shows our claim.

this with

(20), we have

(21) Now take any

p

for any

with

=

and using

the role of

6

These two inequalities

5D.

Let

R to a multiplicative If(0)l ~ m l ( ~ ( 0 ) , 0 ) + s/2.

i = m ~( Id,0).

I £ p £ ~.

Since

~(R,q)~(R,n-l~)

and

{z6:

let

e

inequality:

£ mP(~,0).

show that

F0(R)*.

mP(~,C)

is continuous

in

[]

j = i, 2,...}

n = i, 2,...

~ mP(n,0).

~, we get the reverse

together

group

~ mP(~,0).

(21), we have

lim sup mP(~,0)

every

n = i, 2,...

we could also assume

~, q e F0(R)* , we have

~ ÷ q

on the character

(0))

shown

lim inf mP(~,0) Changing

n = i,

: ml(~(0),0).

mP(D,0)m~(q-l~,0) By letting

for

Hence we have

lim m~(~,0) ~÷Id

~P(R,~)

to a subse-

fn E ~ ( R , < n - l {

for each

if necessary~

lim ml({-l$(0),0) ~÷Id Combining

an

> ml(~(0),0) + e/2

subsequenee

that {fn } would converge almost unformly on function f C ~ ( R , ~ (0)) with llfU1 ~ 1 and This contradiction

passing

be an enumeration

be the character

of

Z(O;R).

For

of the l.a.m.

n n

exp(-

and l e t and

Cn(Z) E ~ ( R , 8 n)

(Cn)0(0 ) > 0.

Since

~ g(zj,z))

j=l

be d e f i n e d by

IOn(Z) I = exp(-~ j =n l g ( z j ' z))

llCnllp =< llCnll~ = i, we see first that

n

exp(for

~ g(z~,0)) j=l

= ICn(0) I £ mP(en,0)

i £ p £ ~. Let

p,

i ~ p ~ ~, be fixed and take any

f E ~(R,Sn).

The

.

206

quotient f0/(Cn) 0 of the principal branches then extends uniquely to a meromorphic function, say hf, on R. Since lhflexp(-[j~l g(zj,.)), being equal to Ifl, has a harmonic majorant, the weak Cauchy theorem (Theorem 4B, Ch. VII) shows that Since

hf • LI(dx)

lhf(0) I = If(0)I/ICn(0) I

and

If(o)l ! ICn(O)I I

and

hf(0) = fA I hfdx.

ILhfllI = NfilI S IIfNp, we have

I~fldx

=

ICn(O)lFIhfll 1

A1

= ICn(O)IllfllI ~ ;Cn(O)lllfH p. As f • ~P(R,8 n) and thus

is arbitrary,

mP(~n,O) ! ICn(O)l

= exp(-[j~ I g(zj,O))

n

mP(Sn,O) for

= exp(- ~ g(zj,O)) j=l

i S p S ~• oo

Since

~j=l

of

R\ Z(0;R),

the

function

verges to

g(zj,z)

we s e e B, w h i c h

~(0)

ml(6(O),o)

If

is that

uniformly {C n}

was d e f i n e d

ml($;0)

convergent

converges in

5B,

on a n y c o m p a c t

almost

uniformly

and therefore

is continuous

in

that

on

subset R

{0 } n

to con-

~, then

= lim ml(@ O) : exp(- ~ g(zj,O)) n+~ n' j:l

: g(O)(o),

which implies, by Theorem 5A, that R satisfies (DCT). Summing up our considerations in §2, we finally get the following characterization of (DCT): Theorem. conditions (a) (b) (c) the form (d)

Let R be a regular PWS with origin are equivalent: (DCT) holds. H~(dx) ± = HI(dx,I/Q(0)). Every simply invariant subspace of

0.

Then the following

LP(dx),

i ~ p ~ ~, is of

HP(dx,Q) for some i-function Q on T. ml(~(0),0) = g(0)(0), where ~(0) is the character of the

l.a.m, g(0)(z) = exp(-[ {g(w,z): w e Z(0;R)}). (e) m~(6,0)ml(~-l~(0),0) = ml(~ (0) ,0) for any ~ • F0(R)*. (f) 6 + ml(~,0) is continuous on F0(R)*. (g) ~ ÷ mP(~,0) is continuous on F0(R)* for every p with i~pS~.

207

NOTES Most of the o b s e r v a t i o n s Hayashi

stated above are based on the idea of

[28] with a couple of m o d i f i c a t i o n by the author.

involving the function

m~(~;0)

is also in Pranger

[55].

A discussion

CHAPTER

The faces sion

EXAMPLES

objective

of this

of P a r r e a u - W i d o m is by no m e a n s

Riemann The

X.

surface

emphasis

we are

going

should

(DCT)

but the

examples

§i we g i v e

surface

of M y r b e r g (DCT).

type

three

type

for which

problem

in the m a x i m a l

shown here

functions positive corona

as w e l l

problem.

the o t h e r ,

i.

OF I N F I N I T E

PWS's

of M y r b e r g

of M y r b e r g

ing

type

surfaee

counting

over

R ÷ ~ of

in

§3 t h a t

of

Ha(R)

is d e n s e

of a l m o s t

~,

are

condition

corona

exist

PWS's

FOR WHICH

(DCT)

for which out more

s u c h a PWS a l w a y s

every

for

spaee

of P a r r e a u -

discussing

PWS

R

as an o p e n

can

that

the

our

the

be em-

subset.

It is

o f all h o l o m o r p h i c

convergence.

theorem

for which

PWS's

for a R i e m a n n

regions

constructed

one h a n d ,

the

Here

for which

PWS's

show that

in t h e

uniform

examples

on t h e

GENUS

Finally

concerning

examples

but not corona

in

(DCT)

in

§4,

the §2 c a n

and,

on

theorem

fails.

PWS's

of i n f i -

HOLDS

Type

for w h i c h

~:

show

of PWS's.

(i)

PWS's

is to f i n d

As a p r e l i m i n a r y

space

satisfy

Our o b j e c t i v e

genus,

function

we

ideal

show,

there

PWS

IA. nite

that

fails.

as n e g a t i v e We

sufficient

in

bordered

of PWS's.

§2 p l a n a r

H~(R)

so as to

problem

sur-

discus-

connectivity.

(iii)

then

PWS's,

that

and

classification and

compact

be t y p i c a l

planar

to be a PWS a n d

in the t o p o l o g y

be m o d i f i e d

§i.

for

is true;

of

foregoing

of c o n s t r u c t i o n :

(ii)

we c o n s t r u c t

(DCT)

corona

types

A remaining

a necessary

every

never

TYPE

examples

our

of i n f i n i t e

holds;

theorem

is false.

Next,

Although such can

on P W S ' s

(DCT)

some

to s h o w t h a t

different

corona

bedded also

a PWS,

at a r e a s o n a b l e

In

Widom

in vain.

for which

theorem

aiming

satisfies

a labor

be p l a c e d

OF P A R R E A U - W I D O M

is to p r o v i d e

in o r d e r

to g i v e genus

the c o r o n a

chapter

type

is e v i d e n t l y

of i n f i n i t e fails

OF S U R F A C E S

in t h i s

(DCT)

the unit which i.e.

multiplicities.

section

holds. disk

makes

• R

each point In

§i

R

is to c o n s t r u c t

A Riemann

surface

if t h e r e

exists

an n - s h e e t e d , of

~

denotes

has

R

is s a i d to be an a n a l y t i c

branched,

exactly

a Riemann

n

full

cover-

pre-images,

surface

of t h i s

209

type and

{~j} = {6j:

over which Theorem.

