Hardy Classes on Riemann Surfaces (Lecture Notes in Mathematics 98)

Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, Z~irich Series: Forschungsinstitut for Mathematik, ETH, ZUrich 9Adviser: K. Chandrasekharan

98 Maurice Heins University of Illinois, Urbana, Illinois

1969

Hardy Classes on Riemann Surfaces

Springer-Verlag Berlin. Heidelberg. New York

All rights reserved. N o part of this b o o k may be translated or reproduced in any form without written permission from Springer Verlag. 9 by Springer-Verlag Berlin - Heidelberg 1969 Library of Congress Catalog Card N u m b e r 75- 84833 - Printed in Germany. Title No. 3704

Contents

Chapter I

General Observations

and P r e l i m i n a r i e s ..............................

2

Chapter II

The T h e o r e m of Szeg8 - S o l o m e n t s e v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

Chapter III

A C l a s s i f i c a t i o n P r o b l e m for R i e m a n n Surfaces .......................

34

Chapter IV

B o u n d a r y Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

Chapter u

V e c t o r - V a l u e d Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

Bibliography

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

-

2

-

Chapter

General

1. Some r e m a r k s originally analytic H

~)

concerning

introduced

Observations

and P r e l i m i n a r i e s

the t h e o r y of H a r d y classes.

in the

f o l l o w i n g manner.

on the o p e n u n i t d i s k

provided

I

~ = {Izl

Let

< 1}

The n o t i o n

o < p

< +

~

is said to b e l o n g

of a H a r d y class w a s

.

A function

f

to the H a r d y c l a s s

that

P

~o

27f(re i8) IPd8 = 0(1) ,

By d e f i n i t i o n

the H a r d y class

bounded.

The s t u d y of

p th

classical

p a p e r o f 1915

[13].

very extensive recently, afford

treatment

the s e t t i n g

interesting

a topic

of c u r r e n t

reference classes

is m a d e

on R i e m a n n

important

mental

lively

that this

class

surfaces.

of B a n a c h

interest.

on R i e m a n

spaces

of H o f f m a n

~

are

1916

questions

and s u b s e q u e n t l y b y F. Riesz

[21].

whose

The

first

in his t h e s i s notes

the a p p e a r a n c e

treat

in H a r d y ' s

has r e c e i v e d

r i c h and

investigation

systematic [26],

selected

remains

of H a r d y ' s

Congress

paper

in w h i c h m a n y topics

from the

to me r e c e n t l y .

cited above

paper

of f u n c t i o n s

p

of F. and M . R i e s z which

belonging

[36] g i v e n b y h i m positive

funda-

We are e a s i l y p e r s u a d e d

of M a t h e m a t i c i a n s

properties

for u n r e s t r i c t e d

s t u d y of H a r d y

s t u d y of the subject.

(1) the c e l e b r a t e d

t h e o r e m of G. S z e g 5 [31]

and

s t u d y from this p o i n t of v i e w p a r t i c u l a r

or e x h a u s t i v e

the b o u n d a r y

~

initiated

s u r f a c e s w h i c h h a v e b e e n of i n t e r e s t

Scandinavian

(2) the m a x i m a l

was

on

structurally

in the t h e o r y of H a r d y c l a s s e s w e r e o b t a i n e d .

at the

analytic

o f the o p e n u n i t d i s k and, m o r e

(1 4 p 4 + ~)

The p r e s e n t

of a s y s t e m a t i c

following

on

Hardy classes

surfaces was given by Parreau

the d e c a d e

to o t h e r

HI(A),

analytic

setting

For their

introduced.

of f u n c t i o n s

time the s u b j e c t of H a r d y c l a s s e s

is the case w h e n we call to m i n d

[31] p r e s e n t e d addition

of f u n c t i o n s

Since that

to the m o n o g r a p h

is no q u e s t i o n

results

is the class

in b o t h the c l a s s i c a l

examples

notions were

During

(~)

means

of R i e m a n n

t h e o r y of H a r d y c l a s s e s There

H

(1.1)

o 4 r < 1.

and

treats

in

to the H a r d y

for the case

p = 2

(3) M . R i e s z t s

theorem

-

on the c o n j u g a t e

series

of the F o u r i e r

I < p < + ~,

which

classes

[It is to be o b s e r v e d

[32].

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

admits

paper

the t h e o r y of T o e p l i t z

It was subharmonic

in this

(1.1) with

surfaces,

appeared.

fact to serve

the m e d i a t i o n solution

theoretic

Thus

enter

and

to

of the paper

The paper

of Szeg6

of Szeg6 appeals

function

to

admits

on H a r d y c l a s s e s

in F. Riesz's identity

of

for power

powers

of a f u n c t i o n

power

as we

see

We shall u s e this on R i e m a n n (1.1)

subregions

through

and the

[26].

w h i c h we cited.

of S z e g ~ ' s series,

majorant

definition

b y the Riesz b r o t h e r s

treatment

p th

The c o n d i t i o n

of H a r d y c l a s s e s

of "reasonable"

cf. p.35

the

functions.

the c l a s s i c a l

in terms

questions

a harmonic

of s u b h a r m o n i c

problems,

Indeed,

on the t h e o r y of

is subharmonic.

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

results

of a n a l y t i c

[31] of F. Riesz

is apposite.

Ifl P

properties

introduced

these

the Parseval

and the e x i s t e n c e

that

as a basis

to the three given

p there

papers

The r e f e r e n c e

Dirichlet

that the t r e a t m e n t aspect.

fundamental

of an a n a l y t i c

values

of a s s o c i a t e d

We r e t u r n

L [O,2E], P in the t h e o r y of H a r d y

the m e t h o d s

different.

be sure - one m a y p a r a p h r a s e

of m e a n

belonging

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

that

are quite

to the c o n d i t i o n

though-to

of a f u n c t i o n

forms.]

the a i d of v e r y e l e m e n t a r y

motivating

series

in p a s s i n g

of F. Riesz

of the m o d u l u s

is e q u i v a l e n t

-

interpretation

era that the

functions

(0 < p < + ~)

direct

3

has a p r o n o u n c e d

maximal

the n o t i o n

analytic

It is to be noted

principle

function-

for general

of a B l a s c h k e

product,

on a s i m p l y - c o n n e c t e d

region

free zeros.

Part of the a r g u m e n t Cauchy

theory.

given b y M. Riesz

Subsequently,

P. Stein

[35], w h i c h

checked)

of an a u x i l i a r y

because useful

we shall instrument

subject

counterparts

an e l e g a n t

on e x a m i n a t i o n function

of i n v e s t i g a t i o n appear

to c o n d i t i o n s

proof

introduced

as simple

theorem

theorem

of s u b h a r m o n i c also the basic

corollaries

of a t h e o r e m

in the t h e o r y of H a r d y c l a s s e s

more

was

general

are subject.

than The

(easily these

functions results

of the paper

to w h i c h

subsuming

facts

be a v e r y

concerning

those

on the

given b y

We m e n t i o n

study but

considerably

is b a s e d

the s u b h a r m o n i c i t y

into the argument.

the t h e o r y

in our

series

of the M. Riesz

is seen to e x p l o i t

see that not o n l y w i l l

of F. and M. Riesz w i l l functions

for his c o n j u g a t e

subharmonic their

theorem,

given

-

by

Solomentsev

shall tely was

see

that

termed given

[34]

by me

subsumes

"Theorem in

been

paper

sense

results,

definitions

theory,

with

such

e.g.

terms

perharmonic,

as

that

on a R i e m a n n

1

at

of

Q

a .

.

I_~f u

disk has

k

mann

surface

is,

in

fact,

II. We appropria-

for R i e m a n n

,

S

Harnack

lower

with

that

A

convergence

theorem

is a c o n s e q u e n c e

It is d e s i r a b l e

of

to r e c a l l

shall

surfaces

harmonic

for

on

of

function

the

surfaces

definitions

reader

in and

on R i e m a n n

harmonic,

Q

let

normalized

,

and

p

,

classical

sur-

is f a m i l i a r

subharmonic,

harmonic

su-

the

func-

harmonic

a ~ S

,

to t a k e

Harnack

and

funclet

the value

the u p p e r

and continuous

that we have

envelope

inequality

and that

following

qualita-

:

S

on

4 u 4 u(a)N

for m o n o t o n e

that

S

positive

S

Riemann

texts

surface,

t h e a i d o f the c i t e d

holding

with

for n o n - n e g a t i v e

be a R i e m a n n

is s t r i c t l y

[7].

for n o n - n e g a t i v e

counterpart

envelope

of v i e w a r e the

theory.

inequality

functions

point

standard

assume

surface

o f the R i e s z b o t h e r s

elementary

meromorphic,

It is n o w o b v i o u s

inequality

is a n o n - n e q a t i v e

S

harmonic

the

and continuous.

the H a r n a c k

we

analytic,

let

Doob

the u s u a l

to o n e of the

of R i e m a n n

Indeed,

the r e s u l t s

o f J.L.

for g r a n t e d

u(a)A

The Harnack

in C h a p t e r

version

shall be concerned

a qualitative

concluded of

We

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

the c l a s s i c a l

We i n t r o d u c e

It is e a s i l y

form of

and

The general

treatmen~of

is r e f e r r e d

f a m i l y of p o s i t i v e

finite-valued

tive

[27].

surface.

a n d the c o n n e c t e d n e s s is

treated

principle

a n d the p a p e r

take

in the c o n t e x t

tions

the

be

f r o m the p o t e n t i a l - t h e o r e t i c

local uniformizer,

o n the o p e n u n i t

denote

[IO]

shall

the r e a d e r

tions

Q

and

[1],

taken

We r e c a l l

interest

and preliminaries.

of Weyl-Rad6

face

that numerous

and HSrmander

for w h i c h

Szeg~ maximal

will

[18].

Of s p e c i a l

of G~rding

2. B a s i c the

given.

the

of S z e g 6 - S o l o m e n t s e v " .

It s h o u l d b e r e m a r k e d have

-

for t h e c a s e o f t h e u n i t b a l l ,

it a l s o

the

4

,

then

(2.1)

.

sequences

of harmonic

functions

on a Rie-

(2.1).

the n o t i o n

of a Perron

family

and

its u t i l i t y

in the

-

study

of harmonic

majorants

of

subharmonic of

may,

that

the uniformizers

a

open

unit

disk

domain.

~

stant

- ~)

(~,r)

- Pq~sson

dition be

. Let

that

given

as ~

their be

such

v

=

center be that u

and

6 ,

the

A More

with

convenience

conformal

u

and

the ,

on

let

S

for

suppose,

all

have

(possibly

O ~r

function

while

we

structure

subharmonic

is m e a n t

family

~(9) If a

harmonic upper

conclusion

9

by

of

41

~

defined points

a function 9

of

the

functions

containing generated contains

function

h

by

~

v

envelope

of

there

of

on ~(9)

S

that

on

.

S

,

and

on

as w e

the

the

Then

of

S

con-

by

the

by

the

con-

~ [~(O~r)]

it

the

open

circular v

is

(2)

given

an

allowed

of

u

other

is o n e

A

in

C

family

of

Let

provided

~

fact

~

and

for

convenient

fundamental

and

with

uniformizer

a member

equally

disk,

subharmonic.

a Perron

is a l s o

unit

disk

#

that

each

defini-

concerning

theorem:

of

the

followinq:

the

constant

- ~,

S on

S

Perron

this

for

term

trichotomy

on

< I

We

available.

family

all

It is

generates

families

minimal

a Perron

containing

family

9

family is

~(9)-

itself

that we mean

by

the

- ~

addition,

a

Perron

.

a member harmonic

functions

that

9.

standard

subharmonic

intersection

kernel

It is

are

harmonic

Izl

open

following

a Perron

,

the

observed

family

the

denote

modification

is to b e

a Perron

Poisson

shall

u,v ~ #

It

envelope

we

the

subharmonic

~ I of

,

a(a~Q).]

- Poisson

is g i v e n

family

family

exists

be

For

u[a(reiS) ]k(e,z)d~

whenever

( r

notion

precisely,

by

(~,r)

+ ~,

Perron

the

O

upper

constant

Q

~ #

the

families

The the

its

uniformizer

u

=

functions

u,v

then

for

Perron

u

it a g r e e

[In g e n e r a l ,

of

satisfying

tions

Let

Re[(eiS+z)/(eiS-z)]

radius

family

(I) m a x

E

r

is

~z~ ( r a

a

functions.

by

k(8,z)

~(O~r)

-

defining

of

S - a[~(O~r) ]

v[~(rz)]

Here

S

an allowed

modification

in

5

other on

S

than

constant

satisfying

satisfying is m a j o r i z e d

is a l e a s t

the

harmonic

u ~ h,

u

~ 9,

v ~ h

is a P e r r o n

by

and

h

function

on

and,

S

then

family

is h a r m o n i c .

in

the

set of

containing We

are

there

led

sub9

to

and the

-

which term

majorizes it the

instead this

each member

least

of the

least

paragraph

may

correspondingly nic

functions

~)

For

ant b y

mv

S

,

sk

has

harmonic

such a

of

dually

~ of

for

having

fact the u p p e r

. u

When

~

has

which

we

denote

families

of the q r e a t e s t

as t h a t

v

-

It is in

majorant

be r e c a s t

as w e l l

~

ma~orant

the n o t i o n

of a given a harmonic

envelope

minorant

superharmonic minorant

#(~)

a sole member by

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

harmonic

of

,

The

we

speak

results

functions.

We o b t a i n

v

(not the c o n s t a n t

its g r e a t e s t

harmonic

now

neither

that

being

a harmonic

first part

Ms 2 ~ h + s I

h

is h a r m o n i c

the c o n s t a n t

majorant,

- ~

so d o e s

of the a s s e r t i o n and hence

Ms 2 ~ h + Ms I

The

The

result

lemma

concerning

Lemma

1 :

there

exists

satisfvina and

I_~f h

on

S

,

and

and

the o t h e r

(I)

Proof:

M(h+)

and

ql

differences

that

that

sI

and

s2

s2 = h + sI

are

Then

subharmonic

if o n e o f the

and

'

which

were

that

M(-h-)

'

Pk $

h+

h = M ( h +)

We n o w r e c a l l tions

then

~

some

q2

-

,

admits

where

'

we observe

that

and hence

harmonic

Pl

and and

following

classical

functions.

harmonic P2

q2

to the

fun c t i o n ~

on

are n o n - n e q a t i v e are n o n F n e q a t i v e

S

,

harmonic

harmonic

on

then on

S S

k = 1,2

and

-h-

M(-h-) The

(2.2)

s 2 ~ h + Ms I

application

of non-neqative

(2) i_~f ql qk

To e s t a b l i s h

follows.

have Since

lemma

fundamental

introduced

Similarly

of n o n - n e g a t i v e

(pl,P2)

(2.2)

to v e r i f y .

just proved

is the d i f f e r e n c e

We n o t e

that

(2.2)

that we have

h = Pl - P2

we conclude

is r o u t i n e

Ms 2 ~ h + Ms I

equality

a unique

h = ql - q2

~

+

minor-

Ms 2 = h + Ms I

The

of

o f a f a m i l y of s u p e r h a r m o -

function

we d e n o t e

Mu

u

We

.

Suppose on

harmonic

of

6

by Parreau

harmonic

majorants.

From

h + = h +(-h-)

h + ~ ql

and

q2

we

-h- ~

'

see t h a t

follows.

concepts in his

concerning thesis

[26]

non-negative Let

h

harmonic

func-

be a n o n - n e g a t i v e

,

-

harmonic

function

decreasing h

.

on

sequence

S

definition

sequence

be n o n - n e g a t i v e

negative

bounded

harmonic

-

guasi-bounded

bounded

is o b t a i n e d

is dropped.) function

that each n o n - n e g a t i v e

tion of the

h

of n o n - n e g a t i v e

(An e q u i v a l e n t

showed

We term

7

harmonic when

We term

on

S

harmonic

provided functions

on

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

singular

majorized

function

that there

h

S

exists

which

a non-

has

limit

that the m e m b e r s

provided

that

of the

the o n l y non-

by

h

is the c o n s t a n t

zero.

on

S

admits

representa-

a unique

Parreau

form

q + s

where

q

is q u a s i - b o u n d e d

and

s

(2.3)

is singular.

This

r e s u l t m a y be e s t a b l i s h e d

very

simply.

We first

show uniqueness.

To that end suppose

that

ql + sl = q2 + s2 where

the

qk

are q u a s i - b o u n d e d

non-decreasing

sequence

and the

of n o n - n e g a t i v e

sk

'

are singular,

bounded

harmonic

k = 1,2

functions

.

on

Let S

(b n)

which

be a

has

li-

+ mit

ql

"

Then

M[ (bn-q2) +] b n $ q2

"

ql = q2

"

(bn - q2 )

is bounded, On taking

n

the g r e a t e s t

decreasing by

h

.

harmonic

letting

the term on the

non-negative,

and is m a j o r i z e d ql ~ q2

is e a s i l y

minorant,

bn

,

treated. of

and has as limit a q u a s i - b o u n d e d

function

that on

S

h-q

is the c o n s t a n t

we c o n c l u d e zero.

tation of the form

is singular.

majorized

m ~ sup b(S)

n ~ ~

by

"

subharmonic.

s2 ,

Since

we c o n c l u d e

By s y m m e t r y

for

by

h-q

We i n t r o d u c e

min{h,n} harmonic

Suppose

we c o n c l u d e

that

b ~ 0

h

that

h-q

b

is established.

m

Hence

and c o n s e q u e n t l y , is singular.

q

,

(b n)

that that

which

is a w h o l e

is m a j o r i z e d

,

being

The e x i s t e n c e

bounded

number

bn + b $ b n + m b

number is non-

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

that

.

for each w h o l e

sequence

function,

and suppose

bn + b ~ m i n { h , m + n } ,

The

that

Then

We c o n c l u d e

(2.3)

left b e i n g

follows.

This q u e s t i o n

harmonic

We a s s e r t

satisfying

s2,b n,

the limit we see that

The u n i q u e n e s s

Existence.

~

On

non-negative, of a r e p r e s e n -

-

We t e r m sinqular

q

of

component

Sums show that

and

a convergent X qk 0

for

the c a s e sk

is

singular

singular

t i o n on

suppose

which

of

qk

,

of

h

and

non-negative

s

non-neqative

statement

harmonic harmonic

holds

with

of

(2.3)

non-negative

non-negative the

harmonic

above

qk

To t r e a t

functions

the

we i n t r o d u c e

b

S

S

,

We n o w

on

S

is

replacing

function

qk

tl~

on

functions

harmonic

Z qk is c o n v e r g e n t . S i n c e e a c h 0 non-negative harmonic functions on quasi-bounded.

functions.

"sinqular"

that

consequently

replacing

S

singular)

be a q u a s i - b o u n d e d

sum of b o u n d e d

'

(resp.

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

Let

component

.

sum o f q u a s i - b o u n d e d

and that

....

0,1

h

a conyerqent

"qu.a s i - b o u n d e d " . =

of

-

the q u a s i - b o u n d e d

of q u a s i - b o u n d e d

quasi-bQunded

k

(2.3)

8

on

S

,

is r e p r e s e n t a b l e ,

the

same

corresponding

we proceed

a non-negative

as

as

is t r u e result

for

follows.

With

harmonic

func-

satisfying QO

b~4

and observe

Z sk 0

,

that GO

(b-Z s k) + ,4 m i n ~ b , s o } , 1 whence

we conclude

is zero.

that

the

least harmonic

Proceeding

inductively

replacing

I

that corresponding non-negative

h $ H .

we

,

see t h a t

Consequently

results

harmonic

Suppose fying

o f the

left

side of this

inequality

Hence

b$

number,

majorant

inequality

b = 0

as w e l l

.

holds

and hence

for

finite

with

Z sk 0

n is

,

an a r b i t r a r y

singular.

sums o f q u a s i - b o u n d e d

It

is

(resp.

whole obvious

singular)

functions.

now that Then

hold

this

Z sk 1

h

h

and

H

are non-negative

is q u a s i - b o u n d e d

(resp.

harmonic

singular)

functions

when

H

is

on

S

satis-

. It s u f f i c e s

to

-

consider

the canonical

The

Hardy

of bounded stand has

the

classes

analytic

functions

majorant.

point

I ~

.

