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
~(x)
of the
= exp(px),x
form
(2.1)
a given harmonic
product
in fact,
log
(resp.
lhl
r R,~(-~)
of T h e o r e m = O.
non-negative
for
u
function
analytic
(6.3)
5 as a p r e c u r s o r
by a quasi-bounded
of a b o u n d e d
of a B l a s c h k e
function
(c)
harmonic
using
on
4.
Q
function
classical
as the l o g a r i t h m
of a singular
is the term
Now
Using
2.
facts of the
the k n o w n
non-negative
harmonic
of the r e p r e s e n t a t i o n
u.
We now s h o w that (2.1)
of the form
(c) o f T h e o r e m
led to a r e p r e s e n t a t i o n
limit b e h a v i o r
function (2.1)
concern
and the r e p r e s e n t a t i o n
radial
with
H
r2n
1
u
p.p.
in
(d) We h a v e
(Mlfl p) (z) = 2 ~ o
We take
of f u n c t i o n s
is a v a i l a b l e
product.
and
We c o n c l u d e
here given
by
(6.1)
has a h a r m o n i c
(c) and g
(d) m a y be s u b s u m e d
is the n e g a t i v e
the i n t e g r a b i l i t y
with
majorant,
F(8)
=
h ~ H
Theorem
of the l o g a r i t h m
of
log
If*(eie) f. (4).
under
The a s s e r t i o n
The
representation
of the m o d u l u s
ff*(ei8) I.
We c o n c l u d e
2.
The
that
term ff{ ~
(c) follows.
Q
of
fhl.
Now
of a B l a s c h k e (2.1) Since
Mifl p
is ~ o Q
is quasi-
P bounded
by Theorem
negative [I0].
tends
to
0
We c o n c l u d e
We r e m a r k on
2 and the m e a n v a l u e
[O,2E],
as
r ~ ~--we
on
C(Osr)
are u s i n g
that the r e p r e s e n t a t i o n
that
(b) o f T h e o r e m
of
MIfT P -
the r e a s o n i n g
(6.3)
Ifl P,
which
of G a r d i n g
is non-
and H 6 r m a n d e r
holds.
5 generalizes:
If
U
and ~ o U
are
integrable
then
I ~o ]I u( eiS)k( e,z)dS,izl<1, u(z) = ~n is such that ~ o
u
has a h a r m o n i c
majorant
(Jensen
inequality)
and
(~ o u)*
= ~~
p.p.
-
The condition
following that
well-known. general
remarks
If*i
~ L
developments
= exp(qx),
[O,2~],
(2.1)
of
log
of w
~(- ~) is
Ifl
O
< p
< q
and hence by Theorem hardly called
for
in T h e o r e m
to
this Ch.,
= ~I (- ~) log
-
are a p p r o p r i a t e .
It m a y b e p r o v e d b y a p p e a l
~l(X)
that
q
29
lhl
= O,
and
of T h e o r e m f ~ H
(c) and
(b) of T h e o r e m
u = log
f ~ H
q
5
then
we m a y p r o c e e d Ifl,
Q
satisfies
This r e s u l t
5.
the term
the
is
In terms o f the ~(x)
= exp(px) ,
o f the r e p r e s e n t a t i o n
has a h a r m o n i c
To be sure,
function
(~).
q
With
lying at hand.
mediating
also
thus.
~ (loglhl)
(4).
situation
4 the i n t e r v e n i n g
f
< + ~,
rhlq = ~I
4 w e see that
in the o b v i o u s
If
majorant
the u s e o f T h e o r e m However
4 is
it is to be n o t e d
n e e d not b e the u p p e r
envelope
of
as it is here.
7.
An application
the H a r d y class tions on
~,
In fact,
of the e x t e n d e d
H2(~)
f
at m o s t
quasibounded In this which
H2(~)
one.
into
Further
component
of
implies
the r e s u l t
with
such a m a p
m log(I/Ifl)
~
then 8f
such that
f
is n o r m p r e s e r v i n g
= O,or equivalently
as w e l l
analytic forms,
8f : g ~ fg
is an a n a l y t i c
a theorem concerning
just q u o t e d
It is w e l l - k n o w n
thei t h e o r y of T o e p l i t z
function on
itsself,
s e c t i o n w e shall c o n s i d e r
of S z e g 6 - S o l o m e n t s e v .
to b e a r on the s t u d y o f b o u n d e d
in c o n n e c t i o n
is a c o m p l e x - v a l u e d
c l o s e d u n i t b a l l of modulus
m a y be b r o u g h t
in p a r t i c u l a r ,
if
Theorem
the s o - c a l l e d
as a c o r r e s p o n d i n g
< p
< + ~.
Related
A function t a t i o n of the stand
that
analytic
form
S
is t e r m e d a
exp
exp(- ~)
function
will have
on
theorems will be met
is
significance
o u
is PL,
where
O.
A
PL u
PL
in
one
PL for
Chsl. IV and V
is s u b h a r m o n i c
for the t h e o r y of a n a l y t i c
ference w e cite the tract of R a d o on s u b h a r m o n i c
We fix fine
~(G)
non-negative
a ~ S.
For each
as the value reals
at a
PL f u n c t i o n of
and i n t r o d u c e
MG. 0F(G) _
H
G
functions.
= I.
Since
a represenwe under-
the m o d u l u s
concerning
of an
PL f u n c t i o n s
By w a y o f g e n e r a l
re-
[29].
a harmonic majorant
be a m a p o f a
if the
(S),
As t h r o u g h o u t
that r e s u l t s
F
of
functions
that it a d m i t s
S.
possessing
We let = FG,
on
functions
G
~
of these notes.
is s u b h a r m o n i c .
it is to be a n t i c i p a t e d
the
P
function provided
function
on
[12].
if M ( I f l ) 2
i
O
cf.
if and o n l y
if a n d o n l y
func-
maps
functio~
that
S
PL function.
we de-
into the set of X
is to d e n o t e
a
-
subset
of
Theorem fies
{~(G)
6:
F ~
harmonic
(a) 1.
4 I}
containing
I__ff F
(b)
An
majorants
a member
is c o n t i n u o u s F
such that
into itself
30
has
on
not
S
OF
-
the c o n s t a n t
and 0 F ( X )
maps
c X,
the set of
the p r o p e r t y
O.
We s h o w
then
PL
F
i__ss PL
functions
G
and satis-
having
that
(7.1) for all m
such
log(l/F)
G
i__ss O,
i__ss PL,F 4 1, holds
if and o n l y
and
see that
FnG
the H a r n a c k
F
i_~s PL,
or e q u i v a l e n t l y , (7.1)
for all a l l o w e d
Proof:
if
holds
F $ 1,
if and o n l y
for one a l l o w e d
and the q u a s i - b o u n d e d
if
F
G,
i_ss PL
component
an____d MF = 1.
not the c o n s t a n t
zero,
(c)
If
then
of F
(7.1)
G.
(a)
Let
G
~ X
for e a c h n o n - n e g a t i v e
inequality
be a m e m b e r
of
X,
not the c o n s t a n t
whole
number
n.
O.
Let
By i n d u c t i o n
b ~ S, G(b)
we
# O.
By
we h a v e
M(FnG) (b) = 0(1) ,
whence F
we c o n c l u d e
is continuous,
number
n,
that
F(b)
4 I.
Since
the set of a d m i t t e d
we c o n c l u d e
that
F 4 I. Since
FnG
is
b
PL
is dense
on
S
for e a c h p o s i t i v e
and whole
we see that
log F + ~ log G n is s u b h a r m o n i c
on
S
continuity
F,
that
of
for e a c h F
such
n.