~

j = i, 2,...}

is branched.

Let

R

fined above.

is the sequence

surface of Myrberg type over

Then the following conditions

R

(b)

The set

(e)

The sequence

in

~D

Then the following holds.

be a Riemann

(a)

of points

D

as de-

are equivalent:

is a PWS. H~(R)

separates {%j]

the points

of

R.

satisfies

the Blaschke

(i-l{j[)

< ~.

condition,

i.e.

j=l The proof will be given in IG below,

after some preliminary

obser-

vations. IB.

Lemma.

Suppose that a bounded analytic

separates

the points

satisfies

the Blaschke

Proof.

We define,

in

~-i(~,)

where

~-I(~)

function

for every

function whole

in

= {al,..., of

n

~* • ~ \ {~j}.

on

R

Then

{~j}

< • D \ {%j},

= (-l)kak(f(al),...,f(an)), a n}

and

~k

variables.

~ \ {~j)

~.

f

condition.

Ak(%)

metric

for some

function

denotes

Clearly,

the k-th elementary

Ak

and thus can be continued

The definition

of

Ak'S

sym-

is a bounded analytic analytically

to the

implies that

f n + (A I o ~)fn-I + ... + A n O ~ = 0 holds on

~ - i ( ~ \ {~j})

discriminant

and thus everywhere

of the equation

see that the function

D(~)

and only if the equation arates the points the points

{j

in

iC. verges,

condition.

(z E R: g(a,z) choiee of distinct

to

X

n

has

D({j)

D(~) = 0.

function

H~(~)

B(~,a) > e}.

D(~) = 0

roots.

be the

in

X.

D(~)

~ 0

Since

f

We if sep-

does not vanish identically. Namely,

D(~)

on

~j's ~.

So

At

are among the zeros {~j}

satisfies

[] integral

f0 B(~,a)d~

is the first Betti number of the region Since the convergence

a, we may assume that the set points.

Let

and that

distinct

Next we want to see when Widom's where

R.

X n + AI({)xn-I + --- + An(~) belongs

~-i({,),

we have

of the bounded analytic the Blaschke

in

on

We may also assume,

conR(e,a)

does not depend on the

~-l(@(a))

consists

by applying a conformal

of

n

transfor-

=

210

mation

of the

~-i(0)

disk

contains

if n e c e s s a r y , n

distinct

h(z) Then

we h a v e

h(z)

= -log

function

(see

harmonic

everywhere

h(z)

closed

u(z) and

region

follows

that

u(z)

For e a c h first such we

Betti that

find

on

e > 0

number

let

of

{z E R:

S

S .

I~(z)I

a constant

l~(z)I find

~ r.

R.

Lemma.

Let

Proof.

< r}

is c o n n e c t e d .

such

Moreover, Aa,

Let

> 1

any

such

I~(z)I

S e = {h(z) {I~(z)I

in

S

e -e.

by

(i),

R ( ~ / A , a I) all

of

ID.

S

cannot .

By use

of

$ log p

on

see that

log I~(z)l

is a p o t e n t i a l ,

.

and

number

B ' ( ~ , a I) r,

< 0 it

the

0 < r < i,

By H a r n a c k ' s

inequality

S A g ( z , a I) a' E R

with

l%(a')l

< r < I,

that < Aa,g(z,a, )

Then

S

> ~}

< r}

and

e < h(z)

and

=C R ( ~ / A , a l )

= {[~(z) I < e-S},

which

includes

the

so is c o n n e c t e d .

I~(z)I

Conversely,

is c o n n e c t e d

< e -~,

take

any

=< A g ( z , a I)

have

any

compact

Let

0 < e < - log r.

i.e.

z E se.

z E R and

with

thus

complementary

r
e}

fix a p o s i t i v e

R ( e A , a I) ~ S

connected

hand,

Since

: {z E R: h(z)

AaTlg(z

z E R

I~(z)l

we h a v e

We

A > i

a constant

for any

that

We set

: h(z) + log I~(z)I

Letting

other

on

form

0 < p < i.

curve

~ p}.

On the

local

u(z)

with

functions

A - I g ( z , a I) S h(z)

for

we a s s u m e

a = a I.

~ 0, as c l a i m e d .

(i)

we

R.

~ h(z)

the

that p

on the

I~(z)I

and

an

g(z,ak).

any

$ log p

Namely,

= 0.

In fact,

for h a r m o n i c

0 ~ u(z)

~(a)

al,...,

I) shows

Take

{z e R:

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

Ch.

R.

u(z)

principle

n [ k=l

=

19(z)l.

6A,

on

is p o s i t i v e ,

the m i n i m u m the

Theorem

that

points

< As

it i n c l u d e s

211

Proof.

We claim that

relatively have

compact

S

in

3U ~ 3R(~A,al).

therefore cycle

g(z,a I)

y

gion

in

If

has no components were

It would follow that bounds

in

U, which are

such a component,

on

then we would

g(z,a I) : ~A

on

3U

U, a contradiction.

and

So, if a

S , then it should bound in the re-

This shows the first inequality.

be shown similarly.

The other half can

[]

We thus have for any •-log r I B(~A,al)d~ which

U

would be constant

R(eA,a I)

R(~A,al).

\ R(~A,a I)

S .

0 < s < -log r i-log r

~

B'(~,

al)d~

r-logr B(~/A,al)d~,

~

implies A_iI-l°g r ~

B'(~,al)d~

~As

Hence,

converges

We now estimate

suppose that the circle the double

S

of

a meromorphic

S

say

We see that

~.

It is then clear that

makes

order to obtain a further Riemann-Hurwitz

relation:

branched,

= e}

Let

contains

B(~'al)d~"

~, of S

B'(e,a I)

does.

0 < ~ < -logr

and

~j's.

~(z)

We then form

can be extended to

onto the extended

an n-sheeted,

branched,

complex plane

full covering of

is exactly the genus of

information

about

if a compact

full covering

(2)

S

f0 B~(~'al)d~

no

so that the function

function, ~

if and only if

f0 B'(~'al)d~" {l~I

¢.

sheeted,

~ A;c/A

"s

f0 B(~'al)da

IE.

r-log r

I-l°g r B(~,al)d~

In

B'(~,al) , we need the

Riemann

of another

S •

surface

compact

F

surface

is an nF0, then

X = nx0 + V,

where and

X V

(resp.

X0) is the Euler characteristic

of

F

(resp.

is the sum of the orders of all the branch points of

Nevanlinna

F

Fo) (see

[44], p. 324).

Lena.

(3)

B'(~,a I) ~ b(e -e) + I - n,

where

I~1


O.

For

0 < r < r0

{m • V:

I~(w)I

< r}.

subset

U

R, e a c h

of

in

M(H~(R)).

Ur

(reap.

r0.

@

The

M(H~(R)). borhood

6.

spect

set

in the

set

is d e n s e

Theorem. (a)

Let

a subregion

(b) of

is compact.

onto

the

on

R.

M(H=(R)). R

that

0

in

Vr

for

is o p e n ~) m a p s

each

0 < r
0 Pri(v)

subset the

exists on

CO

quasibounded.