< + ~

on

f

is t a k e n

question

vantage

the q u e s t i o n of the

Lemma

2: L e t

1~

< + ~

p

.

and

that

on

< p

< + ~

(S)

,

~f~

Hp(S)

in h i s

However,

as w e

reference

HD(S)

Banach

standard

A more

with

is j u s t

subharmonic

in the .

(S)

by

is a c o m p l e x

introduced sup

the

H

thesis shall

function

to e x h a u s t i o n s

pointwise

space

see, of

Ifl P

space when

in t e r m s

now

set

we u n d e r -

interesting

a Banach

the

manner

question

from

structure

when

of m e a n - v a l u e s

is it p o s s i b l e

S .

We t a k e

to

ad-

lemma.

be non-neqative

It s u f f i c e s

be the

the p r o b l e m

on a r e g i o n where

to c o n s i d e r

sole case

superharmoni 9 functions

S

on

and

let

(u I/p + v I/p) p

of

C

either

only

considered

factor

by differentiation

the c a s e w h e r e

in the a p p l i c a t i o n

by differential is z e r o or wl/P

we obtain

H .

S

(which w i l l

cases

H

and

Then

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

trivial

H-h

By d e f i n i t i o n

by Parreau

inequality. without

.

,

for w h i c h

to be

w =

Proofz

S

are

h

p, 0

of e n d o w i n g

elementary

v

on

is t r e a t e d

internally

following

u

Given

by a scalar

the a i d of the M i n k o w s k i

approach

S .

is t h a t

for

1 4 p 4 + ~

analytic

f

of v i e w

This

,

-

(2.3)

It is r o u t i n e

of a m e m b e r

a technical

with

(S)

p

and multiplication

a n d the n o r m

p

H

set of f u n c t i o n s

a harmonic

addition

decompositions

9

w(1/p)-I

to

= u (1/p)-I

w

p = 1

differentiation

(~-

(with r e s p e c t

v

are both

o f the Lemma) Further

Starting

harmonic

a n d to t r e a t

we put

aside

the

with

,

z) + v(1/p)-I

u

Z

and by a second

and

considerations.

= u I/p + v I/p

(with r e s p e c t

u

Z

to

v

, Z

~)

I ) W ( 1 / p ) - 2 1 W z ~2 + w l / P - l W z z

(2.4)

-

10

-

= (!p _ I) [u(I/P)-21Uzl2 + v(i/p)_21Vz121

Writing

(2.5)

(2.4) as 1

1 -1

w 2 P ( w 2p

1

1 -1

1 -1

+ v2P(v 2p

Z

UZ)

and applying the C a u c h y - S c h w a r z - B u n i a k o w s k y 1_2

wp

1

w ) = u2P(u 2p

I

--

lwzl2 .~ up

v ) Z

inequality, we obtain 1

-2

--

-2

lUzl 2 + ~

Iv~.l2

,

and applying this inequality to (2.5) we see that

Wzz .~ 0 .

Hence

w

is superharmonic.

Given longs to

f ~ Hp(S)

H (S) , P

,

The lemma follows.

I ~4 p < + ~

,

we define

hf

as

M(If} p)

If

g

also be-

then

h~/p + hl/P ~. Ifl + I gl ~, i f + gl g From

(h~/p + h gI/p) p

,If+gl

and the superharmonicity of the left side we conclude that

h~/p

+ h I/pg ~ hf+gl/P

It is immediate that

f + g ~ H (S) P

and that

(2.6)

H (S) , 0 < p < + ~, is a vector space over C when the standard P pointwise definition of addition and m u l t i p l i c a t i o n by scalars is used as we see with the aid of the inequality (a + b) p ,4 2 p(a p + b p) ,

-

a

and

b

which

being

yields

following

non-negative

a triangle

real

-

numbers.

inequality.

B u t we h a v e

Given

q ~ S

obtained

as the

much

(q-)norm

of

more f

in

(2.6)

we p r o p o s e

Parreau

Jlf[I

As n o t e d ,

(2.6)

(i) ilfJl = O verify.

11

yields

if a n d o n l y

Thanks

reference

the

triangle

if

q

yields

inequality.

f = O

and

to the q u a l i t a t i v e

point

= [hf(q) ]i/p.

(ii)

(2.7)

The

flcflt =

remaining

Iciliffl, c ~ C,

Harnack inequalities

an e q u i v a l e n t

norm

and

norm

( 2 . ~ we

that

H

conditions, are

routine

see t h a t

(S)

i.e. to

changing

is a B a n a c h

space

of the in the

P sense

of each

There

norm when

remains

this

to be

is the c a s e

shown

that

for

H

some norm.

(S)

is c o m p l e t e

in the

sense

o f the n o r m

(2.7).

P Suppose

that

(fn)

is a C a u c h y

sequence

in the

sense

of this

norm.

Let

urn, n = M[ (fm - fn )p]"

From

Ifm

the

fact

that

inequality S.

of

in fact,

let

~

m,n

is s m a l l

m,n

(2.1), w e

We p r o c e e d

and,

u

u

to s h o w is the

~

for

sequence

of whole

harmonic

function

m,n

that

f

number ~ v.

numbers, v

n

q

see t h a t

on

and

(fn)

m

in the

let

~, as

we conclude

that

f - f

v

such m ~ ~.

~ H

(S) p

'

and

n

be

of

of

that

n

large,

(fn) ,

whole

the r i g h t compact

is a m e m b e r

topology. number

~ u

we m a y

(u (m),n)

tends

see at o n c e

and

on e a c h

the n o r m

a positive

for e a c h

We

are

Cauchy

limit

sense

if - fnlP ,4 v n,

whence

,n

is u n i f o r m l y

see t h a t

say S

when

(fn)

We

f n Ip ~ U m

the p o i n t w i s e

'

limit of

be a p o s i t i v e

(q) ~

at

-

To

such

select

that

H p (S)

t h a t end,

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

(S). P

of

of

an i n c r e a s i n g

that

f ~ H

subset

that

n )~ ~,

and hence

Harnack

Further

-

llf-

for

n

~ v.

Given

the a r b i t r a r i n e s s

12

-

f n IIp ~4 Vn(q)

of

~,

4 n,

the c o m p l e t e n e s s

of

H

(S) P

follow.

is s e e n

to

-

13

-

Chapter

The Theorem

I. T h e de

la V a l l ~ e

Poussin

decreasing

function

~

values

satisfies

the

real

and

line

R

the inverse ~(- ~ ) } . max

~ o ~

{~,x},- ~ ~ x

restriction

Poussin

of

as

for

and were

~ ~

to

la V a l l 4 e

we cite

the w o r k

A proof

of Theorem

here,

which

Theorem exists

i: L e t

Poussin

u

families

fying:

(I)

constant

Given We n o t e s ~ u l ~ A,

zero.

be s h o w n

surface. that

that

of

that

( A

the lim

family of SUpqS ~

the o b v i o u s The upper

the u p p e r

envelope

has

[38]

of

a harmonic

u

FA =

of the

integrability

[II].

For

o n the w o r k

the u s e functions

of Doob.

to be g i v e n

we

taking {u(q)

is a P e r r o n for

on

S.

Then

there

majorant.

start by

real

= A} on

and ~k

the c a s e

family majorizes at e a c h p o i n t

~A =

and

( / ~)

(u-A) l~ A. FA

greater

satisthe

of a R i e m a n n

subset

of

intoduc-

values

family containing

for t h e c a s e o f an o p e n

limit zero

the

la V a l l 6 e

The proof

subharmonic

family given

this

b y de

=

=

and

subharmonic

is b o u n d e d

introduce s

: ~(x)

Functions

function

on a parameter

functions

{x

~,

arguments.

harmonic

case where

to the

~ o ~(x)

cf.

paper.

~

to i n t r o d u c e

for u n i f o r m

and

real

~ > - ~

introduced

depending

cited

"internal"

O, q ~ FA,

changes

envelope

paper

we

and

~ = - ~.

also by Nagumo.

oossesses

of a Perron

when were

of

~ = max ~

s t u d y of h a r m o n i c

depending

( + ~,

of

condition

non-neqative

trivial

functions

inf u

(2)

is c o n c a v e

by

where

non-

non-negative

on its d o m a i n w h e n

in Y a m a s h i t a ' s

~ o u

the

< + ~},

sufficient

proceeds

takes

a continuous

be convenient

on the d o m a i n

in the

with

(a) the r e s t r i c t i o n

requirement)

and

condition

aside

x

for t h i s p u r p o s e

The definition

is to b e u s e d w i t h Riemann

A,

map

( + ~}

is g i v e n

such

ing a u x i l i a r y

{~ $

which

It w i l l

is c o n c a v e

be a quasi-bounded ~

< + ~}

= +~.

[6] a n d Y a m a s h i t a ' s

I below

Putting

.

to

~

a necessary

of Doob

an a l l o w e d

{u(q) > A}

~(x)

identity

( x

is v e r y d i f f e r e n t ,

i n f u.

@

Further

employed

Proof:

than

-I

( w i t h o u t the c o n v e x i t y

formulating

shall be concerned

two conditions:

of

{~(- ~)

We

- Solomentsev

{- ~ ~ x

lim+ x

is the

( + ~.

subsequently

o f the de

(b)

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

Clearly

same kind

domain

following

is c o n v e x ,

of Szeg6

condition.

with

II

.

surface

of a

It w i l l

now

To t h a t end, w e

-

consider

the interior,

ter" w h i c h

w,

is a J o r d a n

of a small

arc lying on

and of a " s e m i c i r c u m f e r e n c e "

where

m a y have a s t a t i o n a r y

u

the b o u n d e d "diameter" ference"

harmonic less

envelope

the upper

sA

envelope

We see that

s~ {Ms A}

boundedness

of

sA

of the

is also. This

the p r o p e r t y

We define ing

in q u e s t i o n

u.

with

stated

S - ~A

The

It w i l l be seen

that

t

and has

~.

q

of the

boundary

of the

"semicircumbehavior

of

restriction

to

~A

is just

S - QA

is the c o n s t a n t

zero.

{s A}

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

and

family limA,+

tA

Ms k = O,

part

in a m a n n e r

Thus we take

We take

We i n t r o d u c e

thanks

in c o n s t r u c t i n g

to the quasian a l l o w e d

I.

subharmonic

u

to

[The case

to follow.

whose

p l a y a fundamental

function

replacing

is seen

restriction

as a non-

at each p o i n t

The a s s e r t e d

S

FA

into regard.]

at each p o i n t

in q u e s t i o n

of

of a "diame-

for its endpoints.

continuously

slw.

consists

a given point

save

vanishes

each

and w h o s e

frontier

is to be taken

with domain

in T h e o r e m

family of functions

FA

~A

limit u(q)

family

and

in

s A ( u.

the a u x i l i a r y

is subharmonic

which

fact will

(2) lim SUpq t ~ O, q ~ F AtA

~

as the f u n c t i o n

is subharmonic

the f a m i l y

on

It m a j o r i z e s

of the P e r r o n

We define

lying

and has

whose

and c o n t a i n i n g

point

on

its e n d p o i n t s

n o t an endpoint.

the u p p e r

having

function

-

"semicircle" FA

endpoint,

14

on

tAl~ ~

similar

tAI(S

S - ~

- ~)

majorant.

The

in d e f i n -

as the u p p e r

and satisfying:

as the c o n s t a n t

as a h a r m o n i c

to that u s e d

zero.

envelope

(I) t ~ ul(S

- ~k) ,

We see that also

families

{tA} and

{Mt A}

are n o n - d e c r e a s i n g .

The

function

vA

(2) vAI(s

- ~k)

limit

at each p o i n t

k

is r e a d i l y The

as the f u n c t i o n

is the least p o s i t i v e

concluded

following

is d e f i n e d

of

with

FA.

harmonic

The e x i s t e n c e

satisfying:

function

with

(I) v A l ~ A domain

of such a h a r m o n i c

the aid of P e r r o n methods.

The

function

= ul~ A,

S - ~A

function vA

having

on

S - ~A

is superharmonic.

equality

u = th + vh

holds. point

It suffices q ~ S - ~,

to check

(i.i)

that the two sides

the c o n t r o l

for points

of

of ~A

(I.I)

take

the same value

being

a trivial

at each

consequence

of the

-

definitions u(q)

of

- tk(q)

es on

Hence

all

this

non-negative

(I.I) ,

hand,

hand,

we

see u s i n g

O ~ u(q)

~ tA(q).

function

having

the

that

limit

The

- vk(q)

asserted

property

4 u(q)

equality

(2) of

and

vA

u - vk

follows

and

that

vanish-

(I.I)

is

We

show

the

limit

compact

we conclude

thereupon,

that

positive

- ~)

of

open

u

is the

to i as number

N

such

that

the c o n s t a n t

limA,+

zero.

Ms A = O.

Suppose

i which

F k.

that

is m a j o r i z e d

F k,

and

of

~ u(q)

for the

argument. of

q The

(1.2).

that

allowed to

we

At

b

by

is a lim v A.

+ ~".

harmonic

in q u e s t i o n . inequality

The

(1.2)

that

precisely

restriction is t h e r e b y

Msk/u

there

limit

of a

S - ~A number

on

of

~ S - 6A

A

is n o w

established.

is s i n q u l a r ,

and

zero.

has ~

- ~A)

of

kb n (q) ~ v A ( q ) , q

lim v k

given

vl(S

that

on

S - ~A

a non-

of a f i n i t e

it is the c o n s t a n t

the q u o t i e n t

More

function

Since

on

property

is the

F A(n) ,

of the u n i o n

that

- ~k)

follows.

by

for the m o m e n t

bl(S

-(n)+F A " PA

exist

majorized

the m i n i m a l i t y

that

We c o n c l u d e

A

zero,

as

function

there would

suppose

non-negative

quasi-bounded,

tends

violate

established

harmonic

Otherwise

We n o t e

q

be

non-negative

complications on

may

the c o n s t a n t

point

for e a c h

u(q)

not

at e a c h

limit

that

by

(1.2)

This would

subarcs

it is a l s o

is a c t u a l l y

(1.2)

q E Fk.

least

as a c o n s e q u e n c e since

least

is the

n

b(q)

Ab(q)

We o b s e r v e "it t e n d s

inequality

unnecessary

b y an o b v i o u s

b = O

The

F A.

which

S,

v A ~ SA,

above

.4 u(q) ,

on

b

on

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

is q u a s i - b o u n d e d .

Ab(q)

continuously

where

u,

{v A}

we have

S - ~,

(b) , n

harmonic

bounded

at e a c h p o i n t

points

family

inequality

S

on

sequence

relatively

bl(S

To a v o i d

the

by

function

are no s t a t i o n a r y

has

on

b = O.

b(q)

harmonic

- ~k).

dropped

that

the

majorized

q ~ S - ~A

we n o t e

and vanishing

using

harmonic

First,

that

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

being

we conclude

Now

on the o n e

conclude

lim vk,

for p o i n t s

negative

we

which

whence

vl(S

- vA(q)

fact we c o n c l u d e ,

events,

Then

Now

On the o t h e r

u(q)

to

limk~+coVA

From

v A.

-

established.

Thanks that

and

% vA(q).

F~.

thereby

tA

15

< i,

the p r o p e r t y there

that

exists

a

-

16

-

M S A (q) > a

u(q) for

q

E ~

.

To

see

this,

we use

the

inequality

u - A 4 sA

and conclude

that

u(q)

for

q ~ ~

.

The

Since such that Using

asserted

limit

l i m A , + M s A = O,

ZM(SAk)

behavior

there

is c o n v e r g e n t .

the p r o p e r t y

of

Ms A

of

exists The

MsA/u

follows.

an i n c r e a s i n g

sequence

sum is, o f c o u r s e ,

established

in the p r e c e d i n g

(~),

positive

lim ~

harmonic

paragragh,

= + ~,

on

S.

we conclude

that

with

f(xl

= inf

{ZMs~(q)

: q ~ Fx},

inf u

( x

( + ~,

we have

lira x X~+Oo

We are now harmonic

in a p o s i t i o n

majorant.

-1

T o t h a t end, w e

inf u lies

its c o n v e x

( x

( + ~.

above

some

~(x)

i n f u. =

e(B),

line with

( ~.

The proof

a given

we define

Otherwise x

K.

It is i m m e d i a t e

e is n o n - d e c r e a s i n g , x 4

hull

e

has

In e i t h e r

of Theorem

= + ~.

to c o n s t r u c t

{(x,y)

and thereupon

f(x)

~

We next that

e

~(x)

~

I follows

e(x)

as

~,

that

inf

Because

inf u

so c o n s t r u c t e d

on n o t i n g

~ o u

has

a

(1.4)

and hence

= e(x),

s a y at

such that

y )/ f ( x ) }

define

slope

~

set

is c o n v e x .

positive by

the

) i n f u,

a minimum, case

an a d m i t t e d

introduce

: x

(1.3)

{y

of

z (x,y)

(1.3)

the

limx~+ x-le(x) ( x,

and we

and

define

~(x) ~(x)

~ K}, set

(1.4) When

= + ~. = inf

e,

= e(x),

x ~ ~,

is a l l o w e d .

the

subharmonic

function

~ o u

is

-

majorized

b y the h a r m o n i c

2. The t h e o r e m

result and

the h y p o t h e s i s

on

S

and

in this d i r e c t i o n

subsequently

to r e g i o n s

lov a n d K u z n e t s o v .

o

thst

~ o u

is a l l o w e d

converse

has a harmonic

in the sense of

and H 6 r m a n d e r

space h a v i n g

to a

type

[I0].

w

case:

~(x)

=

(x+) ~, I ( ~

and consider

on

S

The

where

regular

u

fundamental

first g i v e n b y S o l o m e n t s e v

T h e w o r k of S o l o m e n t s e v w a s p r e c e d e d

the s p e c i a l

majorant this Ch.

Extensions

a reasonably

question

[34]

of the w o r k of S o l o m e n t s e v

character

were

given by Priva-

by work of Privalov which

( + ~.

[I am

indebted

to P r o f e s s o r

.

Lars G a r d ~ n g

for t h e s e b i b l i o g r a p h i c a l

o f this p a r a g r a p h thesis

We n o w turn

is the t h e o r e m o f S z e g ~

by G~rding

of e u c l i d e a n

considered

~

-

ZMs~.

of S z e g 6 - S o l o m e n t s e v .

the c o n s e q u e n c e ~ o f is s u b h a r m o n i c

function

17

[26],

function

appears

in w h i c h

p a p e r o f R. N e v a n l i n n a Solomentsev

type

below]

u

is t a k e n

principal

consequences

not g i v e n

is a s p e c i a l

somewhat more

In w h a t

for the c a s e of R i e m a n n

properties

lemma w h i c h

the s p e c i a l

just cited.

theorem

case

that

surfaces

[Theorem

harmonic.

its i m p e t u s

2 w i l l be b a s e d on a

in the lemma

The p r o o f of the lemma of T h e o r e m

2:

harmonic

Suppose

majorants

that and~

If t in a d d i t i o n ,

~

o u

in fact,

u

[18],

yielding

14of

[Lemma

is simple his

1

and

thesis.