We c o n c l u d e
is a PL
function.
b y a limit argument,
The a s s e r t i o n
stipulation
of c o n t i n u i t y
is dropped,
however
requirement
of c o n t i n u i t y
is r e p l a c e d
b y the r e q u i r e m e n t
the
first h a l f
(b) functions set.
of the c o n c l u s i o n
F
is
having
If
OF
constant
1
PL
in
(7.1)
majorants (7.1)
into
majorant itself
for all a l l o w e d
we see that
(a) is not v a l i d
assertion that
is o b t a i n e d F
be a
PL
the if the
if the function,
redundant.
and has a h a r m o n i c
harmonic
satisfies
being
a valid
using
when0F
since G,
the
maps
the c o n s t a n t
then
F 4 1.
set of
1
Taking
PL
belongs G
to the
as the
-
3 1 -
(MF) (a)
and h e n c e b y the m a x i m u m we
see that
since
principle
M1 = I,
= 1
for h a r m o n i c
the q u a s i - b o u n d e d
MF = I.
functions,
component
m
of
B y Th.
log(I/F)
3, this Ch.,
is the c o n s t a n t
zero.
To p r o c e e d quasi-bounded
in the o p p o s i t e
component
in the r e p r e s e n t a t i o n
of
m
(2.1),
direction,
log(I/F) this Ch.,
of
terms
in the r e p r e s e n t a t i o n s
consequently
M(FG)
= MG,
(7.1). M(FG)
true w h e n
G
all
G
thanks
(c) if
G
to
log F
This
has
o
We h a v e conditions, analytic of the
noted
the c o n d i t i o n
following
component
of
manner
on
&
Returning
real
follows
p,
M(FG
o
) = M(G
of
o
)
and
The
equality implies
for an a l l o w e d
F
that
o f this p a r a g r a p h
is the c o n s t a n t
and c o n s e q u e n t l y
of this
component
O, then
F
M ( F p)
This
= I, 0
z e r o that
functions.
the term of
For
Q
relative
m log(I/F)
is
purposes.
< p
We turn We
the e q u i v a l e n c e
o f the
O, M(Ifl 2) = 1,
for
equivalence one=
< + ~.
component
the a i d of T h e o r e m
section.
is
not e x c e e d i n g
then the q u a s i - b o u n d e d
functions.
section
m log(I/Ifl)
at m o s t one.
is
for our p r e s e n t
are equal,
Q
(b).
component
of this
the
so o b t a i n e d
for h a r m o n i c
the q u a s i - b o u n d e d
PL f u n c t i o n s
to a n a l y t i c
Hence
zero.
sentence
log F
principle
first p a r a g r a p h
at once w i t h
first
and the
is e s t a b l i s h e d .
Hence
from
to
zero.
Iog(FG)
implies
to the
relative
of
at the b e g i n n i n g
appropriate
Q
of m o d u l u s
property
Application considered
in the
m log(I/F)
some p o s i t i v e observation
follows
(7.1)
the m a x i m u m
zero.
the q u a s i - b o u n d e d
functions
and
The e q u a l i t y
the s t a t e d p r o p e r t y ,
zero a n d the a s s e r t i o n
log G
1,
see that the term
is the c o n s t a n t
zero.
on u s i n g
is the c o n s t a n t
we
is the c o n s t a n t
2, this Ch.j (b)
follows
zero,
PL, F ~
not the c o n s t a n t
in q u e s t i o n . ]
to Th.
of
is
G
w e see from the fact that the term MF = I,
log F
F
w h e n we c o n s i d e r
[We see incidentR]ly that = MG,
that
is the c o n s t a n t
corresponding
is t r i v i a l l y
given
of
If
is a s p e c i a l
case
If the q u a s i - b o u n d e d M ( F P)
m log(l/F)
= 1
for
is zero.
This
3, this Ch..
to the
start
study of maps
afresh,
defining
of the type 8f
now
in a
-
Let that
Y
0
< p
is a
< + ~.
subset
Let
of the
f,g
32
-
: S ~ C
"closed
unit
and d e f i n e ball"
8f(g)
(relative
to be
to a) of
fg.
We
(S),
containing
H
suppose
P a member
not
(relative
the c o n s t a n t
to a)
zero.
is c o n s t r u e d
[We are
as the
{g
~ H
allowing
all p o s i t i v e
p.
The
closed
unit
ball
set
(S),
~ ( i g f p)
4 I}.]
P We h a v e
the
Theorem
7:
modulus
at m o s t
no p o i n t
following
(a) I_~f 8f(Y)
of
one,
S.
(b)
consequence
of T h e o r e m
r Y,
then
there
exists
fl'
such
that
f
say 8f
easy
is a m a p
of
H
a
unique
differs
(S)
into
6. analytic
from
itself
fl
function
on
S
o_~f
on a set c l u s t e r i n q
at
satisfyinq
P v[18f(g) Ip]
g ~ Hp(S) , f
i.e.
8f
is an a n a l y t i c
component most
one
for all
of
m
and
if
g ~ H
is- a " n o r m " - p r e s e r v i n q
function
on
S
is zero.
(7.2)
for
holds
map
of m o d u l u s
log(I/Ifl)
(c)
some
(7.2)
= v(IglP),
of
H p (S)
at m o s t
I_~f f
into
itself
1 for w h i c h
is a n a l y t i c
g ~ Hp(S) ~ not
on
if a n d o n l y
if
the q u a s i - b o u n d e d S
and of m o d u l u s
the c o n s t a n t
zero,
then
at
(7.2)
holds
(S). P
Proof: that
from
It s u f f i c e s
the a n ~ l y t i c i t y
is a n a l y t i c of
(a)
at e a c h
point
g
and
F = fg
Ifl p, X = {Igl P for
of the c o m p l e m e n t
some
: g ~ Y}
g ~ Y,
of a p a r t
not
and
the
to o b s e r v e
constant
of
S
which
clusters
that
f
is a n a l y t i c
zero,
f
at no p o i n t
S.
(b) On t a k i n g application shows
that
quently 6.
of
to take
(S).
as the c o n s t a n t
(a) of the p r e s e n t
if
when
Hence
g ~ H
of
g
8f 8f
(7.1)
theorem
is " n o r m " - p r e s e r v i n g ,
Hence
for all
allowed
the q u a s i - b o u n d e d
we
F G
see
with
then
is " n o r m " - p r e s e r v i n g , holds
I
Y the c l o s e d f
is of m o d u l u s
satisfies and,
component
unit
at m o s t
the h y p o t h e s e s
afortiori, of
ball
(7.2)
m log(I/tfl)
holds
= O.
P follows
on a p p l y i n g
(c) T h i s
(b) of T h e o r e m
is i m m e d i a t e
when
6 to
F =
Iff p.
(c) of T h e o r e m
6 is a p p l i e d
to
of
F.
on
S.
(relative one.
An to a)
Conse-
(c) o f T h e o r e m for all
The c o n v e r s e
-
We remark that only if
f
ef
is a constant of modulus
f
does not take the value
itself and is "norm"-preserving. constant of modulus one.
-
is a "norm"-preserving
onto itself and is "norm"-preserving, sequently
33
I.
map of
H (S) onto P The "if" is trivial. When
the constant 0
anywhere,
We conclude that
1
belongs to
and further Ifl = 1
itself if and ef
maps
8f[Hp(S)] 01/f
maps
and hence that
H (S) P and con-
Hp(S) onto f
is a
-
A Classification
I. S t a t e m e n t C. W.
of problem
Neville
in 1967:
and result~ If for s o m e
34
-
Chapter
III
Problem
for R i e m a n n
The p,O
following < p
Surfaces
question
( + ~,
was
the class
proposed H
(S)
to m e b y Mr.
has
a non-constant
S?
We
P member, that
does
there
it is n o t
tion,
obtain
relations faces,
the c a s e
a chain
given
cf.
exist
Ch. IV o f
It w i l l
that
of
in the
a non-constant
bounded
the answer
analytic
is a f f i r m a t i v e
strict
inclusion
standard
accounts
relations
function
for a l l
on
S
supplementing
of the classification
shall
and shall, the known
theory
see
in a d d i -
inclusion
of Riemann
sur-
[I].
be convenient
to
introduce
the
following
notations.
By
0
we
shall
P understand
the
set o f R i e m a n n
surfaces
S
for w h i c h
H
(S)
contains
only constant
P members.