If

and

on

S

is s t r i c t l y

[]

has

one

We

set G

on in

G.

We

S \ G.

v.

This

by

means

majorant

u = v-c

on

on

G

u0

to the

resulting The

that

this G,

the

the

such

function,

LHM of

for a n y

inner v

u1

quasibounded

is t r u e

function

on

is h y p e r b o l i c ;

u0

v.

Thus

is a n o n -

components,

extend The

Since

a harmonic

c}

of its

function

then

unbounded.

2B,

positive.

such

v

in 2A.

is m a j o r i z e d

by

is b o u n d e d ,

of T h e o r e m

harmonic

zero

v

assume

denotes

a harmonic

Hence,

If

{z C S: v ( z ) - >

mentioned

and

is m a j o r i z e d

~(v)

G

0 ~ u0 ~ I

identically

be zero.

we m a y

the h y p o t h e s i s

v

for w h i c h should

S.

and

of

v ~ 0.

such that

a nonconstant

is s u b h a r m o n i c bounded

of

that

so that

c > 0

condition

satisfies

by s e t t i n g

prQ(v)

assume

quasibounded,

a constant

satisfies

Since

whole

2A we m a y

part

should

be

238

2D. polar

Theorem.

set

(Ch.

tion

on

G

such

some

convex

a harmonic Let

be the

bounded monic

that

SO

data on

on

equal

SO.

Theorem

impossible, u

§2.

NULL

3.

for

( U + ~(livU

then

Since

to

F

OF CLASS

u

u

F

be a c o m p a c t

is a h a r m o n i c U

on

G

is e x t e n d e d

to

funcfor F

as

Let

as

%(t)

for

))/2

on

on

S O \ F.

~(iu01) G.

SO \ F

set.

F

and

SO

with

~S0,

v

is also

u0

is h a r -

Then

the

~ ¢((lul + 1vi)/2)

If

u0

should

Hence

be a c o n v e x

function

~

were

nonconstant,

be h y p e r b o l i c .

u0 ~ 0 function.

to

there

on

SO

This

and

is

there-

[]

such

= 0

and

~

as

large

t;

the r i g h t

t ÷ =

exists

We w r i t e t => to,

~(t)

tO > 0

E(t)

both

Let

E(t I) $ 4 for

number

with

property:

-~(t2) > 32

and

in v i e w that

E(t)

of

#(t)/t

t}

and

and

E l(t)

and

tI

be a r e a l

El(t I) ~ 2.

num-

We

set

t I ~ t ~ t2,

_~l(t2) > 3.

t 2 > t I + i, When

the p r o p e r t y :

( n + l ) l o g 2},

}'(t + 0)

way.

~ t I + 2 ( t - t I)

n => 3, w i t h

such

= min{#(t)/t,

so that

by i n d u c t i o n .

the

derivative

nondecreasingly,

a number

t I $ m a x { t 0 , log 2},

~ ( t n _ I) > m a x { ( n - 2 ) t n _ l ,

Then

that

in a n o n d e c r e a s i n g

tn_ 2 =< t =< tn_l,

(i).

l i m t + ~ ~ ( t ) / t 2 = 0.

it has

for

our

satisfying

(i);

t ~ tO. 2t}

function

for all

is c o n v e x ,

t ÷ ~

is a r e a l

m a x { t 2 , 41og 2},

on

contains

for

N~

¢

tends

Thus

construct

that

which

is b o u n d e d

as a h a r m o n i c

satisfies

= m i n { % ' ( t + 0), ~

S

problem

Lemmas

t, w h i c h (i).

of

and

that

is a p o l a r

~(t)

for

imply

limt+ ~ ~(t)/~(t)

t2

(i),

u

~S 0

(A3)

such

where

on

~(t)

to

let

majorant

u 0 = ( u - v)/2

(A2)

We n o w ber

~

a convex

is n o n d e c r e a s i n g

tend

and

a harmonic

Dirichlet

Since

~(t + log 2) ~ 27(t)

for all

= (t) -i

set

F

Lemma.

exists

u.

u0 : 0

2B w o u l d

SETS

condition

[

If

subregion

of the

(AI)

Proof.

has

in

G = S \ F.

satisfying

to

We

Preliminary

there

~

is e x t e n d e d

3A.

Set

be a r e g u l a r

S O \ F,

fore

be a r e g i o n S.

~(iui )

solution

(~(iui) + ~ ( i v l ) ) / 2 then

S

in

function

let

boundary

6C)

function.

Proof. v

Let

I,

E(tn_ I) > n

~(t)

2tn-i and

~(t 2) > is d e t e r m i n e d

=> tn_ 2 + i, _~l(tn_l)

> n,

239

we d e f i n e $(t) where

= $ ( t n _ I) + n(t - tn_ I)

for

tn_ I ~ t g t n,

tn

is a r e a l n u m b e r w i t h the p r o p e r t y : 2 m a x { ( n - l ) t , ( n + 2 ) l o g 2}, E(t n) ~ ( n + i) and of the m o n o t o n i o i t y

property

seen to be p o s s i b l e , ously,

~(t)

for all [0, ~)

so that

is c o n v e x

0 g t g tI with

~(0)

of

for

Z

and

~(t) t ~ tI

t => t I .

and

~(t n)

Zl(t n) ~ n + I.

Zl, t h e s e

is d e f i n e d

operations

for all

In v i e w are e a s i l y

t ~ t I.

Obvi-

it is easy to c o n t i n u e

so as to h a v e a n o n d e c r e a s i n g

convex

~(t)

function

on

: 0.

We n o w h a v e o n l y to s h o w that ties for

t n ~ tn_ I + i,

~(t)

tn_ 1 =< t ~ t n.

Let

possesses

the d e s i r e d

t + log 2 =< t n + i

Then

propertn+ 1

and t h e r e f o r e ~(t + log 2) g ~(tn_ I) + (n+l)(t + log 2 - tn_ I) = ~ ( t n _ I) + ( n + l ) ( t

- t n _ I) + ( n + l ) l o g

2

2~(tn_ I) + 2n(t - t n _ I) =

The c o n d i t i o n

(AI)

2~(t).

is thus

satisfied.

n ( t - t I) + t I ~ ~(t)

For the

~(t)/#(t)

÷ 0

shows t h a t

3B.

E(t) ~ n 2, we h a v e and

t/~(t)

The

~(t)/t 2 + O ÷ 0

as

following

as

t ÷ ~.

result

t

we h a v e

= ~(tn_ I) + n(t - tn_ I)

(n-2)tn_ I + n(t-tn_ As we k n o w that

same

~(t)

I) > (n-2)t.

=> n2t

t + ~.

and

t 2 => n2t.

The a b o v e

Thus

inequality

also

[]

shows the

importance

of the c o n d i t i o n

(AI) or, m o r e g e n e r a l l y , ~(t + log 2) = 0(i), ~Ct)

(3) which (of.

is e q u i v a l e n t Krasnosel'skii

Lemma.

Let

let

be a t o t a l l y

E

and o n l y if Proof.

~

to the A 2 - c o n d i t i o n and Rutickii

be a c o n v e x

in the t h e o r y

of 0 r l i c z

spaces

[41]).

function

diseonneeted

t ÷ ~,

which

compact

satisfies set in

¢.

(i) and Then

(3) and E E N~

~ \ E e 0~.