The

states +

Theorem

in the

g i v e n b y u s in

3 below]

The p r o o f of T h e o r e m thesis~

of a harmonic

a theorem of Szeg6-

2 below]

[Theorem

14 of P a r r e a u ' s

the c o n d i t i o n

[25] and in P a r r e a u ' s

finds

follows w e e s t a b l i s h

than the p r o o f g i v e n b y P a r r e a u section

surfaces

is the m o d u l u s

in this d i r e c t i o n

in our paper.

case of Theorem

of this

u

o f the h y p o t h e s i s

to be n o n - n e g a t i v e

immediate

For R i e m a n n

paper of R.Nevanlinna

The w o r k of P a r r e a u

together with additional maximality

in the s u g g e s t i v e

latter w o r k

is c o n s i d e r e d .

indications.]

has a h a r m o n i c Mu +

and

M(~

is not the c o n s t a n t

majorant. o u +)

-~,

Then

+

u

and

~

o u

have

are q u a s i - b o u n d e d .

then

u

admits

a unique

represeny

t a t i o n of the form

Q - s - g,

where

Q

is the d i f f e r e n c e

is a s i n q u l a r t n o n - n e q a t i v e

of quasiTbounded harmonic

function

(2.1)

non-neqative on

S,

and

harmonic g

functions

on

is a n o n - n e q a t i v e

S, s super-

-

harmonic

function

on

S

satisfyinq

harmonic

majorant

but also

mg

18

= O;

M ( ~ o Q)

There Theorem of

3:

is a c o m p a n i o n Under

subharmonic

maximal

the h~oothesis

functions

v

bounded then

of

~.

When

non-neqative

#

is e x a c t l y

the

~ = - ~

but

o f the

form

o__nn S

We Lemma S,

I

u

(2.1)

start with

with

the

functions

the_____n_n u

Proof:

and

Given

@re

a positive

a

(2.2)

envelope,

H, o f t h e f a m i l y ,

on

S. v

-~, When

it is t h e d i f f e r e n c e ~

(of w I, t h i s Ch.)

subharmonic

on

S

which

of quasi-

is n o t

- ~,

satisfy

{~,v} = H.

-

is t h e

eo,

set o f

subharmonic

functions

Q = H.

lemma.

the h y p o t h e s i s

M ~ ? u

have

o u

the c o n s t a n t

following

(Parreau) : U n d e r

oQ

= M ( ~ o u).

upper

the c o n s t a n t

set o f f u n c t i o n s

is n o t

~

It s t a t e s 2 ~he

o v = M~

M max

When

not only does

satisfyinq

is n o t

harmonic

further

= M ( ~ o MU)

theorem.

o__n_n S

H

and

of Theorem

M~

is a m e m b e r

-

of T h e o r e m

2 if

u

is n o n - n e q a t i v e

harmonic

on

quasi-bounded.

number

c,

there

exists

a positive

number

d

such that

c x 4 ~ (x) + d for all r e a l

x.

With

s

denoting

cu

on comparing

the

singular

the a r b i t r a r i n e s s

of

c,

we

singular

component

4 ~ o u + d \4 M ~

components

cs6

Given

the

of

we obtain

the

inequality

from

o u + d,

o f the e x t r e m e m e m b e r s ,

M~

u,

o u.

see that

s = 0

and hence

that

u

is q u a s i - b o u n d e d .

-

We n o w harmonic

introduce

functions

on

(bn), S

19

-

a non-decreasing

with

limit

u.

the

fact

that

the m i d d l e

term

and

that

M~

o u

o b n ,4 M~

is q u a s i - b o u n d e d ,

M~

of bounded

non-negative

From

o u ,4 l im M~

and

sequence

o u = l im M~

o b

we

o u,

see

that

n

is q u a s i - b o u n d e d .

We n o w t u r n

to the p r o o f

of T h e o r e m

2.

~(x +) ,4 ~(x)

It is o b v i o u s

that

+ ~(o), +

being

non-decreasing

a harmonic

majorant.

and taking

the

fact that

majorant

and

values.

the r i g h t

side

+

.4 ~ o M ( ~

the

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

that

I we conclude ~

that

by Lemma

I we

We c o n t i n u e , constant using u

-~.

we

see

has

that

u+

has

a harmonic

o

(Su +) ] = S ( ~

supposing

for the r e m a i n d e r

has

a representation

o u +)

From

(2.3)

we

infer,

using

o u +) .

is q u a s i - b o u n d e d .

a quasi-bounded

representation

(2.3)

that

M(~

u

o u +) .

is q u a s i - b o u n d e d .

see t h a t

Since

the c a n o n i c a l

admits

Mu +

is n o n - d e c r e a s i n g ,

S[~

Hence

~ o u

that

by Lemma

fact

that

o u +)

o (Mu +) ,4 M ( ~

Hence

We c o n c l u d e

From

u

and

non-negative

(2.3),

of the a s s e r t e d

of

the p r o o f

non-negative

Ch.l, form

that

harmonic

of a non-negative (2.1).

u = M u + -m[ (Mu + ) - u ] - g

u

is n o t majorant,

harmonic

the we

see,

function,

It is to b e o b s e r v e d

that

that

-

where

g

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

t h a t the mass

g

in q u e s t i o n

distributions,

functions follows

cf.

possessing

at o n c e

the c o n s t a n t

20

function on

-

S

satisfying

are just the G r e e n t s F. R i e s z ' s

a harmonic

potentials

admits

on

S

[We r e m a r k generated

t h e o r e m on the r e p r e s e n t a t i o n

minorant.]

The u n i q u e n e s s

from the fact t h a t a n o n - n e g a t i v e

+ ~,

m g = O.

a unique

of the

by non-negative

of superharmonic

of the r e p r e s e n t a t i o n

superharmonic

representation

in p a s s i n g

function

on

(2.1)

S,

not

form

q + s + g,

where

q

is a q u a s i - b o u n d e d

non-negative on

S

harmonic

satisfying

consider

minorant

"q"

equality

(2.2)

non-negative

and

"s"

harmonic

that

on

S.

s

where

(2.4)

We i n t r o d u c e

function

follows w h e n w e

function,

note

is just the g r e a t e s t

majorant ql

is a s i n g u l a r

superharmonic

and thereupon

has a h a r m o n i c

Q = ql - q2

functions

and

invoke

(2.3),Ch.I.

a n d t h a t the q2

a positive

are q u a s i number

e.

satisfying

m)+

c]

> O.

the o b s e r v a t i o n

u &

we conclude

functions,

~ o Q

S,

superharmonic

terms of each representation

c + ~[~(-

Starting with

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

of the r e p r e s e n t a t i o n

superharmonic

We s u p p o s e

number

g

function on

for a g i v e n a l l o w e d

to b e s h o w n t h a t

holds.

be a positive

and

o f the g i v e n

There remains

S,

harmonic

The u n i q u e n e s s

two such r e p r e s e n t a t i o n s

harmonic

c

f u n c t i o n on

m g = O.

t h a t the sum of ~h~

bounded

non-negative

(2.4)

on t a k i n g

~

o

the l e a s t h a r m o n i c

(M~

o u

+ c),

majorant

o f the left side t h a t

Q - s ~4 ~ o (M~ o u + e) ,

from w h i c h

i n e q u a l i t y we o b t a i n

c + q l ~ s + q2 + [c + m ~

On n o t i n g

t h a t the

left side o f this

inequality

o (M~ o u + ~)].

is m a j o r i z e d

b y the q u a s i - b o u n d e d

Let

-

component

of

the

right

side,

we

conclude

Q ~ m~

whence

it

follows

that

the the

arbitrariness role

of

of non-negative The

c

Proof

e

o

(M~

of Theorem

- ~.

remaining

assertions

we

o Q ~4 M ~

conclude

is a n c i l l a r y .

Here

o u + e) ,

the

longer

consider

assert

constant

theorem

- ~

are

are

now

assured

that

@

see

On

It is to b e

purpose,

in t h e

observed

to p e r m i t

comparison

argument.

and

where

so

is

true

~ = - ~

trivially

in t h e

and

u

a member

present

is t h e of

case.

We

#.

The

put

this

o u)]

~

is

~

superharmonic.

o [M(~

If

v

~ ~,

we have

(2.5)

o u)].

that

this

we

note

o [M(~

o u)]

~ ~,

o (M~

o u)]

$ M~

(2.6)

that

o [m~

Using

case

vacuously

o [M(~

H = m@

which

its

appears

the

v ~4 m a x { ~ , v }

To

served

(2.2).

aside.

We

We

equality

it h a s

it no

first

is t h e the

o u + e.

2 is c o m p l e t e .

3= W e

H of

Once

functions,

of Theorem

constant

case

of

harmonic

proof

that

that

M~

Given

2 1 -

o u,

yields

(2.5)

combining

M~

o [m~

~

M~

o u ~4 M ~

(M~

o u)]

~ M~

(2.7)

o u.

we obtain

this

inequality

with

(2.7)

we

o [m~

see

.

(M~

that

m~

o u)].

o

(M~

o u)

~ ~.

On

taking

the

-

greatest v

~ ~.

harmonic The

assertation

established

We Suppose value

t u r n to the

~,

u

on to

S.

second

u.

By

on

It f o l l o w s

the

of Theorem

set of

w

It is to b e n o t e d

now

upper

envelope

see that

see

that

m~

o (M~ o u)

of Theorem

~ v,

3 has been

H ~ Ha

2 we

that

~> - ~, of

H H

quasi-bounded

the c o n s t a n t

- ~.

see

H.

that

Q ~

the c o n s t a n t

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

by a quasi-bounded

H

is n o t

is t h e d i f f e r e n c e

We

taking

non-negative introduce

By the

- ~. the

harmonic

Q

of

(2.1)

first paragraph

non-negative

harmonic

of

function

of quasi-bounded

non-negative

to the p r o o f s

of

the

( + ~.

We

S.

which 3.

that

We

now

fix

~

family

and that

follow v ~ #

on

S

and let which

does

Using

~

satisfy

~ ( ~

last

two

introduce

satisfy

o max{~,v}.

indeed

~ o max{a,v} # .

H

are p r e p a r a t o r y

o w = M~

o max{~,v}

inequality

o f the

is n o t

that

subharmonic

the a i d o f the

Then

is m a j o r i z e d

readily on

u

M~

with

we

statement

We assume

- ~.

of Theorem

H

The developments

9 ,

(2.5)

first

is the d i f f e r e n c e

that

(2.2)

see t h a t

functions

assertions

The

assertion.

H

Suppose

2 we

harmonic

side of

follows.

is the c o n s t a n t

Theorem S.

(2.6)

and trivially

functions

of the right

-

in a l l c a s e s .

that

relative

minorant

22

have

~ ~(a)

a harmonic

+ ~ o v.

the r e p r e s e n t a t i o n

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

in

a.

majorant

We l e t

given by

Further

Ha (2.6)

as w e

see

denote

the

for

the convexity

H

we

of

yields

o H

- ~ 0 (M m a x { ~ , v } )

|

~ +(~)(H u - M max{a,v}) ,

~'+(=)

being

the right

derivative

of

0 = M~

~

o H

at

a.

- M~

From

this

inequality

we

o (M m a x { a , v } )

~'+(~) ( H

- M max{~,v}),

(2.8)

conclude

that

-

and

since

~'+(a)

we

O,

>

obtain

the

H

Suppose

now

that

~

>

conclude

= M max{~,v}.

- ~.

Using

the

opposite

we

find

Suppose we

see

direction,

using

lima~M

(2.5)

that that

that

for

v

is

M~

o H

each

v

~ ~

subharmonic

we

v

~ ~.

satisfying

max{~,v}

and

(2.6)

~ = - ~

be

in

max{~(~),~},

We

now

let

The

the

to b e

and senses

and

third

note

denote

u

have

(a-G),

H.

the

and

equality

(2.10)

of

o v = M~

o M max{~,v}

the

set

the

of

o v.

final

constant

We

functions

of Theorem

the

specified

~

(2.10).

majorant

considered

to b e

obtain

satisfies

S

assertion

is n o t

~

we

on

% is p r e c i s e l y

remains

recall, taken

+

= M max{~,v}.

= H.

is a h a r m o n i c

Hence

(2.10).

There

inequality

~4 m a x { ~ , v }

M max{~,v}

M~

that

obvious

~4 m a x { a , v }

M max{~,v}

Thus

(2.9)

that

H ~

In t h e

-

formula

max{~,v}

we

23

3 is

presently.

of

o

the

a

the

class

~

of

w

M~

+

[~(=)

-

o w = M~

o v

on

S

~

~ R,

which

on

theorem.

let

~(- ~)].

subharmonic

o max{~,v},

from

that

9 =,~

v = ~

S

and

established.

notations

take

~

subharmonic

thereby

The We

conclude

v

assertion - ~.

Since

satisfy

Here,

and ~

H =

we are

to

- 24 -

where We

v

let

Using

is a g i v e n Q

member

now denote

the

third

of

~.

the t e r m

assertion

We n o w let

in q u e s t i o n

o f the p r e s e n t

M~

which

is a c o n s e q u e n c e

o f Th.

o

H

the u p p e r

in the r e p r e s e n t a t i o n

theorem

Q = M~a

2, t h i s C h . ,

denote

a n d the

fact

of

envelope the

form

of

~ .

(2.1)

o f v.

that

V,

o

we conclude

H

= M max{a,Q}.

m*

9 (M~ o v) ~4 m ~

that

Since

and

Q

~ ~,

we conclude

We write

Q = ql - q2'

functions

on

bounded

S,

harmonic

a

is l a r g e

where

ql

and introduce functions

on

and

and negative.

with

We c o n c l u d e

each member

Suppose associated majorant.

We c o n c l u d e

Q

that

Q

~ H

S

limit

n

Hence

is

that

v

is s u b h a r m o n i c

non-negative

sequence

For a given

harmonic

of non-negative

n

we have

~ max{~,Q}

for

n

q2"

such

a

the inequality

) , H a )tH

and thereupon

~

to

are quasi-bounded

a non-decreasing

with

of

is e q u a l

q2

(bn),

qi - b

holds.

o v),

that

ql - b

when

o (M~

that

We

Q -- Ho

see t h a t

the Q a s s o c i a t e d

H.

H.

It is n o w i m m e d i a t e

From

on

S

and of the

~ o v ~ ~

. H

we

form

(2.1)

see t h a t

~

and that the o v

has

a harmonic

that

M~

o v = M~

t h a t i n the c a s e w h e r e

~ = - ~

o Q = M~

and

u

o H.

is n o t

the constant

- ~

the

-

family

#

consists

The proof

member

o f the

of Theorem

The

following

of

9

Corollary:

exactly

when

Given

we

q

let

of Theorem

3 permits

that

~ > - ~.

as

For each

follows.

It s u f f i c e s

1, C h . I .

3. A c o m p a r i s o n in w

for the

9

that

v ~ #

Q

Q = H.

us

to r e l a t e

H

to an a r b i t r a r y

not

the c o n s t a n t

- ~,

we have

taken

relative

to

v

in the

sense

of

(2.1)

that

retains

If t h e r e

~1

has

a harmonic

Proof: case where 3.

We put

exists

aside

in t h i s c a s e

let

q.]

that

~1 on

assigned

v ~ #

= ~ + M[ ( Q - ~ ) + ] .

satisfies

the c o n d i t i o n s

that

has

S,

~ o u

to it in the p r e c e d i n g

such

that

~I

o v

has

a

imposed

a harmonic

on

majorant,

section.

harmonic

majorant~

then

majorant.

with

To c o n t i n u e ,

suppose

of

is s u b h a r m o n i c

~ = max{x

Indeed,

that

We

the m e a n i n g

4:

o u

significance

u

Theorem

see

with

q + ~,

With

to n o t e

theorem.

t h i s Ch.,

and that

we

(2.1)

q = M[ (Q-~)+].

L.

2 and

form

~ > - ~.

M max{~,Q}

The

of the

3 is c o m p l e t e .

corollary

i.s d e f i n e d

Proof:

[cf.

-

functions

H=

where

25

the

trivial

: ~(x)

#1

= ~(- ~ ) }

being ~ c #1"

Q(v)

case where

taken

the

to

~I

this case

term

u

or

is an i m m e d i a t e

relative

We p u t

denote

= - ~

either

Q

and

v

v

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

as

~

is to

- ~.

of Theorems ~

a n d u,

aside.

of

(2.1)

taken

relative

to

v.

We

have

(3.1)

u ~4 M m a x { ~ , Q ( v ) }

by Theorem

3.

Using

the r e p r e s e n t a t i o n

(2.1)

for

v

we obtain

m a x { ~ , Q ( v ) } ~4 m a x { ~ + , v }

+ s + g

-

and conclude

26

-

that

M max{~,Q(v)}

,4 M m a x { ~ + , v } .

Since

91

91

o max{~+,v}

has

we

see

o u

that

4. R e m a r k .

91

More

a harmonic does.

w , this Ch.~save

in p l a c e

Here

Q

the r o l e

of

are n o n - n e g a t i v e However,

we

therefore

5. The

shall

leave

has many

principal

HI(~)

admits

convergence p ~ I,

results

a Poisson

(order p)

continuity

"boundary

functions"

Now

the aid of Th. extended Ifl

to has

result,

- Lebesgue

- Lebesgue

a quasi-bounded

celebrated

may be fact

boundary

taken

that

let

stipulation

that p-q,

on

From

~

general

1916

p

in

constant. and

q

= O.

hypothesis

as

r ~ I of

and

which

say

Let = O. q.

of F.

belonging conclude

where f,

is e a s y

the

to the c l a s s the m e a n

f ~ H

p

(~)

o

a n d the

annihilates in

and M.Riesz

for p r o v i n g

s u c h an

and analytic

9(- ~)

paper

as a b a s i s

in q u e s t i o n

majorant,

be n o t

m min{p,q}

it one m a y

function

f ~ HI(~). that

9

where

and

a function

circumference

representation

harmonic

imposed

(3.1)

indication.

: 8 ~ f(re i8)

r

continuous

In fact,

the c o n d i t i o n s

form

of this m o r e

The

on the u n i t

of f u n c t i o n s

of the

summary

is t h e

f

and by

= + ~ we require

use

which

of the F a t o u

b y the

satisfying

representation.

family

o M max{~+,v}

is q u a s i - b o u n d e d

and M.Riesz.

o f the p a p e r ,

2, t h i s Ch..

Ru{- ~ }

S, q

above

of a measure

the P o i s s o n

on

with

of the

9

x-I 9(x)

to m a k e

a central

91

is e s t a b l i s h e d .

occasion the

o v + 91(~+) ,

so d o e s

over by a function

o f F.

a characterization

absolute

that

but

~ 91

Hence

consider

limx~+

functions

not have

Theorem

aspects,

other

of

the m a t t e r

classical

theorem

one m a y

is t a k e n

harmonic

majorant~

The

generally,

o max{~+,v}

the

4.

to e s t a b l i s h

u = loglfl

and

We c o n c l u d e

by Theorem

with

let 9 = exp 2

From

-q \4 R e f ~4 q,

we

see t h a t

Ref

is the d i f f e r e n c e

of q u a s i - b o u n d e d

non-negative

harmonic

functions

on

-

the u n i t disk;

the same is true

representation functions Lebesgue

on

A

are p r e c i s e l y

integrals

Szeg6's

H2

setting with

Theorem.

The

fact t h a t

functions

that

p

on

A

a Poisson

non-negative

- Lebesgue

harmonic

g i v e n b y the P o i s s o n -

This theorem was given originally by Szeg8

form of this theorem,

We s u p p o s e

admits

integrands.

the a i d o f the t h e o r y o f T o e p l i t z

the e x t e n d e d

f

that the q u a s i - b o u n d e d

the h a r m o n i c

with non-negative

Maximal

-

Imf.

follows w h e n w e o b s e r v e

6.

gave

for

27

forms.

Subsequently,

w h i c h w e shall c o n s i d e r

is finite and p o s i t i v e

a n d that

[36] in the F.Riesz

[31]

immediately.

f ~ H

(~)

but

f

is not

P the c o n s t a n t recalled

that

functions, ~(x)

O.

such an

f*

denote

the F a t o u

f

admits

a representation

as we see r e a d i l y w i t h

= exp(px),

for b o u n d e d

x ~ R, ~(- ~)

analytic

the 1916 paper) function on measure

Let

which

~,

zero,

limits p,p..