Here
and throughout
to m e a n w e a k
inclusion
to d e a l w i t h
strict
inclusion.
in t h e c o u r s e
inclusion
Given
p
( q
that
H
(S)
) H
P
0
< p
of these
we
Since
and we
shall
"c"
shall have
employ
'~"
is a l w a y s frequent to m e a n
taken occasion
strict
from
< x q + 1,
(S)
< + ~.
notes
in t h i s c h a p t e r ,
~ + ~,
xp
we conclude
this chapter
0 ~4 x
and hence
q
0
< + ~,
c 0 P
O = u -p o
0
.
We
introduce
q
q
and = N P
By
OBA
we
shall understand
non-constant
We of our
shall
the
analytic
also consider
classifying
hyperbolic have
bounded
Riemann
chain. surface
0 P
q
set of Riemann
functions.
a "null"
We r e c a l l S
surfaces
Clearly
class which
Op
w i l l be
Lindel6fian
there do not exist
c OBA.
that a non-constant
is t e r m e d
on which
s e e n to a p p e a r
meromorphic
provided
that
at the
function for a l l
lower
f
a ~ S
on a we
en
- 35 -
Zf(s)__ w n(ssW) gs(a)
where
n(slf)
w i t h pole
s [15].
LindelSfian istic.
is the m u l t i p l i c i t y The
stipulation
is e q u i v a l e n t
In special
be L i n d e l ~ f i a n
of
f
at
case o f a n o n - c o n s t a n t
Given
S
hyperbolic
either
are c o n s t a n t
by
LA(S)
LA(S)
either
parabolic
complex
We define
OLA
on
S.
The
is G r e e n ' s meromorphic
function
f
l~glfl
have
that
from
the
function function
for on
Nevanlinna
a harmonic functions
S S
be
character-
the s t i p u l a t i o n
the set of a n a l y t i c
It is o b v i o u s
that
f
majorant. on
S
which
inequality
x ~o,
as the set of R i e m a n n
or else are h y p e r b o l i c
constants
gs
it h a v e b o u n d e d
analytic
we u n d e r s t a n d
or LindelSfian.
~ Hp(S).
and
that
to the c o n d i t i o n
l~g x ~ xP/p, that
s
w ~ f(a)
that a n o n - c o n s t a n t
to the r e q u i r e m e n t
is e q u i v a l e n t
< + ~,
and such that
inclusion,
OLA
( 0
,
surfaces
LA(S)
S
consists
which
exactly
are of the
holds.
P The p r i n c i p a l strict
inclusions,
result
developed
in this c h a p t e r
the first of w h i c h was
established
is s i m p l y
the f o l l o w i n g
chain of
b y Parreau
in his thesis
[26,
p. 90]:
~ No
OLA
Uo
We o b s e r v e which
there
such that
exists
that the
last
inclusion
no n o n - c o n s t a n t
for each p there
exists
bounded
(I.1)
OBA-
asserts
the e x i s t e n c e
analytic
a non-constant
function
member
of a R i e m a n n
but which
of H
surface
nevertheless
(S).Our c o n s t r u c t i o n
S on is
(w
P furnishes
such an
belonging
to
S
for w h i c h
there
exists
a non-constant
analytic
function
on
S
No
Indeed,
the c o n s t r u c t i o n
Ch. II, r e g a r d l e s s
of w
of its rate
shows
that given
of g r o w t h
there
~
exists
satisfying S ( OBA
the c o n d i t i o n
of w
such that there
of
exists
-
f
analytic
on
S, not constant,
one can refine of R i e m a n n monic
the
last
surfaces
majorant,
f
Not m u c h tive p l a n a r
S
and such
inclusion
theory.
of
that
~ o Ifl
null c l a s s e s
is a n a l y t i c
S
f
about
conclude
has a h a r m o n i c
by introducing
with very rapidly
to be k n o w n
We shall
-
(I.I)
such that w h e n
is constant)
appears
36
on
growing
the c o u n t e r p a r t
this c h a p t e r
with
and ~
of
majorant. 0
(the set
~ o Ifl
is not
(1.1)
has a har-
settled.
in the m o r e
some simple
Whether
remarks
restric-
concerning
this question.
2. OLA ~ No
Oq.
We give
an a c c o u n t
on P. J. M y r b e r g ' s
original
which
a non-constant
there
exists
example
theory
example
when
constructions
was
startling
o f the s u b s e q u e n t
We c o n s i d e r for w h i c h
cf.
the a n a l y t i c
[19],
[37])
The c e n t e r points order
map
exactly equal
Riemann
"from
2
surface
Riemann
surface
belongs
to
OBA
is h y p e r b o l i c . shows
sections
that there
into the z-plane"
The analytic
is b a s e d
belonging
to
directly OBA
but on
It is no e x a g g e r a t i o n its impetus
to
to M y r b e r g ' s
role of M y r b e r g V s
idea
in the
w i l l be apparent.
Gebilde
in the sense of Weyl,
but does
each
a
homeomorph
by removing
with
a Lindel~fian
valence
is termed ~ 1/2}
of the c l o s e d
the b o u n d e d analytic
and has
harmonic
having
element.
ramification
with respect u n i t disk. from ~ .
the p r o p e r t y
functions
valence
function
boundary
"transcendental
one of these c o m p o n e n t s
bounded
J
two and has r a m i f i c a t i o n
such p o i n t
[Izl
b y P. J. M y r b e r g
admit n o n - c o n s t a n t
S
of
d
one K e r ~ k 3 a r t o
each
in q u e s t i o n
The p r e i m a g e
last fact t o g e t h e r on
has c o n s t a n t
structure
,
genus w i t h
the o d d integers,
reasons.
is o b t a i n e d
exists
The
of this c h a p t e r
of infinite
of the type c o n s i d e r e d
This
owes
structure~(analytische
surface
for o b v i o u s
S
surfaces
which
n - cos ~z = O.
center m a p has two c o m p o n e n t s , Riemann
of R i e m a n n
surface
function.
it was published.
at the p r e i m a g e s o f
to one.
hyperelliptic"
~
harmonic
result
annihilating
w
It is a p a r a b o l i c
of a R i e m a n n
bounded
say that the c l a s s i f i c a t i o n which
[24]
of P a r r e a u ' s
and,
to the
The d e s i r e d It is a that
it
of course,
o f the c e n t e r m a p
(namely,
the r e s t r i c t i o n
-
to
S
of the center
f ~ Hp(S).
map associated
We i n t r o d u c e
for
I/2
with < Izl
%b(z) = [f(s)
where
{s,t}
is analytic
is the p r e i m a g e on its domain.
of
z
From
the
fact that
Ifl p
-
~ ).
Hence
< + ~,
the
- f(t)]
with
{I/2
< Izi
tion
in a p u n c t u r e d
that
~
< + ~}.
+
majorant
h
see that
vanishes
the center m a p a s s o c i a t e d on
C.
For
I/2
using
singularity
@
< Izl
with
< + ~
applied The
{s,t}, to
~,
inclusion
Remark. the t h e o r e m
as above, that relation
inequalities
at
~.
0
We note
that
we c o n c l u d e
the i n e q u a l i t y
harmonic
function
of a p o s i t i v e
for the L a u r e n t ~
takes
Let
K
denote
that
on harmonic
coefficients
the value
O
where
f
of ~,
at each odd
the r e s t r i c t i o n
f = f o E
func-
to
S
of
is a n a l y t i c
is the p r e i m a g e and,
+ h(t)]/2.
of
z.
consequently,
of P a r r e a u
We conclude,
f
are constant.
using
the r e a s o n i n g
Hence
S ~ N
o
0 . q
is established.
of
~
and
This
;
near
observation
~
m a y be i n f e r r e d
affords
a second
with
approach
the aid of for show-
f.
U o < q < + ~ Oq ~ OBA.
s h o w that w i t h
S
Since
We c o n c l u d e
of S z e g 6 - S o l o m e n t s e v . of
map.
we have
The b o u n d e d n e s s
ing the c o n s t a n c y
3.
f
on
the local b e h a v i o r
If(z) ip ,4 [h(s)
where
given by
+ h(t)]~
identically.