S i n c e the n e c e s s i t y

is t r i v i a l ,

we o n l y

show that

[ \ E 6 0~

if

240

implies V 2 E not

E E N}. and any

vanish

and

u

~(u)

~ U

LHM of

on

now divided

both

u = lim

into

and n~

Let

and

parts. that

are

uA n

and

U

E

U : lim

are b o u n d e d ,

means

of T h e o r e m

2D o r o t h e r w i s e .

monic

on

V

and

that

quently,

f

is b o u n d e d

even

(ii) In t h i s We

part,

set

Since

we

Hence,

suppose

assumption

We may

assume

case

analytic V \ En,

curves

and

now

loss

that

where

2C, Ch. II,

uA n

and

functions

and

by

U are harConse-

therefore

is h o l o -

to be p r o v e d . E

is not

F = [\ V

including (i)

shows

that

E

is n o t

of generality

a polar

is a p o l a r

F

and

that

f

set. set.

f E H ~ ( S \ F).

f ~ H~(S).

is a c o n s t a n t

of

a polar

that

region

is c o n t i n u o u s

n $ i, be a n e x h a u s t i o n

boundaries,

and

that

i.e.

in

both

by c o n t i n u i t y .

E

as w a s

implies

is a J o r d a n f

seen

5C,

is

Our

function.

f E H~(V).

suppose

V

V E 0G,

by T h e o r e m

Since

u

V \ E

proof

by Theorem

that

does

see t h a t

The

as h a r m o n i c

of

follows,

is a r e g i o n

S E 0~

without

in w h i c h

V

V, as

our proof

that

Finally,

on

Then,

Thus, V \ E.

to

f C H~(V),

S

set,

set.

It f o l l o w s

in w h a t that

Then

it is t r i v i a l (iii)

~ U

}(u).

f on

We e a s i l y

L H M of

on

extended

any region

that

}(log+Ifl)

IB).

V \ E.

in a n e i g h b o r h o o d

suppose,

is a p o l a r

original

the

E.

S = [ \ E. F

Hence

on

So we

~(u)

on UA n

n~

UA n

morphic

are

is the

We take

of c o u r s e

L H M of (see

is a p o l a r

quasibounded

these

suppose

v \ E

in f a c t

three

[ \ E E 0~.

be t h e

on

suppose

U

that We m a y

U

log+ifI

V \ E

First

u

suppose

identically.

the

(i)

So w e

f E H ~ ( V \ E).

~ ~ V.

bounded

up to t h e

V k E

E =C En+ I =C En =C V,

set a n d

We b e g i n

by a f i n i t e boundary

by J o r d a n

n => i.

V ~ 0 G.

of

V.

regions

For a n y

with

number

of

Let

with

analytic

z @ V \ En

we

have

f(z)

1 SaV f(~) ~ = ~ d~

i IDE ~f({) r ~ d~.

= 2-~

- 2-~

n

The

first

and

spectively

by

respectively. that

g(z)

the

second

g(z)

members

and

F r o m the

hn(Z) , which fact

is c o n t i n u o u s

f(z)

by a c o n s t a n t

on

V \ E1

and therefore

define

on

[ \ E.

We t h u s

f(z)

lh(z)I

~ M+

positive C2

on

If(z) I

constants

M > 0.

there. CI

and

right-hand are

C2

(z) of

see a l s o

a single

o f the

on

that

on

property

denoted

and on

re-

[ \ E n,

V \ E and

all

follows n so is b o u n d e d

(z) n function,

V \ E

in s u c h a w a y t h a t

are V

on V

holomorphie

= g(z) - h(z)

By u s e

n

boundary

We

side

defined

: g(z) - h

up to the

in m o d u l u s

have

of the

h

and

(3) of

coincide h(z),

consequently ~

we f i n d

~(log+lhl)

£ CIU+

V \ E.

Let us

fix a point

z0 C V \ E1

and

denote

by

Bn

(resp.

v n)

the

241

harmonic

measure

at the

point

z0

with

(resp.

V \ E ) for n = i, 2, . . . . n a constant C > I s u c h that

find

see this,

take

a Jordan

E 1 C ~ C CI(~) a region

region

_C V \ {z0}.

D, we

for

n = i,

have

Green

and

V \ E = Un~ I V \ E n.

most

~.

in

E

So,

there

each

exists

fixed

g(a,z:D)

a polar

by a s s u m p t i o n

): n = i

boundary the

set,

for all

DE n

set,

such

Green

2,.

that

}

we can n.

To

that

function

the

g(w,z0;¢\

sequence

for

{m n}

g ( w , z 0 ; ¢ \ E) ;V\E) g(w'z0

principle

and

¢\ E

V \ E 1 C V \ E 2 =C ...

to

we

g(w

see that

z0;V\E)

al-

E n) ÷ g ( w , z 0 ; V k E )

E n) ÷ g ( w , z 0 ; ¢ \ converges

E)

uniformly

to

< oo.

i < C < ~ such = < C g ( w , z 0 ; V \ E n) for all

n, the m a x i m u m

V \ E

theorem,

tends

a constant

E n)

both

n ' "" In p a r t i c u l a r , g(w,z0;V\

max wED~

g(w,z0;¢\

analytic

by

by use of H a r n a c k ' s

Similarly,

Consequently,

i.e.

with

is not

We k n o w

V\E.

D~.

Hence

on

¢\ E n

a polar

g ( w , z 0 ; ¢ \ E n) g(w,z0; V \ E n )
a}

a constant

by

Then

t.

on

lel 4~a ~ A ' U l ( e ; w ) ' where

D 2 = {Izl

in

Denote

DO.

and

D. whose boundary ] elsewhere. Clearly,

Ul(e;z) on

F(a)

set i n c l u d e d

,

and any n o n n e g a t i v e

on

function

on

function

in a n o n d e c r e a s i n g

Zl, z 2 • F(c)

to

~ A ( a / b ) . - - ~2~a

element

exists

closed

is a region.

with respect

nondecreasing

a/b

for any

the b o u n d e d to

the p o i n t at i n f i n i t y

be a b o u n d e d

> a} \ F ~

z • F(a)}

is a finite, Let

F

D O = {Izl

at the p o i n t

m a x {d ~ (z): where

> a}, we i n c l u d e

set.

we see for

lwl = c

lel i + (a/b) I/2 ~ 2~a i - (a/b) I/2 e.

'

So

lwl = c} ~ A . m i n { U l ( e ; w ) :

lwl = c},

where A = A(a/b)

From

(5) f o l l o w s

el,

B(e I) ~ A ~ ( e 2)

if

and the f u n c t i o n problem

e2

max{

on

F(a)

lel]

w i t h the

= le21 , b e c a u s e

U0(ej;z) ,

F(e)

(z):

lwl : e} ~ A - m i n { U 0 ( e 2 ; w ) :

on

for the r e g i o n

U0(ej;z)

"

at once that

max{U0(el;w): for any arcs

= 2A'(a/b) .I + (a/b)i/2 i - (a/b) I/2

and to

z e F(a)}

same length. ~(ej)

j = i, 2, is the

{Izl

> c} \ F 0

(z):

This

shows

= U0(ej;~),

solution

w i t h the b o u n d a r y

elsewhere.

~ A.min{

lwl : c}

j = I, 2,

of the D i r i c h l e t d a t a e q u a l to

Hence z • F(a)}

that

~ A-#--(-~

.

244

3D. class

The next

l e m m a is the key to our c o n s t r u c t i o n

of null

Lemma.