=

functions asserts

The

facts are,

limit

and that t h a n k s

on

A

2

that the F a t o u r a d i a l takes

of bounded o f course,

applied

of

analytic standard.]

lit is to be

of bounded

analytic

u = loglfl ,

limit

theorem

F. and M. R i e s z

limit

the value

f.

to

to the F a t o u

a n d the t h e o r e m

zero,

function of

as the q u o t i e n t

the aid of T h e o r e m

O,

not the c o n s t a n t

such a quotient

radial

(also from

function of a bounded

analytic

zero at m o s t on a set of

functions

on

~

possesses

Fatou radial

T h e S z e g 6 t h e o r e m m a y be f o r m u l a t e d

as follows:

Theorem F

) 0

5: o nn

(a)

log

[0,2E]

If*(eie) I

and

such that

log F

ff*(e i8) Ip and

Fp

are

inteqrab!e

are i n t e q r a b l e .

on If

[0,2H]. G

(b) G i v e n

is analytic, on

and s a t i s f i e s

log

then log

G

~ H

(A) and P If, (e is) I . T h e n

IG(z) I = 2 K

IG* (e i8) I = F(e)

p.p..

I f I .~ l h l .

log F ( e ) k ( e , z ) d S , r z l

(c)

Le___tt h

< 1,

be s u c h a

G

(6.1)

with

F(e)

=

{6.2)

-

That

is,

h

is m a x i m a l

modulus

of the Fatou

maximal

functions

of m o d u l u s

in m o d u l u s

radial

is just

one.

limit

28

-

in t h e _ f a m i l y function

aqrees

the set of functions

Our m a i n

implies

that

on A .

We are

concerning modulus

of T h e o r e m u

is to i d e n t i f y 2 as

so c h o s e n

analytic

on & ) w e

of

log

]fl

and

is m a j o r i z e d

functions

having

p

(4)

If*l. ~h

for w h i c h

the

.

The s e t of such

where

is a c o n s t a n t

see that,

If~(ele) IPk(e'z)de' Izf O}

onto

the a b o v e

itself

of

E3+ k

segments

- {0},

distinguished;

and a l s o

(4.1)

,

positive

of

- {O}

less

z ~

J

integer

given

[I,(3/2) p]

consists

A(O;2)

A(O;2)

, have been

to be

the

: Rez

>

is d i s t i n g u i s h e d .

the

image

less

of

restricted,

are

of

A(O;2)

0},

The u n d e r l y i n g - {O~

{2 -(n+l) : n = O,I,

images

(e 2 ~ i k / m

further

by

{ ( e 2 K i k / m e x p (n~---L o g z) ,z,k)

=

a copy of

E1 ,

n = O,I,

, m - I,

segment

the

(z,2)

it s u f f i c e s

H

analytic

3 2~ik/m,

a given

E3+k

for w h i c h

that

part belongs

identity

[2-(2n+2),2-(2n+l)],

9~ ~

distinguished;

real

of the

construction,

[e2~ik/m

k = O,

the r e s u l t

In fact,

a th p o w e r

our

the s e g m e n t s

E 2, a c o p y

has

yields

= I.

To p r o c e e d for w h i c h

which

in the c a s e

is the

satisfying

S

just made

I

and

with .~

(3/2) p

with

set of

respect

with respect

to

respect to

exp(L~

(4.2) P

The

surface

slits

S

is o b t a i n e d

distinguished

(4.1), joining

k = O, of

E2

in c o m m o n ., m - I.

to

by joining

E1

for

E1

E2

to

E1

and

E2

and by joining

The p r e c i s e

should

be c l e a r

technical from

the

in the

meaning

standard E3+ k

manner to

along E2

to be a t t r i b u t e d

developments

of

w

the

along to the

- it c o n s i s t s

in

-

stating

what

6(a~r)

when

a

[2-(2n+2),2-(2n+l)]

the segments

of such a segment discerning

enter

the

but

44

-

is of the form or is of the form

is n o t an endpoint.

6(e2Kik/mx,2;r)

where

I ~ x 6 3/2

1 < x

with

(x,2)

The joining

6 (e2 Hik/mx I/p

for

(x,l)

of

x

where

a p o i n t of one of x

to

E3+ k

is n o w a point E2

consists

in

and the

, x,k~r)

(4,3)

((3/2) p

When

I ( x

( 3/2

we

take

for

r

6 (e 2 Eik/mx, 2 ~r)

small

as the u n i o n

of the

image of {Iz - xl

with

respect

image w i t h

respect

(e2Eik/mz,2)

to

(4.2)

{Iz -xl

with

respect

(4.4)

~ O}

to z ~

and the

< r, Imz

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

(4.5)

of the p r e i m a g e

~ r,

Imz

of

(4.6)

( 0},

of

(4.7)

z ~ exp(L~ P to a d i s k c e n t e r e d for

(4.3)

and

'Imz

( O'

or

3/2

x = I

image w i t h

is g i v e n

xp

on w h i c h

(4.7)

on r e p l a c i n g

'x'

'Imz

(4.6).

by

~iO'

as the u n i o n

respect

restricted.

at

to

(4.2)

in

of the

is univalent.

by

0x I/p' The

of the p r e i m a g e

For other p Q i n t s

of

E3+ k

, 'Imz

set

image of

), O'

by

6(e2Kik/mx,2~r)

a(x~r) of

The c o r r e s p o m d i n g

with

A(x~r)

respect

- {X}

~Imz

definition

( O'

in

is d e f i n e d to

(4.5)

with respect

(4.4) ,

for

and the to

(4.7)

as

we define

6 ( e 2 ~ i k / m e x p ( LOg w) , w,k~r) P for small

r

as the image o f

The p r o j e c t i o n

map

~

A(w~r)

on

with

3+(m-l) U Ek o

respect

to

is d e f i n e d

(4.2).

as the m a p taking

each p o i n t of

-

3+(m-l) U Ek o will

into

be d e n o t e d

We

fix

= C - {O}.

union

of

We

first component.

by

K.

A conformal

as the

smallest

m

~(E3)

the

its

When

p

u I/2,

show

that

-

For

convenience

structure

integer

exceeding

., m - I,

S E O

.

Let

than

I.

of the p o i n t s

Applying

at p o i n t s

tation

the

same

that u s e d

of the

El

the a r g u m e n t

having

paralleling

of

f E H

form

and of

E2

w

on

S

2p.

When

p

paragraph

EIS

as in w

< I/2,

m = I

.,m - 2,

and

and

the

C - {O}.

(S).

whose

We

introduce

where

g

first components

this Ch.

this Ch.

g o ~

following

fl,

the r e g i o n

of

S

P

first component

in w

the

~ ~, k = O,

is

P consisting

after

is i n t r o d u c e d

~(E3+k) N ~ ( E 4 + k )

K(E3+k) , k = O,

first

45

and

we

we

see

further

conclude

is an e n t i r e

that

are of m o d u l u s

flfl

is b o u n d e d .

that

f

function.

admits

less

takes

the

Using

an a r g u m e n t

a unique

It n o w

same v a l u e

represen-

suffices

to s h o w

that g(z) z

large,

for t h e n

g,

= O(Izl) ,

and consequently,

f

will

be c o n s t a n t

so t h a t

S ~ O

. P

As an a u x i l i a r y

step we consider

the b e h a v i o r

at

~

of a f u n c t i o n

F ~ H

(D) P

where D = {Rez

B y the S z e g O - S o l o m e n t s e v Applying

standard

theorem

estimates

) O,Izl

(Th.2,

) 2P}.

Ch. II) w e

to the P o i s s o n

(4.8)

see

integral

that

MIFI p

is q u a s i - b o u n d e d .

for a h a l f - p l a n e

we

conclude

that

IF(z) Ip = O(Izl)

for

z

large

and

satisfying

We o b s e r v e

that a Poisson

M I F Ip

{Rez

to

Applying k = O,

IArg

integral

zl 4 a K / 2

where

c

(4.9)

to

restricted

., m - I,

where

representation

> c}

f

(4.g)

a

is g i v e n

is a v a i l a b l e

satisfying

O

( ~

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

> 2 p.

we c o n c l u d e

to the

image

of

D

that

z) ]I p = O ( I z l ) I g [ e 2 K i k / m e x p (Log ~

with

respect

to

(4.2),

( i. of

-

for large near of

I,

z

satisfying

we c o n c l u d e

IA r g

that

46

-

zl 4 ~ / 2 ,

k = O,

Ig(z) lp = O(Izl P)

,m - 1.

for

z

Taking

a

sufficiently

large and t h e r e u p o n

the c o n s t a n c y

g.

We n o w s h o w that follow.

For

K E H

k = O,

q

(S) ,0 < q

, m - I

{Rez

with respect

to

(4.2)

and

Ck

the

to the same map.

let

> O,Iz

{Iz-

with respect

we

< p,

-

whence

~3+k

the a s s e r t e d

denote

(3/2)PI

the

> (3/2) p -

property

of

S

will

image o f

(1/2) p}

image of

(3/2)PI

=

(3/2) p -

By the o b s e r v a t i o n

(1/2) P}

o f the

first p a r a g r a p h

o f this

section

Hln k ~ Hq(nk) ,

k = 3, {Arg z note

.,2 +m,

since

z ~ e x p ( ~ Log z)

< P ~ ~}. K We denote

U34k42+m~ k

the f u n c t i o n w i t h d o m a i n

and w h o s e

restriction

ishing c o n t i n u o u s l y

to

on

~ C

S

positive

to

harmonic

sufficiently IKl q.

5.

S - ~

~.

that

on

~

q

1

having

number,

E E H

and

U34~42+mC k to

M(IKl~lq). introduce

p l a n e o n t o the s e c t o r

by

S - ~

non-negative

We t h e r e u p o n

is the c o n s t a n t

function

the r i g h t - h a l f

restriction

and d o m i n a t e d b y

large p o s i t i v e

We c o n c l u d e

whose

~

is the l a r g e s t

itive v a l u e at e a c h p o i n t of striction

by

maps

C.

v

function

function

restriction

limit

at e a c h p o i n t o f

then

u + Av

u

with domain

and w h o s e 1

let

u

is the c o n s t a n t

harmonic The

We

to

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

~

on

~

takes S

If S

zero

on

is the C.

de-

vana pos-

whose

re-

smallest A

is a

and m a j o r i z e s

(S).

~ 5 . The c o n s t r u c t i o n o f this s e c t i o n is the m o s t e l a b o r a t e of the chapter. P P It c o m b i n e s the u s e o f " R i e m a n n i a n sectors" of the sort i n t r o d u c e d w i t h the a i d of the E3+ k

0

in w

by their

together with H

images,

we take

E1

the a t t e n d a n t

and parameter

as in w

concern

control

distinguishing

to c o v e r

a deleted

of the sort u s e d

the same

neighborhood

of

in w

segments.

W e let

(qk)

denote

-

a decreasing

numbers

is not important.

We take

E2

satisfying

We let

as a c o p y of

n = O,1,

-

sequence of p o s i t i v e numbers w i t h limit

quence of p o s i t i v e 1

4 7

.,

v

e

denote

~(012)

o

= 1

and

p

and

lim ek = 2.

- {O}

of the form

The

Ejk

for w h i c h the segments

sectors".

For

j = O ....

and d i s t i n g u i s h

the "parameter"

qk We define

S,

proceeding

S~.

We see that

and

Ejk

The surface

Sp

map

E

S

and

k = 0,1,

,

we

(j,k)

(5.1)

segment

to

sections, E2

along

in the earlier

P

with

k = O,

joining ~jk'

.,p.

E2

E2

to

E1

the p r e c i s e

along

technical

sections.

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

is o b t a i n e d b y joining

to

surface E1

S

P

and its

as above and

The s u b r e g i o n

S* P

Ejk

is o b t a i n e d

from

S the images w i t h r e s p e c t to z ~ (z,2) of the segments ~jk,k P (S~) is an i n c r e a s i n g sequence of regions e x h a u s t i n g S.

We shall now show that i n d e p e n d e n t surface

will be, as

qkl.

for each w h o l e number

but now o n l y for

by removing

E2

(j,k))

as in the e a r l i e r

[2 -(2n+2) ,2 -(2n+1) ]

We also introduce

E2

(2n+2) ,2- (2n+1) ] ,

, (e2k+l)

sense of the p r o c e d u r e b e i n g that indicated

to

-

The points of

v - 1,

(e2~iJ/Vexp L ~ qk

[ (e2k)

subregion

[2

2qo.

as the image w i t h r e s p e c t to

{Re z > O7

the segments

as

o

,

are d i s t i n g u i s h e d .

z ~

of

e

(z,2).

"Riemannian

introduce

The choice of

se-

and the segments

,~ - 1, k = O,1,

above,

an increasing

the least p o s i t i v e w h o l e n u m b e r exceeding

ajk = [ e 2 E i j / v s

j = O,

(e k)

is a m e m b e r of

belongs

to

To show that

Op

of the choice of an a l l o w e d sequence

and that w i t h a suitable choice of

Hp(S).

We c o n c l u d e

S ~ O

we show that if P

that

(ek)

> P-

(ek) the

the p r o j e c t i o n

Op ~ Op.

p

< q < + ~, S ~ O . q

G i v e n such a

q

we

-

fix an index paraphrase entire {IArg

1

such that

of the a r g u m e n t

and b y e x a m i n i n g zl

< ~/2}

conclude,

- {e2k

as in w

fixed and shall (Inls *Ip P

has on

qk

,e2k+1 g(z)

the

a harmonic S

of the

sequence

choose

< q"

Now

qk

-

let

preceding

the b e h a v i o r

that

The p a r a m e t e r

function

ql

48

of

}, 0

f

< ~

= O(Izl)

(ek) . e2k+3

f s H (S). Afortiori, f ~ H (S). A q ql section shows that f = g ~ E where g is on the image

< I,

with

and hence

We shall k e e p

recursively

majorant

and)

in a n o n - d e c r e a s i n g

respect

to

(5.1),

the c o n s t a n c y

of

the

even

ek

to o b t a i n

the sequence

fashion.

of a sector

with

It then

f.

follows

being Hence

indices

a sequence

(M(IKIs*IP)) p

k

(S~) tends

that

I, we S ~ 0 . q

and

el

such that to a h a r m o n i c

r H

(S).

We shall

P denote

the p r o j e c t i o n

We note suffices We

let

h +1

IK tp P

to p a r a p h r a s e hp

denote

tends

- 1,

that

map associated

involves

M(In

e2N+3

a mild

Ip)

to

to

~on

S~

K . P

majorant,N

of the

e2p+2.

complication

by

= 0,1,

last p a r a g r a p h

and show that w i t h

hp

tends

S

a harmonic

the a r g u m e n t

pointwise

as

has

with

of the

preceding

that

held

in w

fast

= O,

that the p r e s e n t

studied

it

section.

., 2N+2,

(e2~iJ/~e2p+2,2),j

It is to be o b s e r v e d with

To see this,

ek,_ k = O,

_less the p o i n t s

in c o m p a r i s o n

....

since

o

~

situation

the

K

are P

not bounded,

however

We b e g i n Here

C will

j = O, denote

With U

by adapting

be the u n i o n

.,~ - 1

of

the c o m p o n e n t

In m a k i n g

the special

of

this u n d e r s t a n d i n g

the c o n s t a n t taking

1

% +1- C

values

of

V

= r},r

we

introduce V

denote

restriction

further

fixed,

that U

to

limit

El

1

on

let and

is the s m a l l e s t

e2p+3.

to o b t a i n Let

D

~2

of

some

denote

We let

~1

SN+I-

~1"

denote '~'

same n o t a t i o n s

(e2Kij/Vz, 2),

z ~

is r e p l a c e d

context,

S + 1 whose

at each p o i n t

success.

and positive.

in the p r e s e n t

it is d e s i r a b l e

on the p a r a m e t e r

small

assure

some of the

to the m a p s

= (x+) p

function ~k

will

using

and we

~(x)

taken the

of w

respect

containing

and h a v i n g

proceeding

the d e p e n d e n c e

of the a r g u m e n t

we understand

and w h o s e

of the c o n s t r u c t i o n

of the images w i t h

Sp+ 1. We let

positive

Before

aspects

{Iz - ~2N+21

the p a r a p h r a s e

has d o m a i n

nature

save

'~'.

that here

restriction

harmonic

by

function

to on

C is ~k

C, k = 1,2.

information the u n i o n

concerning

of the

images

-

49

-

of A(e2p+2sr) with respect

to the maps

to see that

VlD

z ~ (e2Hij/Vz,2)

tends to

We introduce

A., 3

1

pointwise

union.

the p o i n t

We let

Q

(e2~iJ/Ve

limit S

fr(~ 2 - A)

p+l."

2N+2

harmonic

0

,2)

'

We let

~2p+2

R

denote

is the c o n s t a n t

1

Q + cR

It is e a s y

e2p+2.

2~+2 ~r) ]

,v - 1, S +1

~2 - ~

A

denote their

whose restriction

to

Q2 - ~

2 - 2) Ip) fr(Q 2 - 2),

and w h i c h takes the value

the function w i t h d o m a i n to

numbers

U + dV

IHp+ll(~ 2

c

and

d,

S

limit

1

independent

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

O

else-

whose restriction

p+l

S + 1- fr(Q 2 - A)

function on this d o m a i n having

and

and let

majorized by

and w h o s e r e s t r i c t i o n

There exist p o s i t i v e

such that

paragraph.

tends to

j = O,

function on

at each p o i n t of

smallest p o s i t i v e h a r m o n i c fr(fl2 - 2).

e2p+3

--I[A (e2nij/v e

M(~+ll(n

w h e r e on

as

d e n o t e the function w i t h d o m a i n

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

and having

of the p r e c e e d i n g

the c o m p o n e n t of

~p+l containing

- {e2p+2}

to

is the

at each point of of

e2p+3

near

and s a t i s f y

- 2) IP ~4 (Q + cR) l(f12 - A)

and

III+ 1 I f21 1p ,% (u + dV) In I. Hence we c o n c l u d e

that

Q +cR

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

m a j o r a n t of

with domain

Sp - {(e

sequence of

h +I

2~ij/~

+U

+dV

IHp+llP

e2~+2,2)

( the p a r a m e t e r

and therefore

: j = O, e2M+3

of

,v - I}

tending to

h + 1.

As in w

be the p o i n t w i s e

e2~+2).

Here we have

we let

H

limit of a

-

5 0 -

U(s) 4 h (s) ,~ H(s) ,%

I~

(s) + dV(s) , s ~ ~1 ~ ,s ~ D.

It is to be o b s e r v e d

that

R(s)

s ~ Sp - { ( e 2 ~ i j / v e 2 ~ + 2 , 2 ) follows

along

the

same

H(s)

for a d m i t t e d

tends

to

mutatis

e2p+2,s

as

O

in w

We

and conclude

as

e2~+3

,~ - 1}.

that

fix

that

tends

The

hp(s)

a E El, for

S

the

is the

repeat

to

remainder

Thus we conclude

and thereupon

admitted.

mutandis,

to

: j = O,

lines

s

tends

so d e f i n e d ,

o f the a r g u m e n t

equality

limit

the

e2p+2,

of

last

of

h

hp+l(S)

(s)

~ ~ H

(S).

and

as

two p a r a g r a p h s Hence

e2~+3 of w

S ~ O

P The

6.

strict

Given

O

< p

inclusion

< q

asserted

< + ~,

we

see on

co

p The

inclusion

7.

It w o u l d

plane

We

be

theory

The let

planes

are

(1.1)

interest

strict.

logarithmic The

that belong

a removable

to

H1

of

map

R

on t a k i n g

whether

Then

~

having

at

Now

F

< q,

account

in c o n s t r u c t i n g

zero

there

follows.

that

the r e s u l t s

of

f

appropriate

that

~

to the u p p e r

half-plane(in

the

sense

in 01 - OLA.

measure

but

for m e m b e r s h i p

in

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

lower

o f Ch.I.)

: x ~ f(x + iy) ,a ~ x 4 b,

in the

examples.

region

Lebesgue

as a c a n d i d a t e

(We n o t e

of w167 - 4.

inclusions

is a p l a n e

1-dimensional

~ = C - E

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

~.

< s

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

is L i n d e l S f i a n .