5.
n o w that
If(t)fP]
a non-negative
disk and the C a u c h y
has a r e m o v a b l e
integer w e
represents
We conclude,
~
to the c e n t e r
2p[ff(s)ip
has a h a r m o n i c
side of w h i c h
function
Suppose
,
respect
l~(z) p/2 ~ 2P[h(s)
the r i g h t - h a n d
2
S ~ OLA.
inequality
l~(z) fp/2 4
and the
37
Given
as d e f i n e d
~
satisfying in w o
the c o n d i t i o n s
this Ch. OBA 9
imposed
in w
of
Ch. II, we
-
3 8 -
When log ~( )_.x.
lim x~+o0
= + ~,
log x we h a v e
O
cO
,
0
< p
< +~.
P
The strict that
inclusion
r OBA.
O
of the first
We s h o w that the
To that end, we i n t r o d u c e along) the
segments
and d i s t i n g u i s h
along)
n = 0,1,
, where
E 3 + n, a c o p y of running
~(O~3
through
in the c o u r s e
distinguished
manner,
E3+ n
increasing,
and d i s t i n g u i s h whole
surface
in the s t a n d a r d
joined to
constructions
to the E2
(along)
e
> 1
o
(a slit along)
numbers.
S
manner
lower
along
The
e
b y joining
- {0},
the segments
and [e2n,
n
lime
n
~2n+1 ],
will be
= 2. the
further
indices
restricted
(i.e.
E
to E 1 along their c o m m o n 2 the u p p e r edge of such a slit of a
edge of the c o r r e s p o n d i n g
their
common
distinguished
in the case of the c o n s t r u c t i o n
w e shall c o n t e n t
just e m p l o y e d
- {O}
understand B y the
and also
4(012)
(slits
slit of the other
slit also
copy)
in the s t a n d a r d
n = O,1,
that we h a v e ~(012)
and distinguish
! E2, a c o p y of
segments
is s t r i c t l y
the d e s i r e d
Let us be s p e c i f i c maining
- {0},
e2n+l],
the n o n - n e g a t i v e
slits
c o p y being
and joining
+ n)
It is immediate
of the c o n s t r u c t i o n .
We construct
given
(c n)
~(Os2)
= 0,1,
the above
follows.
is strict.
a c o p y of
[i/22n+2,1/22n+l],n
(~slits
of this p a r a g r a p h
inclusion
E 1,
[e2n,
n
sentence
the
with respect image of
"joining"
g r o s s o modo. to the m a p
~(O13
+ n)
of the p r e c e d i n g
ourselves
with
By the c o p y z ~
(z,n),
with respect paragraph
lying at hand.
the i n e x a c t E
n
to
z ~
we understand
descriptive
we u n d e r s t a n d
n = 1,2~
For the re-
b y the c o p y
language
the image of E3+ n
(z,3 + n), n = O,1,
we ....
that we are c o n c e r n e d
with
-
a l-complex formal lying
dimensional
structure set of
manifold
rendering
S
S,
-
to be p r e s e n t l y
a "natural"
is t h e u n i o n
39
projection
r not
as
follows.
(1)
< Izl
< 2}~
[(z,2)
: 0
< [zl
< 2~z ~ 2-(n+1) ,n=O,i,
: Izl
< n+3,z
exceed
the d i s t a n c e
set w e d e f i n e
a
with
analytic.
: 0
of t h i s When
C
{(z,1)
n = 0,1,
(a,k)
into
endowed
a con-
The under-
of
{(z,n+3)
Given
map
described,
~ e 2 n , e 2 n + l },
....
[(a,k~r)
is n o t on a s e g m e n t
from
a
to the u n i o n
}s
for s u f f i c i e n t l y
distinguished
of
the
segments
for
small
~
and
distinguished
positive r
does
for
~,
it is {(z,k)
(2) W h e n not
k
is
1
an e n d p o i n t ,
the e n d p o i n t s
or and
2
and
r
: Iz - al
a
is a p o i n t
is no g r e a t e r
o f the d i s t i n g u i s h e d
{(z,k)
< r}.
than
segment
of a s e g m e n t the m i n i m u m
containing
: Iz - al
< r,
Imz
distinguished
for
of the d i s t a n c e s
a,6(a,klr)
E1
from
but
a
is the u n i o n
to of
~ O}
and {(z,3
(3) W h e n is at m o s t
k
is
I
and
the d i s t a n c e
distinguished
for
a
- k)
: Iz - al
is an e n d p o i n t
from
E1,6(a,IIr)
a
to the
< r,
Imz
of a segment
< O}.
distinguished
set o f the r e m a i n i n g
is the u n i o n
{(z,1)
: Iz-
{(z,2)
: O
al
endpoints
for
El ,
of the
and
r
segments
of
< r}
and
(4) W h e n
k = 2
or
3+n
and
< Iz - al
a e [e2n,e2n+l],
< r}.
we proceed
as we h a v e
just done
in
(2)
-
and
(3), m u t a t i s
which
dimensional component, ing
~
such a c o n f o r m a l
mapping
~(O~rl/2)
say
consisting of w h i c h
f,
counterparts
in case
S ~ OBA.
and s h o w that
the v a l e n c e
is
2.
Paraphrasing
= f(t).
f(s) E
= If(s)
of
H
The
functions
of
~2"
vanishes
f ~ ~I
It follows
that
f ~ ~2 f ~ ~3+n
say,
{Izl
value
at e a c h p o i n t of the p r e i m a g e ~
and h e n c e
< 2, Imz
and
so is
It r e m a i n s S / O~. o I~I
For
f.
on
C
to be s h o w n
that p u r p o s e
has a h a r m o n i c
into
(1),
its first
on
S render-
that
S
if w e w i s h
is
and con-
(2)! the c o n t i n u o u s
~ a + z2
and
a bounded
in c a s e
image
< Izl
to EI~
of
(3)~
z.
The
for
= O,1,
E
F
is b o u n d -
....
We con-
o f the r e s t r i c t i o n n
,n = 0,I,
....
at each p o i n t o f the d o m a i n
It is n o w c o n c l u d e d of
at e a c h p o i n t
function
take the same v a l u e s
of a g i v e n p o i n t
of - I ( ~ )
introduce
the i n v e r s e
distinguished
f ~ ~i
A -{O},
function
< I,
2-(n+l),n
~n'
analytic
the c o m p o n e n t
argument we
take the same v a l u e s
C
that
at e a c h p o i n t of, f
takes
and t h e r e u p o n
By the t h e o r e m of L i o u v i l l e
~
the same
that
f = f o
is c o n s t a n t
S r OBA.
that g i v e n
it suffices
majorant.
0
introduce
and b o u n d e d .
Consequently
here
denote
has
- f(t)] 2,
< 0}, n = 0,1 . . . . .
is a n a l y t i c
~I~
at the p o i n t s
We t h e r e u p o n
S
structure
~[e(z)]
~
the M y r b e r g
less the u n i o n of the s e g m e n t s
n
Let
The m a p
is the p r e - i m a g e w i t h r e s p e c t
clude
to
E 2.
a l-complex
We s u p p o s e
in c a s e s
satisfying
it is c o n s t a n t .
and
that
~16(a,k~r)
To that end w e c o n s i d e r
E1
and f u r t h e r
is a c o n f o r m a l
S
(4).]
of
ed and a n a l y t i c ,
each p o i n t of
[We m a y be s p e c i f i c
6(a,k~r)
family of allowed
topology renders
~, taking There
of the
onto
This
b y the
up to an e q u i v a l e n c e .
of points
{s,t}
where
C.
structure.
F(z)
where
map
onto
the i n v e r s e s
We n o w s h o w that S,
S
and it is d e t e r m i n e d
and the a p p r o p r i a t e
on
for the topology.
and the p r o j e c t i o n
sider a~ u n i f o r m i z e r s 8
is taken as that g e n e r a t e d
an i n t e r i o r m a p o f
endowed with
maps
S
is a b a s e
manifold
analytic
-
mutandis.