Let

number

0 < a2/b

of b o u n d e d

< a0 < a < b < ~

closed

1 < i < k, is a r e g i o n i, 2,...} ~(n)/n

subsets

including

be an i n c r e a s i n g

: o(i),

n ÷ ~, and

FI,... , F k

{Izl

> b}

such that

all

F.

with

j ~ i.

sequence ~(n)

and let

of

of p o s i t i v e

> n I/2

a0exp(2~ji/n) We set polar

and

Kn, j = { I z - Wn,jl

numbers

for all large

K n : u.n]_l Kn,j.

Suppose

set and that we c o n s i d e r

in the d i s k

{Izl

Let

Pn

~

touthe

and

regions Then,

such that

for

that e a c h

only large

F.,] n's

> a 0} \ F

for any

and

s > 0, there

exists

B = B(a/b)

(6)

< E,

is a c o n s t a n t

We first

and t h e n c h o o s e

(l+s')-ip(F(a0))

where

~'

(resp.

s' > 0

< ~'(F(a'))

~") d e n o t e s

{Izl

w. ~K n

Un(Z)

with

N = N(s)

o n l y on the r a t i o

F : > 0

a/b

in a

support

the G r e e n

on

potential

Un(Z)

b = i.

Let

c >

< 2-1~(F(a0))-l~ so c l o s e

> a'} \ F

to

a0

as to have

< (l+s')~(F(a0)) ,

measure (resp.

at the p o i n t {Izl

be the G r e e n

We d e n o t e

> a"} \ F).

function

for the

by ~ the p o s i t i v e m e a s u r e n and w i t h u n i f o r m d e n s i t y on ~Kn, and

of

= I

that

with

< ~"(F(a"))

= l o g ( l l - z ~ I / I z - w I)

of m a s s by

is i n c l u d e d

i ~ j ~ n,

the h a r m o n i c

o p e n unit d i s k w i t h pole at with

s'}

< a 0 < a" < a

to the r e g i o n

g(z,w)

i

Kn

1 ~ j ~ k,

loss of g e n e r a l i t y

choose

0 < a'

(7)

Let

i =< j =< k, is not a so that

< g,

depending

m a x { ( l + s ' ) - (i+~') -3,

with respect

for

:

fashion.

We may a s s u m e w i t h o u t

be given.

w

n,j j = i,..., n.

an i n t e r g e r

~n(~Kn, j) ~ ( B n ) - l ~ ( F ( a 0 ) ) ,

Proof.

Let

n > N

I ~ ( F j ) - ~n(Fj)l

nondecreasing

n =

such that n.

(K n U F) , r e s p e c t i v e l y

[\

'j~(F(ao) ) - p n ( ~ K n ) l

where

~ \ Fi,

{~(n):

< a} and K .'s, i ~ j ~ n, are m u t u a l l y d i s j o i n t . n,] be the h a r m o n i c m e a s u r e s at the p o i n t ~ with respect

{Izl

Uj~ I Fj.

~ a0e-~(n)}

be a f i n i t e

each

Let

=

0

sets of

N~.

Vn'

i.e.

g(z,w)d~

~K n

(w) n

245

on the disk

{Izl

~ i}.

harmonic

{Izl

< i} \ Kn, and v a n i s h e s

tation

on

shows

Clearly,

Un(Z)

is e o n t i n u o u s on

F(1).

on

{Izl

~ 1},

An e x p l i c i t

compu-

that i

n

~ g(z ) j:l 'Wn'j

if

z ~ K

if

z E K

n'

Un(Z): { i

where of

Rn,i(z)

{£(n)},

(ii)

F(a')

there

seen that

K

n Let

for

i

|2~ J"

--2~

0

exists

n > N'. = (resp.

u'

(i)

U (z) n

n,

i'

By use of the p r o p e r t y converges

uniformly

on

to the value ,a0eit)d t g(a'

log

a0

N' = N'(e')

i__ a0
a'} \ F F(a')

(resp.

(resp.

[ \ (K n m F ) )

aK ) and to

with the b o u n d a r y 0

elsewhere.

data

Take

N" =

n

N"(c')

so large

that,

for

n > N"

K '

Izl

a"}


( 2 B N ' ( n ) ) - l ~ n I(~D~) = "

(c) that

for any

n

(16)

~(~D~)

~ (2B)-n(N'(n)N'(n-I).''N(1))-IB~(F(a)) : (2B)-IN(n)-I~(F(a)).

We set

250

E = and d e n o t e

by

DE

to the r e g i o n

¢\

the

Izl

harmonic

[Cl(

U Ks)] s:n

measure

at the

point

~

with

respect

(EUF).

Our c o n s t r u c t i o n K I' =C {6a


n

of

> 0.

problem

Thus

Moreover

theorem,

with

BE(E)

we h a v e

un

The

and

that

function

u

the b o u n d a r y

= u(~)

> 0

and

v n (~)

= ~ n ( Z K n) => 1 - (I+s)B(F)

is seen data

to be the

equal

consequently

to

1 E

solution on

E

is not

>

of and

to

a polar

251

(17)

~E(E)

= u(~)

: lim Un(~)

~ lim sup Vn(~)

n~

n~

= lim sup ~n(~Kn) On the other hand, and therefore with

(17),

since

~E(F)

proves

shows

is included

$ B(F).

(12).

~E(Fj) -B(Fj) which

E

Thus

~(F(a))

~ ~E(F) -~(F)

show finally

that

E E N@.

nonzero

H ~ ( [ \ E)

element

in

To show that

is included

n $ 0

f

and

v

of

K' n

D' = {Iz- w' I £ r'}

(a) says that the annulus

to the annulus

Let

~w

{r' < Iz - w' I < p2r'}

is bounded

A'/(4~r')

on

where

3C.

Iz -w' I < p2r'},

f

we have for any

the convex

function

and by

and analytic w

f

be a

}(log+Ifl)

we first note that Let us fix ' K n.

in

measure w

inequality

is the constant

is bounded

log+If(w)I

equality,

F(w';r')

A' = A'(p -2)

Since

of

n = 0, i, . . . .

at a point

then Harnack's

F(w';p2r'),

(3), we

Let

{r' ~ Iz- w' I £ p4r'}

r(w';pr'),

by

majorant

be the harmonic

on the circle

Applying

together

satisfies

be one of the members

and

K'n \ D'.

%

function,

for

the property

of Lemma

~ ~(Fj)

~ su(F),

[ \ E E 0}.

a harmonic

meet

F

Since

3B, that

is a constant

in the interior

and let E,

ZE(Fj)

~ BE(E) , which,

: D(F(a)) - ~ E ( E )

in view of Lemma

[ \ E.

{Izl ~ a},

(ii).

We will

E

in

We also have

have only to prove, on

~ ~(F(a)) - e~(F).

does not

with respect

in it. implies

A'/(4~p2r ') appearing

Then

If

w

that

is

d~w/dS

on in the proof

on the annulus

{r'
b} \ F

such

that

measures and

{Iz[

and {Izl

at t h e

F

be a b o u n d e d

> b} \ F

let

is a r e g i o n .

point

~

with

respect

> a} k F, r e s p e c t i v e l y .