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

singularity

into

at all e v e n t s ,

We p r o p o s e

on

s, p

section

-q

succeeded

s h o w that,

subset

f s HI(~).

not

of this

. P

co.

s

follows

We h a v e

capacity.

identity

~8

-s

now

introducing

to d e t e r m i n e

remarks

be a c o m p a c t

01 - OLA. Suppose

of

following

E

positive

relation

at the b e g i n n i n g

now

converges

halfand

f

has

in the

Y mean equal

of o r d e r p.p.

segment integral

of

I

as

(in fact R

y ~ O take

containing

formula

for

(resp.

y T o)

the v a l u e E

f(w),w

f(x)

in its

a n d the p.p.).

interior.

~ C

not

a point

{Rez

~ ~,IIm

zl

respective

Here

-m

We obtain of

4 h},

< a with

limits < b

in the m e a n

< + ~.

Let

~

are be a

the a i d of the C a u c h y

-

h

positive,

from which

we

on taking conclude

the that

limit f

as

h

51

-

tends

is c o n s t a n t .

to

O,

a representation

It f o l l o w s

that ~ ~

for

f(w) ,w ~ C - ~,

01 - OLA.

-

52

-

Chapter Boundary

I.

In this c h a p t e r

classes

given

a Schottky

our p r i n c i p a l

on the border

doubling,

concern

of a c o m p a c t

we m a y and do assume

the c l o s u r e

of a r e g i o n

~ of a c o m p a c t

of a finite

number

of d i s j o i n t

ists a u n i v a l e n t mapping

~ onto

functions

(>o)

anticonformal

S-~.

L

Problems

w i l l be the study of functions bordered

Riemann

regular

consists

(F) in terms

Riemann

surface.

that our c o m p a c t surface

analytic

m a p ~ of S onto

Our p r o g r a m

in the class

IV

bordered

Jordan

itself k e e p i n g

in o b t a i n i n g

of functions

H

tions

involving

extension sequence

S-~,

Hardy classes

of M . R i e s z ' s

conjugate

[21].

the c l a s s i c a l

representation

point duce

the S c h o t t k y

symmetry

Parts 2.

The T h e o r e m

real-valued

finite

we

and there

ex-

of F fixed

and

for c o m p l e x - v a l u e d

(F) d e n o t e

just r e f e r r e d functions

func-

ques-

to be d e v e l o p e d

and ~=~, w h i c h

is a con-

in the c l a s s i c a l with

and of the

on S w h i c h

case.

study of

the aid of

so-called

are a n a l y t i c

of F. The u n i t a r y

is an

unitary

at each

functions

were given

I < p

in our p a p e r

< + ~. G i v e n

re-

It w i l l be seen that

w i l l be v e r y a d v a n t a g e o u s .

of this c h a p t e r Let

to b o u n d a r y

will be e s t a b l i s h e d

in the c l a s s i c a l

setting

(Hp version).

plane,

and is s t a n d a r d

one at each point

products

in the

of the m a t e r i a l

let

theorem

to be d e v e l o p e d

of m o d u l u s

present

of M . R i e s z

curves

(~) and r e l a t e d

representations

the e x t e n d e d

to w h i c h we have

Blaschke

these

to ~. The r e p r e s e n t a t i o n

on S, i.e. m e r o m o r p h i c

of ~ and take v a l u e s to just the

S=~,

series

The e x t e n s i o n

functions

applying

pertaining

one w h e r e

Hardy classes

meromorphic

is

P

and t h e r e u p o n

of the c l a s s i c a l

surface

F = fr~ is the u n i o n

representations

of the class

to the use of

Riemann

each p o i n t

P tions w i t h d o m a i n

Thanks

S such that

closed

of v a r i o u s

[17].

F ~ Lp[O,2~]

and

-

The conjugate

series

t h e o r e m of M. Riesz

series of its F o u r i e r that there e x i s t s

53

series

a positive

-

states

is the F o u r i e r number

C

that

for e a c h

such

series of a f u n c t i o n

such that

F

the c o n j u g a t e

{ ~ L [0,2~] P

for all a l l o w e d

F

and

we h a v e

({) ,( c ~ (~). The

i n f i m u m of such

general

p

Here

theorem

concerning

Theorem

the Riesz

is e q u i v a l e n t

Hp(A)

II

is t a k e n

1

(M. Riesz

il

is t e r m e d

constant

associated

with

p.

Its value

for

has not b e e n d e t e r m i n e d .

The above theorem

C

to

(and is e a s i l y d e r i v e d

w h i c h w i l l be e s t a b l i s h e d

in the sense of w

H

theorem):

Ch. I j w i t h

Given

I < p

from)

b y the a r g u m e n t

the f o l l o w i n g

of P . S t e i n

S = A

and

q = O.

< + ~.

Let

u

[35].

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

P harmonic

f u n c t i o n on

majorant. f(O)

Let

= u(0).

there e x i s t s

f Then

A

such that

lul p

b e the u n i q u e

analytic

for all

u,

a positive

such

number

C

the

B e f o r e we assures on

A

us that and that

are e x a c t l y

is s u b h a r m o n i c

function function

on f

A

on

A)

satisfyinq

is a m e m b e r

has a harmonic Ref = u

of

H

and

(A). F u r t h e r P

such that

iffli

for all a l l o w e d

(which

.< C[(Mlul p) (0) ]I/p

u.

turn to the p r o o f we c o m m e n t u

is the d i f f e r e n c e

Mlul p

of quasi-bounded

is q u a s i - b o u n d e d .

the f u n c t i o n s

that the t h e o r e m

non-negative

It is r e a d i l y

g i v e n b y the P o i s s o n - L e b e s g u e

~-~

U(8)k(8,z)de,

of S z e g 6 - S o l o m e n t s e v

Izl

concluded

harmonic

functions

that the a l l o w e d

u

integrals

< I.

o

where

U e Lp[0,2~]

a n d is r e a l - v a l u e d .

present

question with

the s t u d i e s

Theorem

1 a n d the M. R i e s z

These remarks

of C h a p t e r

conjugate

series

s h o w the c l o s e

II and i n d i c a t e theorem.

relation

a connecting

of the

link b e t w e e n

-

Proofs u

that are n o n - n e g a t i v e

c

-

The p r o o f w i l l be c a r r i e d out in three stages.

consider only let

54

u

taking

and

p

In the first w e c o n s i d e r

further r e s t r i c t e d b y the r e q u i r e m e n t

s t r i c t l y p o s i t i v e values,

the e x c l u d e d case b e i n g

denote a real number w h o s e v a l u e w i l l be r e s t r i c t e d

We introduce

the a u x i l i a r y

p ~ 2.

allowed Here w e

trivial.

We

in the c o u r s e of the argument,

function

e = Ifl p - cu p

(this is one of the k e y steps of the a r g u m e n t of P~ Stein)

and o b t a i n b y e l e m e n t a r y

calculation

~ e = 4ezz = p

= p

Ifl p

If'12[Ifl p-2

~< p 2 1 f ' 1 2 u P - 2 [ 1

We take

c = p/(p-1)

- p(p-1)cuP-21fll

and o b s e r v e

that

(1 -

)cu p-2]

- (1 - p)C]~

e

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

4.

on

Consequently

Ifl p

has as a s u p e r h a r m o n i c m a j o r a n t

e + p

M(u p)

and lJfllp ~

< ~P

The t h e o r e m

is e s t a b l i s h e d

e(O) + ~ - 1 M(uP)(o)

M(uP)(o)

for a l l o w e d n o n - n e g a t i v e

We n o w s h o w that the r e q u i r e m e n t condition

I ~ p ~ 2

the a n a l y t i c

persisting.

function on

~

that

u

u

when

be non-negative

To that end w e i n t r o d u c e

w i t h real p a r t

Uk

1 < p ~ 2.

satisfying

N o t i n g that lul p = (u+) p + (-u-) p,

m a y be dropped,

U1 =

ek(o)

M(u +),

U 2 =

= uk(o),

the

M(-u-)

k = 1,2.

, Fk

- 55 -

we c o n c l u d e w i t h the aid of

Th.

2, Ch. II that

Up

and

Up

have h a r m o n i c m a j o r a n t s

and that

M(,u, p) -- M(u ) +

Using the r e s u l t of the p r e c e e d i n g conclude

p a r a g r a p h w e see that

f = F 1 - F 2 ~ Hp(~).

We n o w

that

llfoP 4 ("Flll + DF2")P

~( 2P-l(,Flll p + ~F211P )

.< 2 p-I p M(lul p) (0) . p-1

The a s s e r t i o n of the first sentence of this p a r a g r a p h

There remains r e s p e c t to sign.

the final stage w h e r e

is established.

2 < p < + ~

and

u

is not r e s t r i c t e d w i t h

It will be r e f e r r e d b a c k to the case w h e r e

p

is r e p l a c e d b y

p/(p-l)

w i t h the a i d of the C a u c h y integral

f o r m u l a and w e l l - k n o w n p r o p e r t i e s

of linear

tionals on

denote the i m a g i n a r y part of

We c o n s i d e r

polynomial

L

spaces.

P

We let

function w h o s e

real p a r t b y

U.

v

i m a g i n a r y part

V

takes the value

Let 0 ( r < I. By the C a u c h y integral

0

f. at

O.

funca

We d e n o t e its

formula we o b t a i n

~

u(reiS) V(e i8) d 8 =

v(reie) U(e i8) d8

and t h e r e u p o n w i t h the aid of the H~Ider i n e q u a l i t y and the r e s u l t s of the p r e c e d i n g p a r a g r a p h we c o n c l u d e

that

v(reie) U(ei8) de

.~

~

u (re ie) i

-

56

-

[M( lu ip) (o) ]I/p c[~(lulpP-_i)(o) ]~-1 , where

C

is independent of allowed

u

and

U.

We now conclude using well-known facts

L

spaces and the possibility of approxi-

concerning the norm of linear functionals on mating members of

Lp/(p_l)[O,2n]

in the mean of order

II~_K~?o~iV(reiS) iPd8

p/(p-1)

by allowed

U(e 18) that

I/p

4 C[M(Iul p) (0) ]!/p Thereupon we conclude with the aid of the Minkowski inequality that

(1

~--~oElf(re i8) IPd8

It follows that

f ~ H (A) P

+ C)PM(Iul p) (0)

m

and that flfflP ~ (1 + c)PM(Iul p) (0).

Theorem I is now established. It is to be remarked that the exclusion of the cases not accidental.

The

p = 1

and

p = + ~

was

theorem does not hold for these cases.

Our object in introducing Theorem 1 is to prepare the way for the representation of a complex-valued member of

Lp[C(Osl)], 1 < p < + ~

boundary functions belonging to

Hp(A)

ference with center

r.

satisfying

a,

radius

Izl > r -1. ]

and

A(~r)

,

Hp[A(~I1)]. is the set of

as the sum p.p. of Fatou [C(asr) z

denotes the circum-

in the extended plane

We begin with a uniqueness theorem which is valid in the

H1

setting. Theorem 2: C(Osl),

Let

then

f ~ HI(A), g ~ H I [ A ( ~ I ) ] , g(~) = O. f

and

g

I_~f f*(z) + g*(z) = Op.p.

are identically zero.

This result may be arrived at almost immediately on noting that

on

-

57

-

~

iKf*(eie) ekiSde = O, k = 1,2,

.

,

and ~20Hg*(e i8) ekiSd8 = O, k = 0,-I,-2,

so that all the Fourier coefficients of of the hypothesis of the theorem.

f*(e i8)

and

....

g*(e i8)

are

0

as a consequence

However it is desirable to have a proof that extends

conveniently to the Riemann surface situation, which is our principal concern in this chapter.

To that end we proceed by showing that there exists a function analytic on the

extended plane whose restriction to

~

is

f

and whose restriction to

an assertion from which the theorem follows at once.

&(~l)

is -g,

We shall use the Cauchy integral

theorem and Cauchy formula for an annulus together with the mean convergence of to

f*(e i8)

as

rTl

and the corresponding

property of

g

exact prototype argument for the Riemann surface situation. in the mean of order

p

of

f(re i8) to

f*(e i8)

f(re i8)

in order to exhibit an We recall the convergence

for

f ~ H (~), I 4 P ( + ~ ! it may P be demonstrated simply with the aid of the Poisson-Lebesgue representation of f. [F. Riesz [31! p.651] showed that this result is valid for all positive 0 < r < 1.

[We shall consider

r

near

1

in the Riemann surface situation.]

1 ~ f*(~) d 2~i C(O~I) ~-z

f(z)-

1 ~ g*(~)d~ 2Ki C(O~1) ~-----~

I 2Hi S

g(~) ~-----zd~

I ~ 2Ki

C(O~r)

1 f 2Ei

f(~) d~ C (O~r) ~-z

is

Let We have

f(~) d~ ~-z

d~ f(~) ~------z

I ~i f C 2

C(O~r -1)

r < Izl < I.

p.]

,

(O~r)

An analogous argument shows that the value of the last line of the display

-g(z), 1 < Izl < r

-1

We

conclude that

f

and

-g

are the restrictions to their

respective domains of a function analytic on the extended plane. It is now clear that if

fk (resp. gk )

satisfies the condition imposed on

f

-

(resp. g), k = 1,2,

p.p. o n

C(OII),

fl = f2

and

We n o w turn to the e x i s t e n c e

Theorem

3-

Let

I < p ( + ~

= O,

= f*(z)

The o r d e r e d pair

Ilfll,

Ugll

remaining

assertions

F ~ Lp[C(OII)],

f , ~p(A),

C

is u n i q u e l y d e t e r m i n e d b 7 these r e q u i r e m e n t s .

such that for e a c h ~ l l o w e d

14 ~

is an e a s y c o n s e q u e n c e

the case of c o m p l e x - v a l u e d

F

d e n o t e the real p a r t of

and

F

iF(eie) ipdo

F

we have

i/p

S = ~ ( ~ l l ) , q = ~.

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

o r d e r e d pair a s s o c i a t e d

then there ~xists

+ g*(z)

(f,g)

Ilgll k( C

is taken r e l a t i v e to

The u n i q u e n e s s

The basic r e s u l t is the following

such that

T h e r e e x i s t s a p o s i t i v e number

Here

aspect of the q u e s t i o n and h e r e r e s t r i c t a t t e n t i o n

(a) I f

F(Z)

(b)

gl = g2"

as is inevitable.

I < p < + ~ .

g ~ Hp[~(~ll) ], g(~)

p.p. o_~_n C(OII).

-

and if in a d d i t i o n

then

to the c a s e w h e r e

58

is a l r e a d y c a r e d for.

The p r o o f of the

of Theorem

We first show that

I, this w

m a y be r e d u c e d to that of r e a l - v a l u e d F2

the i m a g i n a r y part.

in the theorem w i t h

Fk

We let

(k = 1,2).

F.

(fk,gk)

We let

Fl

d e n o t e the

T h e n the o r d e r e d pair

(f,g) g i v e n b y f = fl + if2' g = gl + ig2'

serves (a).

for If

F. C

It is immediate

serves

that

f

and

g

fullfill

in (b) for the real case, w e see that

Ilfll

~. IlflU

+ IIf211

the r e q u i r e m e n t s

imposed

in

-

1 ~-~

4 C

4 2C

and that the c o r r e s p o n d i n g

59

-

I/p + C

IFl(e i8) IPd

~I O

iF(eiO) iPde

inequalities

hold

To treat the case of r e a l - v a l u e d

F

~1

IF 2 (e i O) IPd8

1/p

1/p

for

g.

we associate with

F

the P o i s s o n - L e b e s g u e

integral I

u (z)

and let

w

denote

For the functions

the analytic f

and

Tijo

:

g

function w i t h d o m a i n

one p r o p o s e s

f(z)

= [w(z)

g(z -1)

Izl

< 1.

Since

Theorem

3.

3 is v a l i d

positive harmonic

is constant.

v

= u(O).

lul p

has a h a r m o n i c m a j o r a n t i that

F.

We r e t u r n to the setting of w

that will

furnish e x i s t e n c e

Our

results c o n c e r n i n g

section as w e l l as i n f o r m a t i o n

concerning

unitary

minimal

functions w h i c h will be of use on several occasions.

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

The notion was

to be of fundamental readily verified

Rew = u, w(O)

to check w i t h the aid of T h e o r e m

We recall that a p o s i t i v e h a r m o n i c that w h e n e v e r

satisfying

- w(O) ]/2,

functions.

in the following

< 1,

the functions g i v e n b y

for the case of r e a l - v a l u e d

object is to introduce a p p a r a t u s functions w a n t e d

~

we are a s s u r e d that

It is now routine

Minimal positive harmonic

Izl

+ w(O) ]/2,

= [w(z)

F ~ Lp[C(O~I)],

(H61der inequality).

F(eie)k(8,z)ds,

that

function on

introduced by

importance u ~ Q

function

u

on S

R.S. M a r t i n

S

satisfying

if and o n l y if

v ~ u,

then

v/u

[23] in 1941 and has turned out

in the t h e o r y of h a r m o n i c

is m i n i m a l

is termed m i n i m a l p r o v i d e d

functions, u

cf.

[5].

It is

is an extreme p o i n t of

Q.

-

The

following

Lemma of

1:

lemma

Let

A

positive

each p o i n t n + I

of

A(Q)

Gloss: and

denote

as a starting

a continuous

h a r m o n i 9 functions

extreme

number

serves

is the

points

of

u ~ P, A(cu)

functions

is e n d o w e d

and

f ~

+ k(v).

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

topology

on

with

real-valued

functions

on

dimensional

euclidean

spaces

extreme

points

extreme

point

of

A(Q)

of

Ig(s)

S,

A(Q).

following

k

A

whole

P, the set

number.

Then

of the b a r y c e n t e r

of at m o s t

means

c

that g i v e n

is a d d i t i v e

means

a positive

that g i v e n

that the space of c o n t i n u o u s generated

g

on

S

function

on

b y the sets

real-valued

N(f,K,e)

consist-

satisfying

S,

K

is a c o m p a c t

of

A

is r e f e r r e d

subset

of the space

The c o n t i n u i t y

theorem.

I reduces

point of

of p o i n t s There

Q. in

exists

conditions:

and c o m p a c t

that b y the e l e m e n t a r y each p o i n t 4.p.15],

is a c l o s e d

lemma w a s g i v e n

of Lemma

sequence

to

a positive

m a p of

- f(s) I < e,

is a c o n v e x

[cf.

A(Q)

to the K r e i n - M i l m a n

of

functions

the aid of the K r e i n - M i l m a n

an e x t r e m e

n

homogeneous

the t o p o l o g y

real number.

Q

proof

R n,

We u n d e r s t a n d

real-valued

that

This

into

section.

subset

of

S,

to the E e l a t i v e

P.

On noting

with

of this

homoqeneous t additive

to say that

real-valued

is a p o s i t i v e

S,

is p o s i t i v e

max scK

where

of the d e v e l o p m e n t s

positive

on

= cA(u) ~

= A(u)

ing of the c o n t i n u o u s

point

imaqe w i t h r e s p e c t

k

u, v ~ P, A(u+v) S

-

Q.

To say that

on

60

of

and that

support

theorem

in [14].

[cf.

whose

that e a c h

image

a sequence A(U o) = e!

Q [cf. 22. p.

u ~ Q

22. p.

of functions

in

for each w h o l e

of at m o s t

mapped

by

n + 1

A

into an

130], w e c o n c l u d e

It is r e a d i l y

point

as follows. in

in finite

a p r o o f was g i v e n w i t h o u t

extreme

is dense

sets

Lemma

1

131].

the argument.