The t o p o l o g y on 6(a,ksr) ,
40
~
there
to s h o w that
We p r o c e e d
exists (e n)
as follows.
an a l l o w e d
(e) n
m a y b e so c h o s e n
such that that
For e a c h n o n - n e g a t i v e
whole
-
number
v
we
E3 ,
introduce
, E3+ ~
the s e g m e n t s
to
an a u x i l i a r y
E v)
for
-
surface
as above w h e r e
distinguished
are d i s t i n g u i s h e d
41
for
E1
Sv
formed b y joining
E 2(v) d i f f e r s
and the s e g m e n t s
from
[e2k,
E2
E~ v) to
E1
and
in that o n l y
e2k+l],
k = O,
E~ v)
For a given non-negative
whole
number
v
w e let
~
denote
the p r o j e c t i o n
map
V
associated
with
S
defined
as above.
It is trivial
from the c o n s t r u c t i o n
of
S
V*
that
K
V
is bounded.
Let
h
denote
M~
I~
o
I.
It w i l l be shown
that w i t h
V
,e2v+2 O
e
~
9
held
fast,
hv+ 1
.
tends
2v+3
tends
to
We fix
This
2v+2"
o f the p a r a m e t e r s
e
and r e s t r i c t respect
r
to
z ~
fact w i l l
< r
be of f u n d a m e n t a l
< min{2
to be less than
(z,2)
~I
the r e q u i r e m e n t function
on
and that
E1
v
~2'
(~2v+2 '2)
r + s2v+2.
in
takes
The
the value
subharmonic
and
as
V
--
importance
for the d i s p o s i t i o n
[e2v+2,82~+3]
second,
We define
of Sv+ i - C,
Let
u
C
the
be the largest on
C
as the image w i t h
and
indices
being
non-negative
is m a j o r i z e d
specified
by
harmonic
by
I~vl~lf. harmonic
of
C.
function
with respect which
takes
to
u
V,
has the same d o m a i n of its d o m a i n
is s u p e r h a r m o n i c
on
We i n t r o d u c e
of
I at the points V
- ~2v+1 }
= r}.
continuously
at each p o i n t
to be the e x t e n s i o n ~I"
o
~ ~2"
be the least p o s i t i v e
image of
- e2v+2,e2v+2
the c o m p o n e n t s
which vanishes
I continuously
less the
S -{(e2v+2,2)}
of
M~
Let
on --
satisfying
e2~+3
We i n t r o d u c e
not
h V
[Iz - e2~+21
fined
to
n
O
value
pointwise
9
on their
z ~
(z,2). O
as
U,
in
~I"
domain.
taking
two f u n c t i o n s
the value
not
~1
The
the b o u n d a r y with
first,
at e a c h point
domain U,
is de-
of its d o m a i n
is an e x t e n s i o n
of
It is e v i d e n t
that
It is r e a d i l y
S
seen that
v, U
and is
-
U + ~(3
is s u p e r h a r m o n i c and
V
on
its d o m a i n .
hv+ 1
if
H
with
(the p a r a m e t e r
,4 h v ( s ) , h
domain e2v+3
where
V*(s)
tending
= v(s) , s e n 1, a n d
the e q u a l i t y hv+ 1
tends
as a s s e r t e d
even.
of
H(s)
and
pointwise
V*(s)
hV
e
We
a e So,
fix a p o i n t to s a t i s f y
are
n
The the
o f the c o m m o n
surface
image
respect
the v a l u e
+ ~(3
domain
of
U
+ v + l)V(s).
is the p o i n t w i s e we conclude
,~ @(3 + v + 1 ) V * ( s ) ,
= 1
elsewhere
at e a c h
on
whose
the r e q u i r e m e n t
Sv
point
in
limit
of a sequence
that
s e S v -{(e2p+2,2)},
S v - {(e2v+2,2)},
of
and there-
S v - { ( e 2 v + 2 , 2 ) }.
S v - {(e2v+2,2)}
as
e2v+3
It
tends
to
when
n
follows e2~+2
we c o n c l u d e
follows.
component
We is
fix 1.
e
n
We d e f i n e
is
1
(e2n+3)
or
re-
that
< 2 -V + h v
by
(a)
.
S*v P
v. =
.O , 1.
.
.
the r e g i o n
obtained
by removing
from
of
to
at
as
second
is n o w r e p l a c e d
U
with
~4 U(s)
e2v+2),
to be c h o s e n
h v + I (a)
Sv
s
above.
The parameters
cursively
to
- hv(s)
hv(s)
to
+l(s)
S V - {(e2v+2,2) }
O g H(s)
that
for e a c h p o i n t
we have
Consequently,
upon
-
+ u + l)V
Further
U(s)
of
42
a that
z ~
(z,2).
of
M~
~ v + 1[ e 2 p ' ~ 2 ~
The
o IKIS*l v
~ o IKl
has
sequence does
not
a harmonic
(S~)
+ 1]
is i n c r e a s i n g
exceed majorant.
h
~
(a) , Hence,
and exhausts
which since
is less E
S.
than
Since 2 + h
o
(a) ,
is n o t c o n s t a n t ,
s/o 4.
Op ~ _
Op.
The
following
Since
O -p
observation
r O
p
,
it s u f f i c e s
is c e n t r a l
to c o n s t r u c t
in the c o n s t r u c t i o n :
a Riemann
surface
S E O
p
-O
-p
.
-
I__ff 0
< a
< 1,
then
The p r o o f
th@
identity
is an i m m e d i a t e
Iz
map
43
-
of
5=
{IArg
consequence
< Rez
o f the
zl
< aK|2}
belongs
to
a function
f
to
< a
inequality
, z ~.
cos~ 2
We n o t e
that
the observation
on a R i e m a n n given
surface
by Smirnov
where
e
analytic
8(I)
with
of
positive
S = A.
A(O;2)
we
- {0}
, m - I, m
for w h i c h
E 3 + k,
being
the
k = O,
"parameter"
.
map
surface
to be c o n s t r u c t e d
z ~
(z ,1) ,
the
to
z ~
image and
the
introduce:
to n o t e of
{Rez
(S),O
< I,
that
8 o f ~ HI(S) ,
> 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
This
result
surface. for
is, o f c o u r s e ,
We p u t
~ ~ H
aside
the
valid
in the g e n e r a l
trivial
case where
situation f
o f an u n r e s t r i c t e d
is the c o n s t a n t
O.
Riemann
By Th.2,
Ch. II
(fl), P MTf~I p = M [ e x p
where
Q
denote
a non-decreasing
tending
is the t e r m
to
functions this C h . ) NOW
On
Q. on
and
Let ~
so d e s i g n a t e d
v
such
let
is b o u n d e d
sequence n
denote
that
h
n
in
(2.1),
of harmonic the n e g a t i v e
+ v
Ch. I I . , r e l a t i v e functions
On
be an analytic
and
so
0n ~ ~ H
function (~).
on
of a finite
is the r e a l
n
(9.1)
(pQ) I~I p]
o
part
on
By T h . 2 ,
~
to
~,
loglfT.
each bounded
sum of minimal
of an a n a l y t i c
satisfying
Ch.ll.,we
l~
Let
(h n)
above,
positive
function
h~rmonic
(cf.
w
= hn +v. n
have
P M(10n~IP)
= M[exp
o (ph n) I~IP].
(9.2)
-
Given
m
86
-
a w h o l e number,
exp
o (phm) I~I p
lim M [ e x p n-~o
M[exp
whence
o (Phn) l~IP ]
(pQ) i~IP],
o
it follows that
o (Phn) I~l p]
lim M [ e x p n~co
= M[exp
o (pQ) I~IP].
Since
(M[exp
is a n o n - d e c r e a s i n g the m o n o t o n e
sequence,
non-decreasing
by
o (Phn) I~IP])
(9.1) and
sequence
(9.2) we see that
(lle~n(~)ll).
Now
~ ~
llef(~)ll lie
(~)II
is the limit of is c o n t i n u o u s
on
~n Hp(n).
Hence
~ ~
llef(~)ll
is lower s e m i - c o n t i n u o u s
ball b y the cited c l a s s i c a l of a norm,
~ ~
llef(~)ll
m a p s the unit ball of
on
Hp(n).
r e s u l t on lower s e m i - c o n t i n u o u s
is b o u n d e d on the u n i t ball. H (~) P
into itself.
It is b o u n d e d on a
functions.
By p r o p e r t i e s
For some p o s i t i v e n u m b e r
It n o w follows
from Th.7, Ch. II2that
C,ecf f
is
bounded.