Then

Let to

254

log(c/b) log(c/a) Proof. {Izl

Let

u

> a] \ F

elsewhere.

be the s o l u t i o n

F

lies

{Izl

in

u

on

: u(-)

~ c},

Fj,

Lemma.

sets

Let

such that

Suppose

with

~

z • F},

with respect

let

solution

a

Pa(Fj) > 0

on

D a.

strictly

quence

{U(an;Z)}

~\

i

v, so that

e a s y to see that

v

is the

w i t h the b o u n d a r y

d a t a equal

for

{an:

functions.

say

i

and,

measure

> a} \ F.

for e a c h

at the

Then there

is,

such that

is h e l d fixed. D a = {Iz - z01

n = l, 2,...}

Let

u(a;z)

> a} \ F

above, v(z),

sequence

it c o n v e r g e s

uniformly

the p o i n t on

~F.q

z0

[ \ F. and to

se-

is mo-

by H a r n a c k ' s

on any c o m p a c t

problem 0

set

is a r e m o v a b l e In fact,

of the D i r i e h l e t on

of p o s i t i v e

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

Since this

is h a r m o n i c

solution to

[\ F

3F. a n d to 0 e l s e w h e r e , so ] is a s s u m e d to be n o n - p o l a r , we h a v e

is b o u n d e d , v

F.

point.

on

and is b o u n d e d

v

all

be the h a r m o n i c

property

to zero and c o n s i d e r

function,

Since

~

including

0 £ j £ k.

Take any s e q u e n c e

increasing

of

{Iz - z01

< E,

Pj

of h a r m o n i c

( F U {z0}).

singularity

Since

decreasing

to a h a r m o n i c

Let

of b o u n d e d

be any f i n i t e

be the h a r m o n i c

problem

to

number

is a r e g i o n

0 ~ j £ k, w h i c h

data equal

= u(a;~).

numbers

notonically

j,

i

w i t h the a b o v e

of the D i r i e h l e t

w i t h the b o u n d a r y

in

we have

~ log(]zl/e)/log(a/c)

to the r e g i o n

to the r e g i o n

s > 0, a n u m b e r

Take any i n d e x

theorem

elsewhere,

#(F(b)).

set.

Pa

p ( F j ) - #a(Fj)

u(a;z)

u(z)

F = Uj~ 0 Fj

F.± is a n o n - p o l a r at the p o i n t ~ with respect

0 < a < i n f { I z - z01:

that

0

> b} \ F

for this r e g i o n w i t h 0

k, be a f i n i t e

for each

z 0 • [\ F

that each

measure

be the

and to

{Izl

[]

j = i,...,

[ \ F.

j ~ i, and let

Proof.

for the r e g i o n

F(a)

u(z)dp(z).

log(c/b) ~ log(c/a)

5B.

for any

on

problem and to

we s e e t h a t

inequality.

point

problem

i

to the r e g i o n

F(b)

shows the d e s i r e d

closed

to

a < Izl < c, f r o m w h i c h we get

£(b) u ( z ) d p ( z )

with

u

: [ Jr(b)

f

This

of

of the D i r i c h l e t

d a t a e q u a l to

v(r(a))

Since

data e q u a l

Since the r e s t r i c t i o n

is e q u a l to the s o l u t i o n

~ v(r(a)).

of the D i r i c h l e t

w i t h the b o u n d a r y

the b o u n d a r y

for any

~(F(b))

it is for

elsewhere.

~\ F

255

Thus

v(~)

since

= ~(F.). Since U(an;~) tend increasingly 3 =< u(a;~) for a n y 0 < a < an, we have

for all

sufficiently

chosen,

the l e m m a

5C. {bn:

to

U(an;~)

Lemma.

Let

n = 0, i,...}

0 < d < i.

small

a > 0.

is proved.

~

j

and < E

was a r b i t r a r i l y

[]

be a c o n v e x

a sequence

Then t h e r e

As the i n d e x

v(~)

~(Fj) - ~a(Fj)

function

of p o s i t i v e

exist a sequence

satisfying

numbers,

(i) in IA,

0 < p < ~ < i, and

{a : n = 0, i,...}

of p o s i t i v e

n

numbers

and a s e q u e n c e

an+ 1/a n ~ p,

{E n : n = 0, i,...}

a n =< bn,

N~

En =C {6an =< Izl =< a n }

(19)

such that

for

na n < i,

n : 0, i~...,

and

d £ ~ ( - l o g ( n a n ) ) m ( E n) £ i

for

n = I, 2,...,

where

m

is the h a r m o n i c

with respect

to the r e g i o n

Proof.

{dn: n = 0, I,...},

Let

sequence measure {Jz[ an

in

with

limit

d.

at the p o i n t

i/¢(-log(na

na

n

with

E = Un= 0 E n U {0}.

by

with respect

Pn

(resp.

and

~(-log(na

n

decreasing

mn ) the h a r m o n i c

to the r e g i o n

In the f o l l o w i n g

< i

at the p o i n t

0 < d n < i, be a s t r i c t l y

We d e n o t e

~

> an+ 1 } \ (Uk~ 0 Ek)). so small that

~\ E

measure

~\

(Uk~ 0 E k)

construction )) > 0.

(resp.

we shall t a k e

We w r i t e

~

n

=

)). n

In o r d e r a0,

to c o n s t r u c t

satisfying

polar This

E 0 e N~

which

is i n c l u d e d

is p o s s i b l e

by L e m m a

chosen non-polar

E k e N}

following

E

by i n d u c t i o n ,

and

in the a n n u l u s

3A and T h e o r e m in

(C n)

0 < ak < min{bk,i/k} for

4A.

{6a 0 ~

Suppose

{6a k ~

JzJ ~ ak} ,

and

~(-log(nan))

choose

any

a n d also a n o n jzj ~ a0}.

that we h a v e

0 ~ k ~ n, w i t h the

d n ~ k ~ ~ n ( E k ) ~ ek'

T h e n we c h o o s e (21)

an+ I

(22)

so small

O ~ k £ n;

that

d n + l ~ k ~ ~ n ( E k ) ~ ~k' Vn(F(an+l)) is no p r o b l e m a b o u t

and the fact

for

I ~ k ~ n.

0 < an+ I < m i n { b n + l , ( n + l ) - l ] ,

(23)

> 0

0 ~ k ~ n - i; and

(20)

an+ 1 ÷ 0

we first

~ ( - l o g a 3) > 0,

property:

ak+I/a k ~ p

There

our set

0 < a 0 < min{b0,1}

t/~(t)

i ~ k ~ n;

> i/¢(-l°g(n+l)an+l))

= ~n+l"

(21).

is c l e a r f r o m L e m m a

= o(i),

no f a s t e r t h a n

0 < a n + l / a n £ p;

Statement

t ÷ ~, for

(-logan+l)-I

(23)

~ n ( F ( a n + l )) while

decreases

as

i/~(-log((n+l)an+l))

5A

256

decreases

much

Lemma

5B,

Fj

an+l,

and

then

in

{6an+ I ~

faster

than

= E.]

for

use

(20).

this.