This m a y be a c h i e v e d S

the set of

of

of c o n v e x

is the b a r y c e n t e r

In that paper

We indicate

to showing

A(Q)

theory

of c o n t i n u o u s

S. Q,

seen that the

of

A(Q)

is the

Let

(Sk)

be a u n i v a l e n t

Let

e

say

(Uk) ,

number

appeal

be a g i v e n

k, U k + 1

image of

extreme

satisfying is a m e m b e r

point

the of the

-

set

~

of

u ~ Q

in a d d i t i o n

satisfying

A(u)

61

-

= e, u(sj)

., k - I, w h i c h

= Uk(Sj) , j = O,

satisfies

U k + l ( S k)

= max

U(Sk).

u~E k

Clearly,

(uk)

point

of

O

< 1

< t

Q.

possesses To

and

V l , V 2 ~ Q,

number

v(s I)

< max{vl(Sl),

extreme as w e

k

such

Hence

An

positive

of

A(Q)

and we

z --~

ck(8,z),

Theorem only

Let

u

has

if

the p o i n t

of

of

u

t_~o

of

S

takinq

Suppose

F save

b.

We

limit

O

of

then

u

on

of all ~.

(1-t) v I + tv 2

where

If

v I ~ v,

is a l e a s t

But

then

say

i.

Further

condition:

v

is an e x t r e m e

there

Ul+l(Sl)

= m a x u ( s I) u~E 1

is

0

u

that

v

associated

shall

be c o n c e r n e d

harmonic

with

an

function

on

S

= u(a).

the q u e s t i o n situation

with

regions

of the b o u n d a r y is like

functions

~

defined

behavior

the c l a s s i c a l

one

in w

of m i n i m a l for the

are

just

the

functions

harmonic

function

not have

locally

by

of

~.

Then

u

is m i n i m a l

For

u

minimal,

F = fr~.

limit

R e ( z -1)

on

O,

then

in t e r m s

the h a r m o n i c of a s u i t a b l e

if a n d

if

b

is

prolonqation uniformizer

b.

u

show that

at e a c h p o i n t follows.

k(u)

harmonic

a

positive

case

The

does

is q i v e n into

of a m i n i m a l

at all b u t one p o i n t

at w h i c h

of such

number.

be a positive

limit

we

positive

a positive

S - {b}

Proof:

v =

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

special

first

functions

u

O

If

v 1, v 2 ~ E 1 . T h e

section

the m i n i m a l

F

~ v1(sk) ,

the

examine

c

4:

= e.

= A ( v 2) = e.

is the e x i s t e n c e

in t h i s

harmonic

disk where

V(Sk) and

k(v)

follows.

k(Vl)

consequence

see on c o n s i d e r i n g

this Ch.,

as

and

v I = v = v 2.

Henceforth

unit

that

v,

then

v2(sl)}

immediate

point

say

see t h i s we p r o c e e d

whole

violated.

a limit,

To

is p o s i t i v e if of

v F

that

harmonic

is a l s o save

on

a positive

b,

end we make

then use

v

~

and has harmonic

limit

O

function

is p r o p o r t i o n a l

o f the w e l l - k n o w n

to fact

at e a c h p o i n t on u. that

~

of

having The minimality a function

h

-

harmonic for

on

x

{Izl

real,

O

< 1,

Imz

< Ixl

> O}

< I,

admits

h(z)

where

~

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

O

at e a c h p o i n t

A

which

takes

complement t i o n of

O

into 8

v - cu

has

b2

{Izl

the

limit

be d i s t i n c t

let

uk

6 k = 8k({Izl domain point

< 1/2,

6k of

8k((Izl

< x

having

< I/2.

limit

ek(X) , - 1 / 2

we

> O}

that

Taking

there of

F.

into

with

domain

> O})

is the

largest

b y the r e s t r i c t i o n Imz

> O})

if

U[Sk(ei~/2)]

vk at

of

is the

< 1/2

and

Uk(S)

are

subharmonic.

Now

Mu I

is p r o p o r t i o n a l

has

limit

b2

as we

have

limit

at 0

at e i t h e r

constant

O.

at

Contradiction.

b 2.

Of course, assertion

Hence

there of the

Unitary

u

see on n o t i n g

bI

or

b 2,

is a c o n s t a n t We c o n c l u d e

is a p o i n t theorem

functions.

of

F

u

to

- Vk(S), to

u

that

Then

= O

on

domain

a n d the

the r e s t r i c c

such that

O

has

limit

O

described

in

images.

function limit

O

harmonic then

at

with

O

at e a c h

vk

has The of

since

6k O

at

functions u.

uk

Also u

Mu 1

does

O

at all b u t o n e p o i n t

of

limit

so h a s

O.

not

is n o t the limit

not have

and

on

limit

that uI

of

~(x),

function

S~ppose

Mu I

b1

at e a c h p o i n t

limit

is p o s i t i v e

Let

to

b y the m i n i m a l i t y

of

~.

disjoint

s ~ 6 k, k = 1,2.

does

~,

number

and having

< ~/2,

that

u

the v a l u e

with

with

harmonic

positive

multiple

u

S

of the t y p e

u I 4 u - u 2.

Mu I

takes

of

v

function

has

positive

at w h i c h

and

the v a l u e

< ~

for

restriction

u - uk

-~/2

8

and having

6k

takes

smallest

< x

O

= u(s)

harmonic

whose

that

~k(ei~/2),

u

non-negative

and which

verified

limxh

v = cu.

k = 1,2, ~

and

a subset

be u n i f o r m i z e r s

bk,

~

a positive

Hence

8k

O

on

into

exists

of

function

fact,

semicircle

see on c o m p o s i n g

Let

satisfies

form

a uniformizer

positive

F.

of the

and

is h a r m o n i c

is a m i n i m a l

It is r e a d i l y In

hI

upper

at e a c h p o i n t

= 1/2,

- 6 k, k = 1,2. -1/2

S - ~,

values

+ hl(Z)

and

]-1,1[.

taking

Imz

dominated

number

of

u

positive

a representation

the o p e n

points

the

takes

b,

O

paragraph

denote

real

< 1~ Imz

now that

the p r e c e e d i n g We

a subset

-

= R e ( i ~ z -1)

interval

into

to

Suppose and

of the

which

62

The

F.

last

is i m m e d i a t e .

We recall

that

the n o t i o n

of a u n i t a r y

function

was

defined

-

in w

this Ch..]

setting.

We fix

We let

T

a ~ ~

denote

T(z)

existence

of a m i n i m a l

section.

By the t r i v i a l i t y

of T h e o r e m f

we see that

meromorphic

analytic T

4

o f

on

S

elsewhere,

is u n i t a r y

Suppose that

= z-l, z+l

harmonic

of the a b e l i a n

2(u

o 8) zdZ,

and such that

harmonic

is a basis.

differential

being

on

on

in the sense of

functions

vanishing

on

u

of the

~

satisfying

be trivial. u

on

~

We are a s s u r e d

by earlier

homology

a simple

group

pole

s ~ F - {b}.

~.

~

Wk(U)

Q

results

of the of this

and the i n f o r m a t i o n to

~

o f a function

at some p o i n t b It is i m m e d i a t e

of

F,

that

group

denote

locally

of

~

is not trivial

the p e r i o d

in terms

associated

of u n i f o r m i z e r s

and

with

8

by

The m a p

=

(Wl(U),

I.

. .,win(u))

We c o n c l u d e

which

is the

and s a t i s f i e s

~k(U)

paragraph

assuring

homology ~k(U)

given

Lemma

of

of the p r e c e d i n g

goes

that

u

that

since

sum of at m o s t = O, k = I,

over w i t h

the c o n s t a n t m + 1

. . .,m.

obvious

1 belongs

minimal The

positive

final part of

modifications,

is the real p a r t of an a n a l y t i c

the

function

~.

Interpolation the

= O,

We let 6u

harmonic

is a m e m b e r

the a r g u m e n t

having

Ref(s)

u ~ ~(u)

to Q, there

of

function

this Ch.),

in the p r e s e n t

and not constant.

Yk

u

Q

is the real p a r t of the r e s t r i c t i o n

(of w

'Ym

is a d m i s s i b l e

group

n o w that the o n e - d i m e n s i o n a l

Yl'

for

z / ~.

of the o n e - d i m e n s i o n a l u

point

transformation

homology

positive

-

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

the M 6 b i u s

Let the o n e - d i m e n s i o n a l

63

questions.

Another

application

of the a r g u m e n t s

just e m p l o y e d

is

following:

Let Sl,-..,s n ek(o)

=

sk

f be

be an a n a l y t i c n ( ~ 1)

and let

function

distinct vk

points

be a w h o l e

on

~

of

~.

number,

having Let

modulus 8k

less than one.

be a u n i f o r m i z e r

k = 1,...,n.

The q u e s t i o n

Let

satisfying arises w h e t h e r

-

there

exists

a unitary

function

g

64

satisfying

(g o 8k) (J) (O) =

j = O,...,~k! a useful

approximation

We ing w i t h part

k = 1,...,n.

first note analytic

sk

and n o w

that

plane

T

G

analytic

harmonic

functions

placing

g,

on

(3.1).

components

consisting

Sk,

k

=

1

..... nl

k = 1, .... n. functions

~

see that

Thanks f

and

u

transformation

= 1.

We

let

F = T o f

o G

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

following:

of the i n t e g r a l s

the

of

6u

and i m a g i n a r y

u

as the = 1

k

~k(U), along

parts

of

problem

functions

~

having

real

and s a t i s f y i n g from the

o n t o the r i g h t - h a l f

number

of m i n i m a l

replacing ~

f

A(u)

has

~

G

re-

function

g

m + 2n + 2Zv k

k = 1 .... ,m! U(Sk), in

positive

and

of a u n i t a r y

where

2(u

of associat-

functions

fixed paths

sum of a finite

k = I, .... n!

joining

a

to

~ 8k) z(J) (O) , j = O, .... vk - 1,

number

of m i n i m a l

positive

harmonic

and

= A ( R e F),

it is seen that

that this r e s u l t

the P i c k - N e v a n l i n n a

interpolation

problem

of R i e m a n n

having

topological

not p o i n t l l k e

furnishes

function G

G

satisfies

satisfying (3.1) w i t h

G(a) F

= 1.

replacing

g.

It is to be r e m a r k e d

Surfaces

see,

and seek to s h o w that there

F

to

a map

is the real p a r t of an a n a l y t i c

replacing

shall

we take a d i s t i n c t

mapping

(3.1) h o l d s w i t h

to this n o r m a l i z a t i o n , G

Indeed,

such that

of the

u(a)

(3.1).

analytic

harmonic

the sum of a finite

T -1

On taking

positive

to

A(U)

we

as we

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

R e G

the real

such that

(3.1)

is a f f i r m a t i v e ,

real p a r t

of m i n i m a l

To that end we

parts

positive

a M~bius

~,

on

satisfying

which

to treat

corresponding

T[f(a)]

for then

the i m a g i n a r y

having

number

denote

and s a t i s f y i n g

exists

it suffices

functions

conditions let

The answer,

(f ~ 8k) (j) (0) ,

tool.

the sum of a finite

interpolating

-

boundary

finite

components~

yields

qualitative

for b o u n d e d

analytic

characteristics

if i n t e r p o l a t i o n

information functions

concerning

in the setting

and n o n - d e g e n e r a t e

conditions,

finite

(i.e.

in number,

are

-

f u l f i l l e d b y some a n a l y t i c same c o n d i t i o n s modulus

has

limit

given Riemann 4.

are

function

65

-

on the s u r f a c e

f u l f i l l e d b y an a n a l y t i c

I at the a d j o i n e d

of m o d u l u s

function

less than one,

of m o d u l u s

p o i n t o f the A l e x a n d r o f f

then the

less than one w h o s e

compactification

of the

surface.

Provisional

decomposition

theorem

for L

(F) ,1 < p

< + ~.

Our p r i n c i p a l

concern

is

P the e x t e n s i o n

of T h e o r e m

3 of this c h a p t e r

to the L

class

associated

with

F appropri-

P a t e l y defined. (Theor~ in w

In this

5) w h i c h w i l l

of this chapter.

setting

put d o w n L

as w e l l

p

section we obtain

a first p r o v i s i o n a l

serve to h e l p d e r i v e We e m p h a s i z e

L

(F)

in this d i r e c t i o n

(Theorem

of this c h a p t e r

8) to be t r e a t e d continues

in the

in w

(F). We

fix

a ~ ~

as its p r o l o n g a t i o n

F where

final v e r s i o n

that the r e m a i n d e r

and let

C~ ~ a

by Schwarzian

denote

the d e r i v a t i v e

is c o n s t r u e d

is t a k e n as the class

Green's

reflexion

1___ ~ a 2~ 0n on

the

theorem

function

to S. We

for ~ w i t h p o l e a

introduce

the m e a s u r e

ds

as the

of c o m p l e x - v a l u e d

inner n o r m a l functions

derivative

which

relative

are m e a s u r a b l e

to ~.

and such

P that the p t h p o w e r without

reference

question for

is not

f ~ L

(F)

of the m o d u l u s to a and the

important

is integrable.

Of course,

the n o t i o n

can be d e f i n e d

i n d u c e d m e a s u r e b y the use of u n i f o r m i z e r s ,

for our p u r p o s e s .

We

fin

p,1

< p

< + ~,

but

and define

the N(f)

by

P N(f)

=

~

IflP0~a d

(4.1)

0n

T h e genus Given the c o n s t a n t

of S w i l l be d e n o t e d b y g.

f meromorphic +co on T w h e n

on a R i e m a n n

surface

f is the c o n s t a n t

T b y the d i v i s o r

0 and otherwise

the

of f,0f, we u n d e r s t a n d

-

function with

domain

T

minimum

o f the i n d i c e s

is n o t

O,

8

O -Laurent

being

assigning k

for w h i c h

of

g

and positive.] pondingly of

T

provided

each point of the g e n u s surfaces If

b

of

T,

of

Given

T

that there T - {b}

T.

0

1

exists

elsewhere.

If

ing s e n t e n c e

holds when

n

account

suppose

compact

constant o_~f ~

then

and w e

Let

n

with respect

to

its v a l u e

with respect

to

9

and

9,

9,

9

of

gl[~(OIr)

meromorphic

0

we u n d e r -

- {0}], r ~

small

is c o r r e s -

of

in number.

for s u r f a c e s

on

T

function

point

is at least as large as the n e g a t i v e finite

T h e y are a b s e n t

of genus

number

n

greater

point,

the c o n c l u s i o n

as t w i c e

the g e n u s

on

T,

at

b

T.

An excellent

n

~(Sk)

9

to

is not

B y the d i s c r i m i n a n t

the r a t i o n a l

p l a n e w h i c h has

which

of Behnke-Sommer[2].

that

of its v a l e n c e .

and is such that

the g e n u s

which we continue

We s u p p o s e

1.

o f the p r e c e d -

of

in the t r e a t i s e

is m e a n t

n

for

than

exceeding

with pole of order

is g i v e n

[9],

the k t h

is said to be a W e i e r s t r a s s

paragraph.

o f the e x t e n d e d s I, .... Sn,

of

f o 8

at

the c o m m o n v a l u e D

by

T, the d i v i s o r

be meromorphic

following

denoted

z

say

~

of

neighborhood

for e a c h w h o l e

points

coefficient

of the

f, a n a l y t i c

is at least as large

denote

at each p o i n t

b

is a W e i e r s t r a s s

in this a n d the let

are

function

o f the t h e o r y o f W e i e r s t r a s s

Discriminant.

0f(b)

points

point,

b

on

a non-constant

a meromorphic

has no p o l e s

w

a point

a n d are p r e s e n t

is n o t a W e i e r s t r a s s there

compact,

exists

the c o m m o n v a l u e

[To be p r e c i s e ,

in some d e l e t e d

and such that

or

= s.

differential

The Weierstrass

of genus

8(0)

kth Laurent coefficient

G i v e n an a b e l i a n

defined.

s ~ T

the k t h O - L a u r e n t

analytic

s t a n d the c o m m o n v a l u e o f the

-

to e a c h p o i n t

a uniformizer,

coefficient

66

function

distinct ~ ~,

taking

as

pre-images

k = 1,...,n,

the

square o f d e t { [ ~ ( S k ) ] J - 1 }.

The

following

exists

a meromorphic

taking

distinct

on

T - {b}

fact w i l l be u s e f u l z function

values

at

c

on and

w h i c h has a zero at

a i d o f the W e i e r s t r a s s

T

gap t h e o r e m

having d.

c

Given b,c,d a pole

Indeed,

there

and no others.

at

distinct b

exists This

Points

but nowhere a function

of

T, there

else, f

and

analytic

fact m a y be s h o w n w i t h

[2] and the use of a p p r o p r i a t e

generating

the

harmonic

-

functions

or a l t e r n a t i v e l y w i t h

constitutes on

T

an a p p e a l

having poles

the g e n u s o f the c o n s t a n t

T,

b

results.

There

of respective

orders

w h i c h are e l s e w h e r e

zero.

-

the a i d o f the t h e o r e m

to d e e p e r

at

67

It f o l l o w s

are m e r o m o r p h i c n, n + 1,

analytic.

from s t a n d a r d

of Behnke

and Stein functions

where

It is c l a s s i c a l

considerations

[3], w h i c h

n

~

and

exceeds

that

twice

D

[~]

is not

o f poles,

and

~o

that

n-I = >

o

k

o

where

the

~

are m e r o m o r p h i c

(resp.

~o ) d e n o t e s

D

is n o t the c o n s t a n t

[~]

of the e l e m e n t a r y

C

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

functions

we

such

~

~

and have

to

of

~

~).

Indeed

the

this a s s e r t i o n w i t h

equations.

same p r i n c i p a l

m a y be so c h o s e n

(resp.

us to e s t a b l i s h

of linear

w i t h the

a finite n u m b e r

T - {b}

zero p e r m i t s

theory of systems

by rational see that

on

On a p p r o x i m a t i n g

parts

as

~

fact that the aid

the

at e a c h p o i n t of

C,

that

n-1 > (Bk

o ~) ~ k

o w h i c h h a s a p o l e at m o s t function

o f the d e s i r e d

We r e t u r n b ~ S - ~

Lemma

2:

to

S

There Du[W]

exists

We p u t a s i d e

is

A t all events,

let

values

at

c

and

d

and so is a

type.

and

fix a n o n - c o n s t a n t

w

meromorphic

unitary

function

on

S

hav!n~ 9 pol 9 at

the v a l u e

0

at an Z p o i n t ~

the t r i v i a l

case w h e r e

the

set o f

else and such that

D u [w]

z ~ ~

with respect but nowhere morphic

takes distinct

d o e s n o t take

Proof:

nowhere

b,

u

on

S.

We fix

b

but nowhere

and s h o w

such that

1.

at

- {u(b)} to

u.

be c h o s e n There

else w h i c h

function

on

S

w

exists

it h a s

a meromorphic

takes distinct having

on

S

is n o t the c o n s t a n t

so t h a t

values

a p o l e at

b

C(0~I).

the c o m m o n v a l u e

meromorphic

n

at the

s k.

but nowhere

o f the v a l e n c e a pole

at

is n o t empty.

distinct

function

n

having 0

on

else

preimages S

there

else taking

u

but

To see this

s I ..... s n

having

Indeed,

b

of

a p o l e at

b

is a m e r o -

respectively

the

-

values

1

and

O

at two g i v e n

to the d e v e l o p m e n t s

before

functions

a function

we o b t a i n

the value

0

b.

a linear

Taking

at the others

coefficients,

one

of a m e r o m o r p h i c values takes

at the

s k.

a non-zero

Du[W]

on

value

at

D

[w]

on

at e a c h p o i n t

that of

belonging There

meromorphic

{r ~ to

exist on

Izl

having

We show that a n o n - z e r o

S

having

Let

w

we o b t a i n

sum of the m u l t i p l i c i t i e s Our o b j e c t

is

D u [ W ~ + t~] number

e

has a zero so small

in

a pole

at

exist

a pole

t

we achieve at

at

b

w h e n we {r 4 Izl

of such z

and

exactly

at

with distinct

and taking with

distinct respect

to u

zero.

but n o w h e r e

v(w) ,

thanks

the c o n s t r u c t i o n

function

b

< r

< 1,

else and such

the sum of the m u l t i p l i c i t i e s

w

minimizing

so that

Let

Du[Wo]

~(w).

such that

but n o w h e r e

is a n a l y t i c

of the zeros

~ ~ C[0~1]

Wo(C ) = Wo(d ) .

m a y be so c h o s e n

of

be a zero of We introduce

else and such that

~(c)

~

~(d).

that

< ~(Wo),

At the least we k n o w of

of

and has a pole

is not the c o n s t a n t

P(Wo).

b

number

that w h e n

D u [ W O + t~]

show that a small

in t

t

is small,

{r ~ Izl ~ r -1} m a y be so c h o s e n

4 r-l} not of m o d u l u s

1.

the

is

~(Wo).

that

We take a p o s i t i v e

that

A(r

and that there

preimage

just o b t a i n e d

z

exactly

c, d ~ u-l({~})

of the zeros

w i l l be a c h i e v e d

of

u,

products

and the sum of the m u l t i p l i c i t i e s

~ r -I}

a contradiction.