A q u e s t i o n that is c o n n e c t e d w i t h the t h e o r y of T o e p l i t z Let
F
be a b o u n d e d r e a l - v a l u e d
Lebesgue measurable
function on
forms is the following. F.
Let
O < p < + ~.
We introduce
Pl = sup ~
where
~ ~ Hp(~)
and
II~II = I,
FI~*I
and also
M2 = sup ~-~
where
Theorem
u
is
PL
on
and
11= ~I = P2 = ess.
~a'
v(u)
= I.
F u* 6 ~ a '
Then we have
sup. F.
We recall that the essential
s u p r e m u m of
F
is s i m p l y the m i n i m u m o f the set of
-
real
c
p.p.
on
wood's
such that F
equal
limit
theorem.
to b e sure,
-
The
The
reference
existence
decomposition
[This a s s e r t i o n
the u n i t c i r c u m f e r e n c e
to one a n o t h e r p.p..
though,
> c} = O.
is a s s u r e d b y the F . R i e s z
radial
which map
meas.{F(t)
87
taken
to s e c t o r i a l
of
F.
limit o f
functions
u
and Little-
in t e r m s o f u n i f o r m i z e r s
The r e s u l t i n g
in the x n o t a t i o n
limits
'radial'
for s u b h a r m o n i c
is to be c o n s t r u e d
o n t o the c o m p o n e n t s license
of a Fatou
for
u
is not w a r r a n t e d .
u*
are
is h a r m l e s s ,
For an a l l o w e d
u
w e have u*
=
(Mu)
u
there
*
p.p.
on
F.
Further~for
p,p.
on
F.
Indeed,
such a
~
is o b t a i n e d
relative
to
log u,
taking
v
as the n e g a t i v e
harmonic
functions
and thereupon Since
I~I p
on
taking
invoking
~ ~
such that
the g e n e r a l
any restriction
it is t r i v i a l
taking
advantage
G(t)
= exp
~1
case
~
such that
the t e r m
Q
of
= Q + v.
sum of m i n i m a l
we conclude
~'Pl ~ P2"
and
P2
positive
to see t h a t
log F 6
~-~
P2 4 PI"
Consequently
Pl = ~2"
t
the case w h e r e
translating
~2 6 ess.
of the r e s t r i c t i o n
function,
that
we c o n s i d e r
follows by suitably
I~*I p = u*
(2.1),Ch. II,
is the real p a r t o f an a n a l y t i c
logl~l
of
an a l l o w e d
o f a finite
for e a c h a l l o w e d
the c o m m o n v a l u e
lower bounds
the o t h e r d i r e c t i o n ,
u
p
exists
by introducing
Q + v
satisfying
is an a l l o w e d
To d e t e r m i n e a positive
an a l l o w e d
sup F.
F.
F
has
Without
We proceed
in
to i n t r o d u c e
'
F t
e ~.
thence
On a p p l y i n g ess.
sup F
Th. 6, ,< ~2"
Ch.
The
II#to
theorem
~G/~2, follows.
we c o n c l u d e
that
G(t)
4 P2' t ~ ~,
and
-
88
-
Chapter
Vector-Valued
V
Functions
I. In this c h a p t e r w e shall
explore
valued analytLc
It is n o t a m e r e
For one
thing,
no longer
functions. even
assured.
aspects
phenomena
Nevertheless,
the b o u n d a r y
case w h e n one r e p l a c e s
vector-valued
Radon measure
b o u n d a r y we u n d e r s t a n d
on the b o u n d a r y
2.
Basic
definitions.
in
X
harmonicity
of
For the p l a n e
into f
at
case
of the d e r i v a t e
X
extends without
the p l a n e
c a s e we
following
equivalent
+ ~(z)
equation
where
u - = O zz
value p r o p e r t y , sideration
, II
Using
f
the
introduce
~
means:
at
~
in a n e i g h b o r h o o d
of
i. e.j for e a c h
z
= ~I
(3) we c o n c l u d e
a,
u(z
are at our d i s p o s a l .
that
if
f
a
of a
at a
and the
o f a local u n i f o r m i z e r
at
The c l a s s i c a l
n e w ideas.
a ~ C
Continuing
~
at
(3)
~,
near of
~ ~)
of the form
(2) the v a l i d i t y
the v a l i d i t y
o f the L a p l a c e
of the G a u s s - K o e b e
the f u n c t i o n
u
under
meancon-
satisfies
+ reiS)de
We a s s u m e
that the u s u a l
Their
treatment
f
is h a r m o n i c
in
b y a n y one of the
o f a local r e p r e s e n t a t i o n
sufficiently
r.
of
The n o r m
in a c u s t o m a r y w a y - via the e x i s t e n c e
of a n y e s s e n t i a l l y
and
on the
continuous
C.
of a point
of the a n a l y t i c i t y
in a n e i g h b o r h o o d
small p o s i t i v e
space o v e r
a neighborhood
the n o t i o n of h a r m o n i c i t y
is a n a l y t i c
to the
consideration.)
at e a c h p o i n t of a n e i g h b o r h o o d .
(I) the e x i s t e n c e
are
restricted
Radon measure
to the p l a n e case b y m e a n s
intoduction
theory.
limits
IV m a y be e x t e n d e d
of c o m p l e x - v a l u e d
is a fixed B a n a c h mapping
of r a d i a l
function by a suitably
space u n d e r
The n o t i o n s
sense)
u(z)
these m a t t e r s
X
are r e f e r r e d
(assumed c o n t i n u o u s
for s u f f i c i e n t l y
that Let
(in the s t r o n g
o f Ch.
(By a v e c t o r - v a l u e d
analyticity will be defined
apparatus
~(z)
the b o u n d a r y
into the fixed B a n a c h
be given. a
studies
of v e c t o r -
o f the c l a s s i c a l
existence
linear m a p of the space
We a s s u m e
w i l l be d e n o t e d b y
paraphrase
as the p.p.
on the b o u n d a r y .
a bounded
functions
surface
routine
such s t a n d a r d
vector-valued
Riemann
of the t h e o r y of H a r d y c l a s s e s
prelimaries
concerning
is routine.
(in p a r t i c u l a r
analytic)
o n an
-
o p e n set
O
of a R i e m a n n
lytic
0
we have
on
see this,
the s t r o n g e r
it suffices
X
of norm
log lh o f l
I
holds
satisfies
in a n e i g h b o r h o o d
ilfli
the plane
property
which
then
-
conclusion
to c o n s i d e r
s h o w that the m e a n - v a l u e on
surface,
89
of
is s u b h a r m o n i c that
case w h e r e
relative
to
A[f(O) ] =
0
we
log]lflt
O.
i
2~
positive
r.
the s u b h a r m o n i c i t y positive
and
let
This
of ~
is s u b h a r m o n i c
on
We
that
IoglA
o f(re 18) Id8
introduce
at
is anaO.
0
To
and to
a linear
functional
the s u b h a r m o n i c i t y
of
.
logllf(re 18) ilde 0
is the c u s t o m a r y
loglif,
f
is a n a l y t i c
infer
21I
4 I___
for small
O.
9
4 ~
tl
When
llf(O)it. U s i n g
j.
iogllf(O)
f
on
for
f
proof.
analytic
The
is due
be a c o m p l e x - v a l u e d
analytic
log I~ (z) --
l~
following 9
ingenious 9
,
t
to D . S . M l t r l n o v ~ c .
function
on
A(Osr)j r
proof
Let
n
small,
of be
satis-
fying B
i 211 ,
z fl + D)k(8,_)de,
J O
Izl
< r.
Then
monicity Hence
of
f(z)~(z)
,f(z) ifl~(z) I on
Hf(O) iJ < I/l~(O) I.
tend to
The following Riemann
3.