For

j = 0,..., By use

IzI ~ an+ 1 }

~K

denotes

a way

to

[\

(ukn 0 E k u K ) .

of

K

such

Then

Vn(Ek)

~ ~n+l(Ek)

(Cn+ I) is s a t i s f i e d . which

4A we

to

~a

find

set,

= ~n

in

with

a set

a =

K E N~

measure

> ~n+l' at the

it is not h a r d

to

point

find

=

with

a closed

respect

subset

En+ I

that

dn+l~n+l Since

only

and

that

~ ~K (K)

the h a r m o n i c

we have

~ = ~n

of T h e o r e m

in such

Vn(F(an+l)) where

(22)

n,

satisfies

easy

to

(22)

implies

~ ~n(Ek)

for all

for each

(19),

for

By i n d u c t i o n

(C n)

see that,

~ ~ n + l ( E n + l ) ~ ~n+l"

for

can

n ~ i.

fixed

d

we

n,

construct

Set

k,

÷ d.

k = 1,...,

the

property

{En:

n = 0, i,...}

E = Un= 0 E n U {0}.

~n(Ek)

÷ m ( E k)

as

It is t h e n

n + ~.

So,

[]

n

6.

Classification 6A.

and

Let

let

For each

polar

set

that

E

The

{an:

n

included is a l s o

we call

obvious.

By

point

~

in

En

{6an

bounded,

and

with

Izl

=
a

} \ E, ren

speetively.

We use

sification Theorem.

Then

result. Let

~

various Namely

and

~

(AI)

~(t)/~(t -s)

there

exists

E n E N~,

we h a v e

= o(i),

that

t + ~,

set

E

the

function

only

constant

Proof.

Let

0 < p < 6 < I

be fixed.

quence

{bn:

n = 0, i,...}

of p o s i t i v e

and

to o b t a i n

our

clas-

following satisfying

for any

with

H~(¢\

< i/n

in o r d e r

functions

while

b

contains

sets

the

be c o n v e x

a circular

n ~ 0, such E)

circular

fixed

center z -i

s > D.

at the o r i g i n

belongs

to

and with

H ~ ( ~ \ E),

functions. By use

of

numbers

(AI) such

we can that

find

b 0 = i,

a se-

257

0 < ~(log(6-1t)) for

n : i, 2, . . . .

of p o s i t i v e that

By L e m m a

numbers

a n =< bn

and

d ~ ~ ( - l o g ( n a n ) ) m ( E n) ~ i

same m e a n i n g

as in L e m m a

We f i r s t

En

show that

is i n c l u d e d

{an: n = 0, I,...}

{En: n : 0, i,...}

in

En C . {6a n. < Jz . I < a. n } for

n ~ i, w h e r e

m

N}

for

and

such n > 0,

d

h a v e the

5C. z -I E H ~ ( ~ \ E).

~(-log(6an)) As

t ~ i/bn,

5C we get a s e q u e n c e

and a sequence

a n + l / a n =< p,

and

£ 2-n~(log(n-lt)),

Since

a n =< b n, we h a v e

~ 2-n~(-log(nan)) ,

n ~ i.

{6a n =< Izl =< a n] , so we have

in

~(logJz-lJ)dm(z)

=

[ n=0

E

~(log[z-ll)dm(z) E n

£

[

~(-log({a ))m(Z )

n=0

n

n co

< ~ ( - l o g ( 6 a 0 ) ) m ( E 0) + =

[ ~(-log(6a n=l

£ ~ ( - l o g ( 6 a 0 ) ) m ( E 0) +

[ n:l

this shows that

z -I

belongs

to

with

n ~ i.

is i n c l u d e d measure

Since in

F(oa

(24)

).

Harnack's

d~w/dS

where

A'

= A ' ( ~ -2)

We take any [ \ E.

assume

that

f e H ~ ( ~ \ E) f

and

origin.

Namely,

£ A ' / ( 4 ~ a n) ~ A'/(4~2an

u

0 < Jz[ < ~.

)

do not v a n i s h

Since

be the h a r m o n i c at the p o i n t

W

that

F ( a 2 a n ), in the p r o o f of L e m m a

majorant

u

of

is c o n s t a n t .

expansion:

}(log+JfJ)

Since each

f(z)

En

be-

at the

= [j=0 cj z-j

is a b o u n d e d

{a n < [zJ < ~2an} , we h a v e

3C.

So we m a y

can o n l y be s i n g u l a r

f(z)

an

F(an) ,

identically. f

following

n ~ i.

~w

then i m p l i e s

appearing

f

similar

We look at any

{a n < JZJ < @p-la n } Let

on

and a h a r m o n i c

N~, we see that

Let

on

is to s h o w that

we h a v e the

t i o n on any a n n u l u s

4A.

{a n < JzJ < O2an }

inequality

is the c o n s t a n t

Our o b j e c t i v e

longs to the class

a = (~p-l)i/4.

to the a n n u l u s

d~w/dS

(25)

for

n

))

can be shown by an a r g u m e n t

in the p r o o f of T h e o r e m We set

n

2 -n < ~.

0 < p < 6 < i, the a n n u l u s

~ \ E.

with respect

on the c i r c l e

on

used

))/}(-log(na

H ~ ( [ \ E).

The l a t t e r h a l f of the t h e o r e m to the one a l r e a d y

n

analytic

func-

258

l°g+If(w)l a I l°g+If(~)la~w(~)

(26) for any

w E F(Oan) , where

Applying

~(t)

to both

}(l°g+If(w)l)

~w

sides

is the h a r m o n i c and u s i n g

< [I = F(a

+ I

)

Assume

now that

w E F(oa

).

Jensen's

F(o2a

n

measure

)

defined

inequality,

above.

we get

] ~(l°g+If(~)l)dSw(~)-

n

Using

(24), (25), (26), Lemma 3C and the -1 ~ m(E n) ~ d } ( - l o g ( n a n ) ) , and c o m p u t i n g as in the p r o o f n

fact

mn(F(an))

of T h e o r e m

4A, we see that the r i g h t - h a n d

is m a j o r i z e d

by

C%(-logan)

, where

side of the above

inequality

C : A'(o-2)A(o-l)u(~)/d.

Thus we

have (27)

%(log+If(w) I ) / ~ ( - l o g a n ) ~ C

for any

w E F(oan).

is p o s i t i v e ~ ( t 2 ) / % ( t I) N => i

Take

a positive

and n o n d e c r e a s i n g for any

satisfying

tI > tO

for

and any

-loga N > t O .

Then

If(w)l for any

w E F(aa n)

and any

number

t _> t O .

< an

n ~ N.

tO

t 2 > 0. (27)

so large

Then we have

that

~(t)/t

t2/t I < I +

We c h o o s e

an i n t e g e r

implies

-C-I Letting

and w r i t i n g

n ~ N

r =

Oan, we have i ICkl Letting

=

f(z)zk-ldz

in

z -I

small

ek : 0

Let

p ~ i, then we w o u l d

ficiently

z, say

have

¢(log+If(z)l)dm(z)

This

contradiction

constant 6B. expressed

[ n:N'

In terms

k > C + i

k-C-i an

and t h e r e f o r e

be its h i g h e s t n o n z e r o P If(z)l => 2 -1 ICpl Izl -P ~ i

~

with

[ n=N'

N' ~ N.

f(z)

coefficient. for all

suf-

Consequently,

~(log(2 - l l c p l a n - P ) ) m ( E n )

~(log(2-11Cplan-l))/%(-log(na

shows

functions.

for

ok =

c

Izl ~ aN,

E d

< rka -C-I = n

F(r)

n ÷ ~, we see that

is a p o l y n o m i a l If

I

~

)) n

that

p = 0.

Hence

H ~ ( [ \ E)

contains

only

[] of null

as follows:

classes

0~

the p r e c e d i n g

theorem

can be

259

Theorem.