S

to

~(w o) = O.

r, 0

V(Wo + t~) whence

on

be an allowed

o

that

We fix

complex

at a g i v e n

zero we i n t r o d u c e

b y showing

distinct

S,

a pole

respect

By taking

of the so c o n s t r u c t e d

on

Izl 4 r-l}

with

of the type

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

> O.

{r ~

1

z 2.

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

C(0~1).

v(w O)

of Lemma

the v a l u e

having

meromorphic

u

of

for each p r e i m a g e

S

z

preimages

of functions

is not the c o n s t a n t

Suppose

Du[Wo].

which

function

2 w i l l be e s t a b l i s h e d

Du[Wo]

taking

The d i s c r i m i n a n t

w

of the zeros of Lemma

such

-

introduction

combination

function

For e a c h that

the

distinct

68

local

analytic

r {r < Izl

inverses

of

< r -i}

u,

namely

~

and

T,

with

domain

-

a(~IQ)

which

zero.

satisfy

a(~)

= c,

T(~)

It is to be o b s e r v e d

that

u

69

-

= d,

and are

such that

has multiplicity

1

~ o T -- ~ o e

at e a c h p o i n t

of

has

no

F. T h e

function W (D

a zero

~,

at

but

-- W

O"

T

o

O

----

~o

has

o

O

it is n o t

o

T

--

~

o

O"

the c o n s t a n t

zero.

TheSe

exist

arbitrarily

small

t

in

w[A(~Q)] - ~[A(~Q) Let

t

be a p o i n t

of

(4.2)

and

suppose

that

Wo[a(z)] + t e [ a ( z ) ] it

follows

t = ~(z).

(4.2)

Then

Izl ~ 1.

Since

= Wo[T(z) ] + t ~ [ T ( z ) ] ,

that

D u [ W O + t~](z)

The proof

n C(OI1)].

o f the

Boundary

lemma

follows

behavior

on t a k i n g

of members

of

= O.

t

H

in the

(~).

We

set

(4.2)

sufficiently

start with

small.

some p r e l i m i n a r i e s

and

P note

that

if

and

F 2 ~ Hp[A(~Ir~l)],

decomposition F(z) The We

= Fl(Z) case

introduce

h2

of

F

< Izl

< r2}) , 0

where

z F1

+ F2(z),

p = + ~

presentation and

F ~ Hp({r I

rI

F1

analytic < Izl

< r

of the

is h a r m o n i c

< r 2,

form on

h(z) {r I

on

< r 2.

is i m m e d i a t e

r,r I

and

h

< Izl

This

< r 2,

where

A

analytic

assertion

not

enter

+ h2(z),

< + ~},

is a s u i t a b l y

then

are the c o m p o n e n t s

a harmonic

is in

F 1 ~ Hp[A(O,r2)] o f the L a u r e n t

on

A(~r~l),

fact valid

in t h e d i s c u s s i o n

majorant where

to c o n c l u d e

4 2P[IF(z) Ip +

4 2Phl(Z)

Izl

< r 2 < + ~,

~(OIr2) , F 2

= hl(Z)

IFI(z)P

r 4

F2

and will

and

< ri

of

hI

IFIP~

when which We use

is h a r m o n i c

on

F2(~) 0

= O,

< p 4 + ~-

follows. a re~ ( O ~ r 2)

that

IF2(z)IP]

+ A,

chosen

positive

number.

It f o l l o w s

b y the

sub-

-

h a r m o n i c i t y of that

F

O < p

< + ~

to

IFIIP

possesses

r2

that

-

IFIIP ~ 2Ph i + A.

Hence

Fatou b o u n d a r y values p.p. on F(re i8)

that

C(Olr2).

thanks to the fact that the c o r r e s p o n d i n g

On i n t r o d u c i n g S

mapping

onto

of modulus

less than

of

composed with

Hp(~)

that a m e m b e r of

I

H (~) P

stipulation We fix

w

b

that

restricted

~

b

~(b)

over

C

the r e q u i r e m e n t s

and e l s e w h e r e is

g + 1.

to

be a n o n - W e i e r s t r a s s

functions on

S

~

b

F*(r2ei8 )

to

statement

holds

for

~,

8

of an a n n u l u s

F

and m a p p i n g p o i n t s

< Izl

as

for

r

tends

F1

b y the

~

< Izl

< R-I}

in its d o m a i n

from the fact that a m e m b e r

< I}

function

belongs

e L (F), O P

to

Hp({R

< p

< + ~.

p o i n t of

S

lying in

a(~).

< I})

The

The

p o i n t is a m a t t e r of t e c h n i c a l convenience.

stated in L e m m a 2 o f this section. functions on

O.

S.

The first, ~ ,

The d i m e n s i o n o f

~

and taking the v a l u e 5.

O

at

a.

The n o t a t i o n

-g

as a vector

at

~

b

space

functions on

The spa~e "N"

We intro-

c o n s i s t s of

take v a l u e s not less than

c o n s i s t s of the m e r o m o r p h i c

the p r o o f but not the statement o f T h e o r e m

< Izl

~(~).

whose divisors

not less than

The second,

having a pole at m o s t at

We m a y even c o n c l u d e

p

we c o n c l u d e

is r e p l a c e d b y

duce two vector spaces of m e r o m o r p h i c

and

~,

to be a n o n - W e i e r s t r a s s

satisfying

the m e r o m o r p h i c

of

has a Fatou b o u n d a r y

same c o n c l u s i o n h o l d s w h e n

We n o w fix

a component

into points of 8

We c o n c l u d e

[31, p. 651].

a univalent conformal map

C(OII)

F i e Hp[A(OIr2)].

tends in the m e a n o f order

r e s u l t of F. Riesz c i t e d earlier

into

70

S

w i l l enter

b e l o w is g i v e n b y

(4.1), this Ch..

Theorem

5z

Let

I < p

< + ~.

(a)

If

f e L

(F),

then

P

f(s) = q(s) D.D. on

F

for e x a c t l y one

f2[~(a)]

= O.

Dositive

number

(fl,f2,a)

+ f~(s) + ~(s)

where

fl ~ Hp(~),

Here * refers to the F a t o u b o u n d a r y C

such that for e a c h

f ~ Lp(F),

~(q) ,N(f~),N(z,r)

f2 E H p [ u ( ~ ) ] , G

function. we have

,~ cN(f).

(b)

E ~,fl(a)=0

T h e r e exists a

-

The e x i s t e n c e (4.3) made

for

f ~ L

p

part

(F)

along

other

Uniqueness.

f

lines

then

on

F,

if

zero on their follows.

{Q < Izl

we c o n c l u d e

< p-l}

of T h e o r e m

into

proof uses

decomposition

to w h o m

with

S

f

is the c o n s t a n t

respective

domains.

+ f~(s)

the aid of u n i v a l e n t

that

if

fl

zero since

Hence

fl

is the c o n s t a n t

zero.

of

-u

which

also

is the c o n s t a n t

zero and so

We recall has d o m a i n

determined

that F.

functions

n

there

Further exists

A ~ ..... An_ i

F,

The u n i q u e n e s s

f2

maps

The q u e s t i o n

was

then

fl,f2

and

of

(fl,f2,~)

of an annulus

and the C a u c h y

at

We see that

~(b) f2

and

integral

of a m e m b e r

of

and takes

argument

~

to

u

~I

the value

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

it is e a s i l y v e r i f i e d

a positive

number

cI

that

value

of the v a l e n c e

the p r o o f b y o b s e r v i n g with

n-1 = > ~[u(s) 0

domain

C(O11)

to

O ~2

zero on

that

of u.

there

We

exist

satisfying

][w(s) ] k

(4.3)

A k ~ Lp[C(O;I)],

such that

N[A k

k = O,...,n

above

is the c o m m o n

We start

f(s)

s ~ F.

at the o u t s e t

domains.

Existence.

uniquely

zero on

conformal

it is a n a l y t i c

a.

f

m y thanks.

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

at

that

~

+ ~ (s) = 0

of the type d e s c r i b e d

w h i c h m u s t be the c o n s t a n t

suppose

I express

of the

to start w i t h

From

2, this Chapter,

their r e s p e c t i v e

the suggestion,

in [17].

q(s) p.p.

-

the Riesz

R. N a r a s i m h a n ,

We show that

are the c o n s t a n t for g i v e n

following

and to m a k e

to me b y P r o f e s s o r

treated

of the

71

k = O, .... n - I

for all a l l o w e d

f

and that

we have

(4.4)

o (ulF) ] 4 ClN(f) ,

- I.

Suppose

that

F ~ LI[C(OII)]

j

2~ O

and takes

e) F(e I d e

non-negative

values.

We seek to relate

-

72

-

and ~F

ds.

F o (ulr)~ ~n

We observe that because of the special conditions

fulfilled by a unitary function we

have log

where

w(1),...,w(n)

n

1

(u) - ~(k)

are points of

~. There exist positive numbers

c2

and

c3

such that C2~a(t) k = 1,...,n,

for

t

in

~

near

~

F.

~w(k)(t)

~< c 3 ~a(t),

We note that

F (e i e) d e =

F o (ulr) Z b ~ w ( k ) bn

~0

ds

,

and hence conclude with the aid of the inequality of the preceding o ( u l F )b ~ a 0n

c 2 fF L

ds 4

~2 ~

sentence that

" e) F(e I de

O (4.5)

~< c 3

~

uJF

F o (ulF) b ~ a bn

ds ,

We are now in a position to obtain the desired representation Ak, 1

and

Ak, 2

denote the components of the M. Riesz decomposition

the sense of Theorem associated

f

and

3, this Ch.jwith Ak, 2

is the associated

and the existence of a bound ization.

Ak

C

taking the r81e of g.

for of

F, Ak, 1

f. Ak~

We let I~

is the

It suffices to prove the decomposition

for the case where

N(f)

= I.

We make this normal-

We introduce

~1

and note that

~1 e Hp(~)

" s ~ zAk,l[U(S)][w(s)]

k,

s e ~,

and that there exists a positive number

dI

independent

-

of a l l o w e d

f

such that

Continuing,

N(~[)

The

function

~2

satisfies

-

4 d i-

we next introduce

~2(s)

73

~2'

the m e r o m o r p h i c

= ZAk,2[u(s)][w(s)]k,

function on

s ~ a(~)

~(~)

satisfying

- {b}.

an i n e q u a l i t y of the form

I~2(s) ip ,( d 2 h ( s ) e x p [ - p ~ T a ( b ) (s) ]

where

d2

is a p o s i t i v e number

order of the pole of satisfies

h[a(a)]

F i n a l l y we

w

at

independent

b,

and

G

r ~,~

introduce

It is to be o b s e r v e d that

A representation

o

f.

~2 f.

taking the v a l u e s

respective

and

+ ~(s)] = O.

of allowed

of f.

Go

On introducing

>g, G

o

at

and

(4.6)

a basis b

~

~

for each integer satisfying for ~

and a b a s i s

(4.6)

and

G

F

k

as

in-

the

O(O) = b,

k th

is b o u n d e d

w h o s e m e m b e r s have for

~

w h o s e m e m b e r s have entering

in terms of the b a s i s e l e m e n t s

The a s s ~ r t e d b o u n d e d n e s s

It is n o w c o n c l u d e d w i t h the aid of

as

are b o u n d e d on

we see that the c o e f f i c i e n t s

and

fl

~ = ~2 - (~+ao) la(n)'

a uniformizer

O,1,...,-g

representations

where

thanks to

~ 8, 8

of d i s t i n c t orders

independently

a(~)

such that

We note that

Indeed,

of

i n d e p e n d e n t l y of a l l o w e d

b

h a r m o n i c on

is t h e r e b y u n i q u e l y determined.

~ - ~[a(a)]

+ ~[a(a)].

O- Laurent coefficient

~ ~

- [Go(S)

(o ,D) o

as

d e p e n d e n t l y of a l l o w e d

p o l e s at

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

(n-i) times the

of the d e s i r e d k i n d is o b t a i n e d b y taking

~I + (~I~) - ~i(a) , f2

divisors

f, v is

~ i.

lim ~2(s) s~b

Go + ~l(a)

h

of a l l o w e d

(4.6)

p r o p e r t y of

Go

into the

are b o u n d e d

and

~ follows.

that

I~I p ~ H

where

H

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

harmonic

function on

(4.7)

a(~)

such that

H[a(a)]

is

-

bounded

i n d e p e n d e n t l y of a l l o w e d

(fl,f2,G)

satisfies

f.

the c o n d i t i o n s

74

-

Thus we are a s s u r e d that each c o m p o n e n t stated in

(a) of the theorem.

T h e r e r e m a i n s the q u e s t i o n of the e x i s t e n c e

of

- -,- l ~ * ?I P 0 ~ t d s M(I~I p) (t) = 12~

Jr which in

is v a l i d

Hp[~(~)].

non-negative components convergence

for

~ ~ H (~), 0 < p P

The formula harmonic

of

{l~tl

C.

We use the formula

t ~ n

(4.8)

On

< + ~,

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

(4.8) m a y be d e r i v e d as follows.

function on

If

~, w i t h the aid of u n i v a l e n t

< c}, c small and positive,

p r o p e r t y of q u a s i - b o u n d e d

of

harmonic

one for functions

q

is a q u a s i - b o u n d e d

c o n f o r m a l m a p s of the

onto plane annuli,

using the m e a n

functions on an annulus, we c o n c l u d e

from q(t)

where

F(A) = { ~ t

= ~}' ~

= -~

small,

q(t)

We a p p l y this e q u a l i t y to

(

on letting

= ~-~

q*

q = M(I~I p)

ds.

and use the o b s e r v a t i o n

[M(I~I p) -

from w h i c h we c o n c l u d e

that

c i t e d in w

The formula

Ch.

T h a n k s to N(f~) (4.8)

II).

I~[P]0~a on

(~)

~o

M(I~IP) * = I~*I p

(4.8) we c o m c l u d e

AIO, that

for

a(G)

we find that

that the same is true for

(4.8)

follows.

that

l~l(a) l ~ N ( ~ )

N(f~)

N(ulF).

5. The t h e o r e m of Cauchy-Read.

as = o

p.p.(argument

is b o u n d e d above i n d e p e n d e n t l y of a l l o w e d

f.

that

of G ~ r d i n g and H 6 r m a n d e r

4 dI

From

and t h e r e u p o n

(4.7) and the a n a l o g u e of

is s i m i l a r l y b o u n d e d above. Theorem

that

It is n o w immediate

5 n o w follows.

One of the i m p o r t a n t q u e s t i o n s

in the c l a s s i c a l

theory

-

of H a r d y c l a s s e s of

Hp(A).

work

is the c h a r a c t e r i z a t i o n

This q u e s t i o n

of F. and M. Riesz

Riemann

surface

Another

proof

section

we

Theorem

regular

establish

of T h e o r e m

to be t r e a t e d

6

(Read)-

f e Hp(n)

[31].

o f the r e s u l t

shall

decomposition Havinson

with

was resolved

belonqs

analytic

of the Fatou b o u n d a r y for the case w h e r e

border

the r e s u l t s

theorem

Lp(~')

treated

by

bordered

A.H.

b y H. R o y d e n

Read

[30].

[33].

In this

the a i d of the p r o v i s i o n a l

section

and a v e r y u s e f u l

of C a u c h y - R e a d

The F a t o u

of a m e m b e r b y the c l a s s i c a l

for a c o m p a c t

subsequently

of R e a d w i t h

5 of the p r e c e d i n g The

first

function

I ~ p 6 + ~

problem

was

of Read w a s g i v e n

1 ~< p ~< + ~.

to

-

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

below.

Let

75

boundary

Lemma

m a y b e stated

function

f*

M.Riesz

of

as follows.

of a f u n c t i o n

and s a t i s f i e s

S f*~ = O

(5.1)

F

whenever -of -

~.

w

is an a b e l i a n

In the o p p o s i t e

placinq

f*

of a u n i q u e

differen.tial

direction,

for all a l l o w e d functiQn

f ~ H

~, (~).

(on,.some.ieqionof F e Lp (F)

if then If,

F

and

is p.p.

S)

(5.1)

equal

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

F

analytic

is s a t i s f i e d

to t h e

Fatou

at e a c h point with

F

boundary

is f i n i t e - v a l u e d

re-

function

and contin r

P

u o u s t then

Proof=

f u F

That

is continuous.

f* ~ L

(F)

follows

from the d e v e l o p m e n t s

of the next

to the

last p a r a -

P graph

of the p r e c e d i n g

e v e n have

the

section

"Poisson"

Using

w

Ch. II.] the C a u c h y

annulus (5.1)

we c o n c l u d e

that

problem

F

We

the v a l i d i t y

the second

first c o n s i d e r

< + ~.

f*0~t

dF

The case

p = + ~

is obvious.

We

of

of

sensed

ds,

~ ~.

s

bn

f

and the m e a n

is p o s i t i v e l y

lies w i t h

C

1 = 2K |

The u n i q u e n e s s theorem

i ~ p

representation

f(t)

[cf.

when

with

a given

convergence

(5.1)

Fatou

property

for all a l l o w e d

relative

to

~.

boundary o f an ~.

Of course,

H1

function

follows.

function

on an

It is u n d e r s t o o d the b u r d e n

in

of the

assertion.

the case w h e r e

I < p

< + ~.

For t e c h n i c a l

convenience

we

-

take

b

of Theorem u

function

suppose part

harmonic

s

singularity

5 different

at

that

s

on

from

of the present

-

a(a).

S - {s,a(a)},

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

(fl,f2,~)

76

We introduce

having

negative

is the triple

associated

f2

with

positive

singularity F

logarithmic

at

by Theorem

a

~(a).

We

The first

5.

theorem yields

SF havior of

s ~ S - {a(a)},

a normalized

logarithmic

(f~ + o) 6U s = O,

[For the operator

for

6, cf.

w

this Ch.]

S ~ a(n)

- {~(a)}.

By the C a u c h y theorem and the b o u n d a r y be-

we obtain

~F f~bUs = 2~if2(s)

~6u s

= -~ F

for the same admitted

s.

A second application

~F ~6Us =

where

u

is a small p o s i t i v e l y

the function given by

f~Au s

~6U -~y

of the C a u c h y theorem

shows that

S

sensed contour admits harmonic

surrounding

a(b).

We conclude

prolongation

and a posteriori

that

analytic

A

prolongation ~(a)

to

S - {~(b)}.

The so obtained

and has at worst a pole at

cluded by noting

function,

a(b). The behavior

- ~[~(a) ] - ~ 2~i

+ ~-y

u

is a small p o s i t i v e l y

s

at

u(b).

is at least

r

Jy

The m e r o m o r p h i c -g

of

~

takes the value

near

~(b)

O

at

m a y be con-

the formula

a(S)

where

say ~,

and so

The r e p r e s e n t a t i o n

~

sensed contour

G6u

s

,

(5.2)

yt

surrounding

b, and the a n a l y t i c i t y

of

~6u t

s

extension

of

is the constant

(5.2) now shows that

to O.

S

has a divisor whose value We see that

is analytic

at

f2 ~(b)

at

is the constans and so ~

~(b) O.

is constant.

-

The a s s e r t e d

p =

boundary

~.