< p
fact that
~(Osr)
we have
The p r o o f
logHf,
formulation
surface
< + ~,
Fatou
an a n a l y t i c
follows
function
on
~(Osr).
from the m a x i m u m on taking
Using
principle:
logarithms
the
subhar-
ilf(O) itl~(O) I< I.
and t h e r e u p o n
letting
O.
the Hardy class 0
defines
of the n o t i o n
f
analytic
~oo(TsX)
provided
limit
show b y e x a m p l e what complex
T,
Banach
on T
provided
spaces
X
taking
that
The o b j e c t
complications
for
f
analytic
of a Hardy class
the subharmonic
theorems. what
is s u b h a r m o n i c
f
values
in the p r e s e n t in
is bounded,
function
of this
llfiip
section
are to be expected.
is it the case
that
leads n a t u r a l l y
X
setting.
to the Given
w i l l be said to b e l o n g
to the H a r d y class has a h a r m o n i c
is to raise
the Fatou
radial
to
~p(T~X) ,
majorant.
a question
The q u e s t i o n
a
is this: limit
and to For
theorem
holds
-
for a v e c t o r - v a l u e d that
it s a t i s f y
e.g.jthat
analytic
one of the
l~gllflt h a v e
even
persists
for
f
when
i r p
sequences
whole
entering
checked
lim f(r~)
does
We t u r n
shall
throughout
that
not
see
values
in
conditions We
that
for
in norm,
and
X
examine
p = +~o , that,
subject
as t h o s e
shall
f
exist
I 4 P
of b y a s i m p l e
this
the m a p
section
f : ~ ~ 1
is a n a l y t i c for a n y
to the p o s i t i v e
I: G i v e n
with
majorant?
is d i s p o s e d
We c o n s i d e r
is r e a d i l y
-
of
to the p r o v i s o
the c l a s s i c a l
the
question
the F a t o u
in c o n t r a s t ,
theory,
for
property
the F a t o u
need
not
property
< + ~.
p = + ~
numbers.
Theorem
and
f
restrictive
t h a t are b o u n d e d
The c a s e the
same
a harmonic
X = Ip(C) , i ~ p 4 + ~, hold
function
9 0
and
example.
have
as t h e i r d o m a i n
(C)
given
by
f(z)
llf(z) II = l,lzl
~ ~ C(O;i)
It w i l l
as
< I.
is e a s i l y
be a s s u m e d
the =
that
set of
(zk),fzf
< I.
It
Nevertheless,
verified.
theorem.
< + ~-
Let
f : ~ ~ 1
(C)
be analytic
on
A
and
such
that
P
H
l~gllfil
have
Corollary:
Proof:
a harmonic
Theorem
We
introduce
logl~l
= M l~gllfil.
taking
values
values
of n o r m
on
~
when
f
~
4 I.
f
Once
it f o l l o w s
bounded
by
has
analytic
f -I
Theorem
we
by
l i m i t p.p.
~
function
obtain
function
on n o t i n g
on
~
o__n_n C ( O ; I ) .
of Ch.
IV.
on
satisfying
~
a vector-valued
I is d e m o n s t r a t e d
unrestricted[y
analytic
a Stoltz
is r e p l a c e d
, a complex-valued
Replacing
$ I,
Then
analytic
for a l l o w e d that
and hence
~
has
function
functions
taking
is the r e c i p r o c a l
a finite
Stoltz
of
limit
C(O;I).
We denote
persists
of norm
a non-vanishing p.p.
I
majorant.
turn
the
and
to the
kth
ZlfklP
restricted
component
of
case where f(z),Izl
is s u b h a r m o n i c
on
~.
given
b y the P o i s s o n - L e b e s g u e
integral
duce
h = M(ZlfklP)
that
boundary
function
and note s =
Zlf~l p.
We
f
values
< I, k = O,1 . . . . . We
introduce
with
F
of norm Each
hk = MlfklP,
boundary
it is g i v e n let
takes
function
fk
the m a p
of
We
let
is a n a l y t i c
k = O,i . . . . . If~l p.
b y the P o i s s o n - L e b e s g u e
denote
~ i.
[-K,~]
We a l s o
fk(z) on It is
intro-
integral
with
into
set of
the
-
complex-valued
sequences
satisfying
is simply
91
-
the c o n d i t i o n
[-~,E]
into
1 (C). P
Paraphrasing
Ji:
F'(8)
integral
exists.
limit exists
iogllf - f*(eiS) Jl,
is c o n c l u d e d which
2~
E
F(G)
is a L i p s c h i t z i a n
< I.
for functions limit of
f
limit at a p o i n t
exists at e i8
e
by a
i8
for w h i c h a
is b o u n d e d above, w i t h the aid of s t a n d a r d arguments
employed
[-E,~]
(I) each of the
at
k th
component
case.
the b e h a v i o r
representable
function
cf.
F'
f~
~
and h a v i n g L e b e s g u e
We assert that an a d m i s s i b l e 8
is d e f i n e d at F,
105].
E r [-II,~]
exists.
at each p o i n t
of
[16,p.
to e x h i b i t i n g
at each p o i n t of w h i c h
of the
F
of the subharmonic
are fulfilled: 8
Fatou t h e o r e m
by examining
is r e d u c e d
is the subset of
(8,z)dF(8) , rzl
of a Stoltz
in the c l a s s i c a l
Thus our p r o b l e m
i n e q u a l i t y that
[16] we see that the radial
The existence
for the same p u r p o s e
of
= 2--~
the p r o o f of the c l a s s i c a l
Poisson-Stieltjes
measure
of
Further
z
radial
component
Hf~ (e l~) as.
It is r e a d i l y v e r i f i e d w i t h the aid of the H61der
when
k th
~8
J m a p of
that the
choice
of w h i c h the following c o n d i t i o n s e i8
and
(2) the d e r i v a t i v e s
~ f~(e 18)
is the d e r i v a t i v e
of the functions
f~(e I~) IPda, k = 0,1 .....
and
~
exist and are r e s p e c t i v e l y is m a d e to
s
introduced
Lebesgue measure from
8
the
p th
2H.
5"
equal to
(e1~) d ~
If~(e i8) Ip, k = O,1 .....
in the p r e c e d i n g paragraph. ]
Further given
8 ~ E
and
Indeed,
we see that for
- F(8) ] (~-8) -1 - (f~(eiS))
[Reference
the set so d e f i n e d has
~ e [-~,H]
power of the norm of [F(~)
s(ei8).
different
-
92
-
does not exceed the sum of
~-m
f~ (e I~ )da f~(e 10) P
O
and 2P-i
/
i lf~(e I~) tPda
m+lI for each whole number
m.
'f~ (ei8) [P
(3.1)
m+l
Now GO
~
~Sf f~(e ia) fPda
[s(eia)-~off~(eia) fP]da
m+l
Hence
the limit at
O
of (3.1)
is equal to
oo
2p~___ ff~(e i8) Ip" m+l Given the restriction imposed on the points of conclude that
F'(8)
exists and is equal to
E
and the arbitrariness of
(f~(ei8)).
m,
we
The proof of Theorem 1 is
complete. Proof of Corollary.
It suffices to map conformally and univalently a plane annulus into
so that one of boundary circumferences corresponds to a given component of to note that the boundary behavior of
f
F
and
is thereby referred to the case where the
domain is an annulus, which case may be referred to the case of a disk with the aid of the Laurent representation.
The details are readily furnished.
4. Vector-valued harmonic functions with norm possessing a harmonic majorant. point on we shall be operating functions
u
with domain
~
and satisfy the condition that given in w
in the setting of Ch. IV.
From this
For the present we consider
taking values in a complex Banach space which are harmonic llull (resp.
~ o llulf, where
Ch. II, has a harmonic majorant.
~
satisfies the conditions
In this section we shall see that the
-
functions volving
in q u e s t i o n
vector-valued
and that
boundary
the P o i s s o n - R a d o n
vector-valued The
are p r e c i s e l y
(F.Riesz-Herglotz
The [16]
that
defines
approximation
lemma
approximation
property
r
follows
(Lemma
A.