Let

O~

Then

}

and

strictly

We set

~

be c o n v e x

includes

functions

satisfying

(AI) above.

0~.

0p = U{0q : 0 < q < p}

we have the f o l l o w i n g ,

where

the

and 0 + : n{0q: p < q < ~}. Then P i n e q u a l i t y sign < m e a n s the s t r i c t

inclusion. Corollary. (b) Proof. one:

(a)

0- < 0 < 0 + for P P P 0AB , < N{0q: 0 < q < ~},

$(t)/~(t)

= o(i),

(a)

(AI)

t ÷ ~, if e i t h e r

Let ~(t)

0 < p < ~. = e pt - 1

is e q u i v a l e n t

~

or

$

Indeed,

In o r d e r

satisfies

eqt/~(t) Hence

~

satisfy

0~~ < 0p, we h a v e o n l y

0 ~ t ~ 2/p,

ePt/t,

t ~ 2/p. ~

is a c o n v e x

So, by the t h e o r e m ,

= o(i),

function

~(t)

t ÷ ~, and t h e r e f o r e

= te pt

and

(AI) and t h e r e f o r e t + ~, for any

~(t)

0q < 0~

(b)

0~ < 0 .

To s h o w t h e

~(t)/~(t)

Hence

(AI),

so that

In fact,

@

and

0~ < 05.

As

U{0q:

We r e m a r k that, their

following without

proof:

the c o n d i t i o n

and

(AI),

~

= t 2

and

0AB , < 0~

¢

by the t h e o r e m .

t ÷ ~, for any

%(t)

q > 0, so that

= exp(e 2t) - e

are seen to s a t i s f y = o(i),

if two n u l l c l a s s e s

exists

H~(D)

~(t)

=

0q
p, so that

0
0 with

(6)

[el m I t l / 2

and therefore

sin 2 8 ~____t_t~ Bt 2.

So, for

6 £ t ~ 7/2,

l-sin

t~

P(r,8 - t)J £

4(1 - r ) t < Const. t-26a(6) I/4 sin3((t - 8)/2)

and thus IIlJ £

i712 ~6

[-sint

~

F(t) P(r,e -t) I s--i-~- F'(0) dt

£ Const.6¢(6)l/4IT/2 ;6 Finally, let 7/2 £ t £ ~. in view of (6),

]~-t

By (5),

t-2dt : Const.a(6) I/4 i - r £ 60a(60 )I/4 £ 1/2

and so,

P ( r , 8 - t)[ < 4 ( 1 - r ) [ s i n ( ( t - e)/2) I = i- r = 16sin4((t - e)/2) Isin3((t - e)/2) I < C o n s t . ( l - r) < Const.a(6) 1/4

265

This i m p l i e s

that

1~21 ~

I ~/2

+tP(r,0-t)llF(t)-F'(0)sintl

< Const.s(@)i/4(maxt

Combining

these

estimates,

IF(t)I + IF'(0)I)

z

so

u(re i8) ÷ 2wF'(0),

tends

A.I.3.

to

i

through

i14

e I £ C o n s t . s ( 6 ) I/4

S(I;~),

as desired.

Corollary.

= Const.s(~)

we get

lu(re ie) - 2 ~ F ' ( 0 ) r c o s If

dt

U n d e r the

then

6 + 0

and thus

E(6) ÷ 0;

[] same a s s u m p t i o n

as in T h e o r e m A.I.2,

let u*(e i0) which

exists

a.e.

on

T

lim r÷l-0

u(rei0),

as a s u m m a b l e

f

(7)

=

P(r,e

function.

Then

.

t)u*(elt)da(t),

-

T

for

z = re ie E ~ ,

Proof.

Since

is the q u a s i b o u n d e d

u * ( e ie)

fines a h a r m o n i c

= 2~F'(e)

function

seen to be q u a s i b o u n d e d .

on

implies

that

Pc

~,

of

we m a y a s s u m e u*(eit)da(t)

u- v that

where ~s

~s = ~ - ~c

positive monic

function,

n.

Set

F'(e)

is d e n o t e d v

is s u m m a b l e , by

v

(7) de-

is e a s i l y part of

function.

For

Since T h e o r e m A . I . I

is the a b s o l u t e l y

continuous

part

P(r,0- t)dPs(eit),

part of u- v

to show that

p.

Since

B

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

is a n o n n e g a t i v e ( u - v) A n

v

= 0

harmonic

for any

= ( u - v) A n . Since v is a b o u n d e d h a r n n the first p a r a g r a p h of the p r o o f of T h e o r e m 2F, Ch. IV

shows that

(8)

v.

is the q u a s i b o u n d e d

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

and t h e r e f o r e

So it is s u f f i c i e n t integer

= I

is the s i n g u l a r

is a l s o n o n n e g a t i v e

function.

u(z).

is an inner h a r m o n i c p

= F'(t)dt

( u - v ) ( r e ie)

and

which

To show that

u, it is e n o u g h to see that this p u r p o s e

a.e.

~,

part of

Vn(rei0)

= I

P(r,e - t ) v ~ ( e i t ) d a ( t )

266

for

re

i8

v* denotes a bounded measurable f u n c t i o n on ~. n to t h e e x p r e s s i o n (8), we f i n d t h a t v* is the n radial boundary f u n c t i o n for vn • Since 0 =< v n =< u - v on D, 0 = < v * ( e it) < u * ( e it) - v * ( e it) = 0 a.e. a n d t h u s v E 0 by (8) as w a s n = n to be p r o v e d . []

Applying

E ~,

where

Theorem

A.I.2

A.I.4.

Let

ekeik8

on

[k~0

nomials

be

the

set

is t h u s following

Then

we

Corollary.

For

any

f E L (d~)

P(T) + P(~)

such

the

Proof.

that

We m a y

for

see

suppose

(9)

have

Ch.

re i8 E ~ . to

ascending IIhnll~ ~

By t h e

f(e i8)

a.e.

sequence IIfiJ

and

there IlfIl

IV,

that

u(rei8)

tends

the

IJfnll~ ~

definition,

of a n a l y t i c

P(T) + P - - ~

~.

(for

on

P(T) T.

Fatou as

2A),

trigonometric set

exists

and

of

f

{fn }

a.e. can

n

If

be

poly-

in

f e H~(d~)

taken

from

P(T).

Set

P(r,8 - t)f(eit)do(t)

theorem

A.2.1

and

Theorem

r + i - 0.

Let

0 < rI < r2
0.

= r JT

the

i

and Using

=

A.I.I~ -.-

u ( r e ie) < i

(e ie) = u ( r eie), n n expansion

the

[

reJee l](e-t),

[ j_~

c . r i J IeiJ 8 jn

be an

so t h a t

we h a v e

h n ( e i8 ) =.

(10)

with

• = c] ST for

j = 0,

formly

on

-+i, . . . . T,

one

Since

can

find

f

sequence have

c.

the

(eiS)

{fn } : 0

conditions thus for

ilhn -

right-hand

and

N[ j:-N

fnil the

)

•.

integer

_ n-• n

fulfills

j < 0

.

the an

n

satisfies

f(elt)e-l]tdo(t

side

> 0

(i0)

converges

such

uni-

that

.. c . r l J i e l] ~ n

< 2Hfl[ /n

requirement.

therefore

of

N : N(n)

f

n

and When

E P(T).

]IfnH~