F ~ LI(F) ,

Here

In this case

yield

the a s s e r t e d

There

Lemma

"Poisson"

result.

e a c h havinq

special

is c o n c l u d e d

parallels

at m o s t

and the fact that,

case w h e r e

from the

the case lemma

Let I,

p = I.

which

trivially,

is continuous, representation.

c a s e o f the P o i s s o n

It w i l l be r e d u c e d

of H a v i n s o n

(Bk)

F

"Poisson"

that of the c l a s s i c a l

f ~ LI(F)-

modulus

representation

In the

the aid of the following

Let

-

follows.

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

3 (Havinson):

o_nn ~,

F

f u F

behavior

remains

with

of

of

the

the d e s i r e d

continuity

the b o u n d a r y

p = + ~

property

77

integral.

to the case

[2 8].

b9 a sequence converqes

of functions

pointwise

on

e

analytic

t__oo B. T h e n

k~

beinq

an analytic

The

lemma

each point the

Bk

and

integrals lemma

of

B,

along

in its

function

F

abelian

follows thanks which {~a

simply

by

permit

= c}, c

small

the factor

f

being

We return is to be o b s e r v e d

is bounded.

of taken

of F,

in

Lp(p_l)_l(F),

Indeed,

of T h e o r e m

2g real h a r m o n i c

that

which

at

behavior

of

by corresponding obtaining I

the

by a

estimates.

there are p a r a l l e l

converge

pointwise,

< + ~.

6 and treat

log(IFl

F

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

1 < p

logl~(t) I - ~-~

exist

along

(2) t h e r e u p o n

of n o r m ~ I

analytic

is analytic

and the b o u n d a r y

but we note

~

there

exists

and

f

F.

in the m e a n of order

and m a k i n g

functions

that there

f

to u s e them,

when

integrals

and p o s i t i v e ,

H (~) P

to the p r o o f

theorem

the

by approximating

at e a c h point

for s e q u e n c e s

its v a l i d i t y

integral

us to r e p l a c e

We shall not have o c c a s i o n results

on some ope n set c o n t a i n i n q

(I) n o t i n g

to the C a u c h y

full g e n e r a l i t y

analytic

differential

on

the r e m a i n i n g ~

+ 1)~t 8n

case:

p = 1.

It

such that

(5.3)

as

functions

on

S - {b}(e.g.

having

-

singularities

of the form

Re(z-k),

uniformizers)

such that the sum of

7 8 -

Im(z -k) , k = 1 ..... g,

log(IF1 + 1)~t

in terms o f suitable

local

as

0n

and a suitable analytic

linear c o m b i n a t i o n of them

(restricted

function w h i c h is a logarithm of such a

T h e r e exists a s e q u e n c e m a x ~IBkl

= 0(1)

and

(~) (Bk)

of functions tends to

established

either b y a P i c k - N e v a n l i n n a

cu k

c

where

p o l a t i o n to

is a p o s i t i v e 1/~

number

on a suitable

1 ~(t)

1/~

~.

to ~ ) The

analytic

is the real part of an

function 1/~

at each p o i n t of

pointwise

on

n.

set of points)

of

k

and

~

such that

This a s s e r t i o n m a y b e

argument which would yield

independent

is bounded.

uk

Bk

of the form

is u n i t a r y

(using inter-

or else b y appeal to the r e p r e s e n t a t i o n

n-1 -~

Aj [u(t) ] [w(t) lj O

The

A

are b o u n d e d

3

number of points o f the same p r i n c i p a l

in a n e i g h b o r h o o d 4.

We m a y a p p r o x i m a t e

parts in

so that the a p p r o x i m a t i n g

of

A

as

functions

Aj

C(OI1)

and have p o l e s at m o s t at a finite

the

A. b y r a t i o n a l functions 3 and o t h e r w i s e a pole at m o s t at ~

are u n i f o r m l y b o u n d e d on

C(OI1).

sequence of functions of the form n-I > Rj [u(t) ] [w(t) ]j 0 furnishes

a sequence o f the d e s i r e d type.

By h y p o t h e s i s we have

~ F(BkW)

for each

Bk

and each a l l o w e d

~.

;m~. w

= O

Hence b y the Lemma of Havinson,

-- O ~

R

having 3 and indeed

The r e s u l t i n g

-

Let to

~

denote the b o u n d e d a n a l y t i c

F/~* p.p. on

Further

F.

~ ~ HI(~)

~

~ Hi(~).

function on

Then the Fatou b o u n d a r y

~

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

function o f

as w e see from the b o u n d e d n e s s

i n e q u a l i t y of the a r i t h m e t i c Hence

7 9 -

and g e o m e t r i c m e a n s

of

~

(5.3)

function equal

is equal to

F p.p..

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

to the integral

entering

of the

in (5.3).

The p r o o f of the theorem is complete.

6.

L (F) d e c o m p o s i t i o n theorem (Final form). Thanks to the theorem of Cauchy-Read, P it is p o s s i b l e to give a m o r e s a t i s f a c t o r y d e c o m p o s i t i o n theorem for L (F). This w i l l P

be a c h i e v e d by first d e t e r m i n i n g L2(F) ,

the o r t h o g o n a l

the terms b e i n g a p p r o p r i a t e l y

thereupon

to

~fF,~

~ G

Inner product.

interpreted,

The terminal

Given

f ~ H2(~)

F, G ~ L2(F)

gonal c o m p l e m e n t be c o n v e n i e n t consider

of

with

H2(~)

to introduce

w e introduce

meromorphic

{~(t) / O} functions

and p r o p o s e

a hyperbolic

is finite. f

ds

a notion g e n e r a l i z i n g T

on

T

H2(~)

and b y applying

w i t h respect

to

this i n f o r m a t i o n

the inner p r o d u c t

i I FG~ = 2-~ u F ?a"

the q u e s t i o n of d e t e r m i n i n g

so c o n s t r u e d w i t h r e s p e c t to

p, 0 < p < + ~,

such that

f*

of

theorem will then be immediate.

1 ~FG~a = ~-{ OF ~n

We "identify"

complement

To that end it will

that of a H a r d y class

Riemann

By the class

L2(F).

surface,

H (T,~) P

and

the ortho-

a

lightly.

a divisor on

we u n d e r s t a n d

We T

the class of

such that

0f + 0 ~ O, and that for some c o m p a c t

K ~ {0(t)

> O},

Ifl (T - K)I p

has a h a r m o n i c majorant.

An equivalent

morphic

T

functions

f

on

definition

H (T,0) is the set of m e r o P

such that

Ifl p e x p [ - p Z 0 ( t ) ~ t

have a h a r m o n i c majorant,

is that

~t

denoting

]

for the m o m e n t Green's

function

for

T

with

-

pole

t.

where

It is u n d e r s t o o d

0(t) ~ O.

Theorem

that

We r e t u r n

7: The o r t h o q o n a l

80

-

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

to the q u e s t i o n

complement

of

definitions

under

H2(~)

H2[~(n) '0b

are m a d e

consideration

with

respect

at the points

t

and s h o w

to

L2(F)

is

(6.1)

Is(n) ]. ~a

Here the m e m b e r s

Proof:

Given

of

(6.1)

f (H2(~)

are

and

identified

~

a member

behavior (6.1)

of

abelian

f

and

is c o n t a i n e d

member

in the o r t h o g o n a l

in the o p p o s i t e

of

and show that

H2(~)

that there

containing

~.

there

exists

point

of

exists

Indeed,

a function

S - {b}

differential

on

on

if

(6.1), we

integral

~

serving

"is a m e m b e r

b ~ S,

to c o m p e n s a t e

s - {b},

an a n a l y t i c

b ~ ~(n)

and o b t a i n

abelian

of

functions

analytic

of"

on

S - {b} zeros.

which

the zeros

w

o

with

= O,

respect

~ ~ L2(F)

and

and poles on

The t h e o r e m

differential

has a simple

integer

zero

powers

of the a b e l i a n

of the d e s i r e d

free type.

--

~

~ia

=

using

o

or w h a t

set that

an a b e l i a n

of such a n a l y t i c differential

from zeros. Thanks

in

We take

to the h y p o t h e s i s

O

of C a u c h y - R e a d ,

follow.

at an a s s i g n e d

JF

in the t h e o r e m

will

to each

gap t h e o r e m

~L~,O

allowed

L2(F).

on some open

b y the W e i e r s t r a s s

S - {b}

to

so that

orthogonal

We form b y m u l t i p l i c a t i o n ,

differential

a differential



(6.1).

abelian

we are a s s u r e d

that

of the b o u n d a r y

4,

for all

as above.

see that

H2(~)

we c o n s i d e r

not the zero d i f f e r e n t i a l ,

functions

theorem

complement

a zero-free

analytic

Fatou b o u n d a r y

We find as a c o n s e q u e n c e

direction

b u t has no other S,

~.

and the C a u c h y

To p r o c e e d

We note

on

differential

~

of

their

o a6 ~ ' a

f$

is an analytic

with

is the same,

-

B y the C a u c h y - R e a d is equal to

~6 ~ a / ~ o

p.p. on

F.

(6.1) p.p.

An application.

Given

We conclude pn

$ ~ ~

~(t)

that

F

for a unique

H2(~)

Since

~

-

theorem there exists a m e m b e r of

function of a m e m b e r of

p.p. on

81

(fl,f2),fl

is a c l o s e d linear

is analytic

F.

~

w h o s e Fatou b o u n d a r y

function

is equal to the FatDu b o u n d a r y follows.

this Ch.) we have

= f[(t)

and

r H2(~)

F,

(6.2)

+ f~(t)

subspace of

analytic

that

The theorem

(of w

at each point of

H2(~)

f2

L2(F),

in (6.1).

appropriate

it follows

strictions

of functions

at each point of

We o b s e r v e

that there exists a p o s i t i v e number

F.

C

that

fl

Let

p,1

It is to be noted conventions and

f2

prevailing.

are b o t h re-

< p < + ~,

be given.

such that

N(q), N(f~) .~ c~(~) for

a ~ ~,

norm d e f i n e d b y P e a s i l y e s t a b l i s h e d w i t h the aid of a b a s i s for ~ members

where

N

is the

(6.3)

L

w h e r e the

ck

are c o m p l e x and

basis c o e f f i c i e n t s

of

G ~ ~

number

of

~.

independent

as a linear c o m b i n a t i o n (6.2).

T h e o r e m 8: where

Indeed,

if

This result is

~l,...,~g+l

are the

are b o u n d e d

The i n e q u a l i t y

of the basis

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

Let

I < p < + ~.

fl ~ Hp(~),

F.

associated

theorem

(a) G i v e n

(6.3)

dN(G)

where

follows on r e p r e s e n t i n g

and d e c o m p o s i n g

d

the

is a p o s i t i v e

a m e m b e r of ~

the b a s i s elements

accord-

in its final form is n o w e a s i l y treated.

F ~ L (F) , P

f2 ~ H p [ a ( n ) , ~ 6 ~ a l a ( n ) ] ,

satisfy

F r o m this i n e q u a l i t y we see that

in m o d u l u s b y

elements

= f~(t)

(b) T h e r e exists a p o s i t i v e (fl,f2)

> O

max ICkl = 1. l~k4g+l

F(t)

p.p. on

this Ch..

of such a basis, we have g+l min N(~--CkGk)__ 1

ing to

(4.1),

there exists

(fl,f2)

unique,

satisfyinq

+ f~(t),

number

C

(6.4)

such that for e a c h

F ~ Lp(F)

the

-

82

N(f~) , N(f~)

The proof (6.2),

is n o w simple.

this Ch..

is r e f e r r e d

the zero c o n s t a n t

on

F,

at each p o i n t

F.

We find that

(b) follows

Theorem for

L

~ )

candidate

from T h e o r e m

5 and

fl

f2

(a)

L2

follows

from T h e o r e m

situation.

are r e s t r i c t i o n s

is o r t h o g o n a l

to itself.

5 and

Thus w h e n

F

of functions Uniqueness

is

analytic

follows.

this Ch..

considered

the o n l y a r b i t r a r y

p a r t of

to the

and

f~

(6.3),

8 is j u s t i f i a b l y

since

~ CN(F).

The e x i s t e n c e

The u n i q u e n e s s

of

-

as a final

element

form of a d e c o m p o s i t i o n

entering

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

theorem

point

a.

P 7.

Linear

functionals

on

H

(~) , I 6 P

< + ~-

Representation

formulas

for b o u n d e d

P linear

linear

functionals

on

H

(~) ,I $ p

< + ~,

m a y be o b t a i n e d

very rapidly with

the

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

extension

be r e f e r r e d

to as "HBBS"

functionals

on

L

(F).

theorem

in c o m p l e x

- and the c l a s s i c a l Thus

if

k

form

F.Riesz

is a b o u n d e d

(Bohnenblust-Sobczyk

representation

linear

functional

for b o u n d e d on

{(f*,~(f)

exists

.- f r H (n)}, P

.~ ~ Lp,,p_l ,/~;

(F)

extend

it b y

(~) ,

linear we

intro-

to

HBBS

L

(F)

and c o n c l u d e

that there

P

such that

(f) =

f ~ H

H

- to

P

P duce

[20])

(7.1)

~r f*$6~a'

(~). P W h e n we c o n s i d e r

vention

of

HBBS

section

a representation

the case w h e r e

b u t using

the F.Riesz

theorem

1 < p

< + ~,

we m a y o b t a i n w i t h o u t

representation

with uniqueness.

and T h e o r e m

Given

F ~ L

the

inter-

8 of the p r e c e d i n g

(F)

we denote

the first

P component that

A o ~

fl

o f the d e c o m p o s i t i o n is a b o u n d e d

linear

of T h e o r e m

functional

on

8 by L

~(F).

(F).

K is a "projection".

We have

(7.1)

We see

again w i t h

some

P ~ Lp/(p_I) (F). ciated

Applying

p a i r and c o n c l u d e

and the C a u c h y

integral

Theorem

with

8 to

~

we i n t r o d u c e

the aid of the b o u n d a r y

theorem

(~I,~2) ,

behavior

the u n i q u e l y

of the e n t e r i n g

asso-

functions

that

A(f)

=

;Ff*~6~a

,

(7.2)

-

f ~ Hp(fl). For

if

Further

~1

is the unique

%U e Hp/(p_l ) (n)

has

83

-

member

the p r o p e r t y

H p / ( p _ l ) (~)

of

having

this property.

that

~r f*~*~a = o, f ~ H

(fl) ,

p

we see on

introducing

w

of the p r e c e d i n g

o

section

that

:~

for all

~

of the C a u c h y - R e a d

a member

of

theorem.

Hence

~*

Hp/(p-1) [~(~) '~6 and b e c a u s e assertion

8.

of this

of Th.

fact we

8, p r e c e d i n g

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

linear

subspace

infer

~

theorem of

H

that

~

is the Fatou b o u n d a r y

function

of

Ta(~) ]

is the c o n s t a n t

O

b y the u n i q u e n e s s

w

for

(~) , 1 < p < + ~. We c o n s i d e r the s m a l l e s t c l o s e d P g e n e r a t e d b y the f a m i l y of functions w h i c h are r e s t r i c -

(~)

H

P tions o f u n i t a r y

Theorem

9:

~=

functions

H

to

fl

and show

(~). P

The p r o o f which

vanishes

on

is e s t a b l i s h e d ~

(Contrapositively, on

~

vanishes

if ~

then

that

on

k

that a b o u n d e d

identically.

~ Hp(~) , there

b u t not i d e n t i c a l l y

Suppose

by showing

For

exists

linear

functional

then HBBS a s s u r e s

a bounded

linear

that

~

functional

on

H (fl) P

= H

(fl). P vanishing

H (~).) P is a b o u n d e d

linear

functional

on

H

(fl)

which

vanishes

P on

~.

whenever and the

We use the r e p r e s e n t a t i o n u

possibility

b y a sequence on ~.

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

(7.1)

of a u n i t a r y

of a p p r o x i m a t i n g

o f such

u

of the p r e c e d i n g

yieldsz

function

an a n a l y t i c

A(B)

= O

to

function

whenever

section. ~,

Since

the lemma

A(u)

= O

of H a v i n s o n

on ~ of m o d u l u s ~ I b o u n d e d l y

B is a b o u n d e d

analytic

function

-

Suppose

n o w that

f ~ H

(~)

but

84

-

is not

the c o n s t a n t

zero.

Introducing

the term

P Q

of

(2.1),

Ch. II, r e l a t i v e

to

loglfl,

MlflP

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

Q*

is equal

Q(t)

Let harmonic by

b

The choice

of the

functions

u

exist

bounded,

h

in terms wk

is not

real

and h a r m o n i c

from the

be a n a l y t i c

on

the real n u m b e r s

ck

is t h e r e b y gously

specified

defined

with

Q(t)

on

S

~

It follows

that

(8.1)

t ~ n.

lying

in

~(~).

Let

singularities

chosen

b

be real

given

locally

k = I, .... g.

W h a t are

such that the p e r i o d

Wkl~

Vk,W k

at

uniformizers,

compelling.

of the

and

(2.2) , Ch. II,

wanted

systems

~H, H

real h a r m o n i c

having

the d e s i r e d

on

are

of the 6u ~.

There

property.

and s a t i s f y g = Q +~-I

function,

uniquely.

,

F.

respectively

systems

are u n i q u e l y

the real p a r t of an a n a l y t i c

of

on

inherently

Vkl~

~

p.p.

of s u i t a b l y

and

loglhl

where

point

that b y

~ Ifl p,

loglf*16~t

and have

for the set of p e r i o d drawn

o (pQ)

logif*l

= ~-~

S - {b}

vk

u I ..... Ug

Let

on

im(z -k)

and

form a basis

to

be a n o n - W e i e r s t r a s s

functions

Re (z -k)

~ exp

we observe

CkUk,

specified

and in a d d i t i o n

For e a c h w h o l e

being

b y the r e q u i r e m e n t

replaced

number

let n

h(a)> the

O.

that The

function

h

loglhl function

n

be h

is analo-

by

min{loglf*l , n } 6 ~ t .

We see that sequence

tending

g r a p h back,

f/h

and e a c h

pointwise

we see that

to

I.

h

n

is b o u n d e d

Referring

and further

to the

last sentence

k [ ( f / h ) h n] = O, n = O,I .....

f*

a

= O,

(hn/h)

of the second para-

or e q u i v a l e n t l y ,

n = O,I . . . . .

is a b o u n d e d

-

B y the

9.

lemma

In t h i s

obtain

section

further

topological enter are

of H a v i n s o n

we return

theorems

= O.

The

to the

of the k i n d

characteristics

into

taken

A(f)

85

-

theorem

follows.

"Toeplitzian"

question

studied

in w

developed

in t h a t

section

but where

of the u n d e r l y i n g

Riemann

surface

as w e l l

the a r g u m e n t .

H e r e we

in the p r e s e n t

context.

take

our

surface

TI II w i l l

to be

refer

to

Q.

H

Ch. the

finite

as n o n - d e g e n e r a c y

The notations

(D)

with

II a n d

ef,~F,V

normalization

point

a.

P The

first

semi-continuous that

asserts

int{~(x)

theorem

to be p r o v e d

functions

on a space

that

for

appeals o f the

such a function

~

to the c l a s s i c a l

second

category

theorem

which

there

exists

a real

maps

H

into

concerning

omit

number

lower

+ co, the o n e c

such

that

~ c} ~ ~.

Theorem

10:

Let

1 ~< p

< + co.

Then

8f

(D)

itself

if a n d o n l y

if

f

P is a n a l y t i c

on

Proof:

is i m m e d i a t e .

"if"

fn ~ H

D

and bounded.

In the o p p o s i t e

(~) , n = O,1, ....

sense we

infer

from the h y p o t h e s i s

that

so t h a t

P f ~ NO-d} -oa < c

< d/2.

reflexion

across

F(c) we are

a n d to be h a r m o n i c We assume

family of such

s

Gc

that

c

at e a c h is so

is b o u n d e d

(4.11) (4.11).

as

thanks c

tends

to the b o u n d e d n e s s to

Our a s s e r t i o n

on

s

{l~a(t) l < d } .

It follows,

fix a

c U1~