F.Riesz-Herglotz
2) w h i c h
(Lemma
transplants
We define
O ~ r < I.
a qiven
uniformly
continuous
b y a finite
satisfyinq
K
EIA
Then
Using
valued
continuous
the P o i s s o n
for
by a f u n c t i o n
Q
of the
on
Poisson
Izl
< i}
I gave
for n o n - n e g a t i v e
in
harmonic
situation
is an
we are studying
an
kernel.
functions
generates on C(OIi)
function
f
for the u n i t
C(Osl)
and
then
u f,
sufficiently
d i s k we is the
a dense
A
linear
(with s t a n d a r d
o__n_n C(OI•
7A K w h e r e the 3 z. 3 r ~< I z I < I, all j. 3
f,
to that w h i c h
situation
form
and
then but
subspace
topoloqy).
m a y be a p p r o x i m a t e d
are c o m p l e x
numbers
3
see that if solution
f
is a given c o m p l e x -
of the D i r i c h l e t
which
m a y be a p p r o x i m a t e d
near
I,
problem
uniformly
m a y be a p p r o x i m a t e d
by
uniformly
form n
~ j=i
n
AjKQq(~j)
= ~=i
AjKQ~
(~) , 3
where
exactly.
iql = I.
2'
zl
complex-valued
function
less
-
continuous
integral
function
with boundary
q ~ u(Qq)
into c o n s i d e r a t i o n .
2
{K z : r 4
sum of the
I ~< maxlfl
Proof:
A
set of
as
z
~
3
for
coming
of the c l a s s i c a l
to the
l) of the c l a s s i c a l
space of c o m p l e x - v a l u e d
In fact,
to be stated b e l o w
(~ = A , X = C)
theorem
in the t r e a t m e n t
lq
of t h e
functions case
in-
from the e n t e r i n g
will be seen to be r e l a t e d
not p r e s e n t
q
I: Given
(i.l) m a p
that of the c l a s s i c a l
I -lzl
Lemma
a
to c o n d i t i o n s
the set of h a r m o n i c
of the c l a s s i c a l
An e l e m e n t
z
onto
subject
representations
representation).
for the p r o o f
Given
Radon m e a s u r e s
parallels
exposition
functions.
-
those g i v e n b y P o i s s o n - R a d o n
representation
Radon m e a s u r e s
state of affairs
93
EIA. I ~ maxlfl, 3
for
Q
as above.
(4.1)
-
It
is to b e o b s e r v e d
that
94
-
the r e s t r i c t i o n
on
the c o e f f i c i e n t s
will
be of
service
immediately.
We n e x t = z of
{r O
and
that
a corresponding
result
holds
< I}, O
< r~u
( I. We
denote
Green's
< Izl
let
K
C(Osl)
of L e m m a @O z'
observe
now denote
z
the
inner
I persists
Green's
the
normal
let @
function
function
for
with
derivative
for the p r e s e n t ~
with
z
domain
at t h a t
setting pole
rO
z(~)
function
C(OI1)
point.
when
for an a n n u l u s .
< r
< I.
and note
for
assigning
We a s s e r t
that
To
that
A
is p o s i t i v e
the d i f f e r e n c e Iz
~ I.
in
(4.1)
taining
on
assertion
as w e
now
Let
{O
number C}
of the
annulus
has
so t h a t
Fk
{~
< ,z'
tinuous
= @
and
z
follows
tends
z
that
with
the
uniformly
of L e m m a
1
to
O as
tends
to
of Lemma
+ h z
where z
Izl ~ 1. C o n s e q u e n t l y O
the a i d of the r e s t r i c t i o n
'K'
introduce
uniformly imposed
1 b y the c o r r e s p o n d i n g
as
on the A. 3 'K' p e r -
C
to t u r n
be a u n i v a l e n t
so t h a t n
of e x a c t l y
a component
which
is a c o m p o n e n t
extension
< ~
there
lemma
< I,
to
{~
4
The
level
enumeration
are no z e r o s
components,
frontier
< I}, O
to the d e s i r e d
for
Q.
We r e t u r n
to the
standard
F.
fix a
setting.
has
Fk' k = 1 , . . . , n . We
~
F I .... , Fn
< ~a(S)<
ponent
K
h
point
the c o n c l u s i o n
see this we
@OTA
pole
A.
for the
positive
and
see on r e p l a c i n g
We are n o w r e a d y notation
A
of the p r e s e n t
Our
to
harmonic
with
to e a c h
z
hz
We c o n s i d e r
each
one
fr~.
onto
such
Izl $ 1}, lines
We
%,
of
6~a
in
an a n n u l u s . annulus,
is a c o m p o n e n t
of
of the c o m p o n e n t s
of
introduce
where the u n i t ~a
F .
o ~k
~k
{O
Each component and We ~k'
the
frontier
index
~ C}
is so c h o s e n
of
F
the
such
AI'''" '~n
conformal
is m a p p e d
set
is a c o m -
of each
that under
the c i r c u m f e r e n c e s
We and
the a n n u l i ,
a univalent
circumference are
< ~a(S)
of
map
of
its con-
onto C(O~r) , ~ <
r
< 1.
introduce
h = ~s
for e a c h
s ~ ~
and
show
F,
(4.2)
-
Lemma
2: G i v e n
qenerates
positive.
a dense
o__nn F
(with
Proof:
The
linear
standard
lemma
continuous jth,
c
once
on
F - F
as
s
that tends
F
h s (t)
that
on a g i v e n the
Green set of
~,
- r
4
for
asserted
h
to
F. 3
subset
~.
Izl
< 1}).
(i.e.
of
sense
of this
points
uniformly of
C(O11)
continuous
for
a complex~valued
For
the
annulus.
to z e r o
as
functions
annulus
{A. 3
of
< I}),
~s
tends
~6~(t) ,
to
< 1,
~j[C(O1r) ]
V
on of
the a i d of of a
is a c o m p a c t with
F. 3
uniformly
properties
which
the
O
seen with
and boundary
we h a v e
uniformly
respect
subto
inequality
t e F - F..
< Izl
< 1}
and
let
and z
K
z
be
taken
in
that
(4.3)
~ ~j - @Z
Izl ~ 1.
we c o n c l u d e
< r
O
h s.
a neighborhood
as m a y be
kj ~
to
the
follows.
s
see
F,
F, save p o s s i b l y
o f the a l l o w e d
exists
symmetry
measure
~s
of
tends
s
Indeed,
to
the
< Izl
h
there
~j[C(Olr)],
4 ~max
h
Further
tends and
s ~ Cj({r
of
components
s ~ V).
s
j = 1 ..... q
combination
e>O,
for
of Ch.I,
We
as
on all
given
~
~@j (Z)
tends
of c o m p l e x - v a l u e d
by a linear
the h a r m o n i c
behavior
We n o w c o n s i d e r the
vanishes
We c o n s i d e r
introduce
that
is p o s i t i v e .
s
(2.1)
and
limit
,< c}
space
it is s h o w n
hs(t)
The
of the
uniformly
each
inequality
function
: O < ~a(S)
s
< e, t r F - F., 3
compact
Harnack
{h
which
3 such
set
topoloqy).
m a y be a p p r o x i m a t e d
We note
-
The
subspace
follows
function
95
On
taking
the n o r m a l
that
C h@j (z) log (1/kj )
~ @j.
_
Kz '
derivative
of
(4.3)
at the
-
~j*
being
the
limit
On c o m b i n i n g 1 to an a n n u l u s , vanishes
z
of
~j
the r e s u l t s
we c o n c l u d e
on all c o m p o n e n t s
formly by a linear K
function
of the
that
of
C(O}1) ,
last
save
tends
uniformly
two p a r a g r a p h s
possibly h
s
jth,
.
to
a n d the
continuous
the
of the a l l o w e d
as
Izl ~ 1.
extension
function
m a y be
The p r o o f
O
on
of Lemma F
which
approximated
proceeds
uni-
by exchanging
and
C
log (1/Aj) and observing
The valued
a map,
Lemma
role
on
on an a p p r o x i m a t i o n
of L e m m a
F,
h j (z)
2 is to p e r m i t
linear maps (standard
from
to
f o ~.
us to o b t a i n
C(F) ,
topology)
o ~j
into
the X
Lemma
2 then
a uniqueness
space
lemma
of c o m p l e x - v a l u e d
(possibly
C).
follows.
for v e c t o r continuous
Indeed,
if
A
is s u c h
we have
when
3: l_~f A ( h s) = O
positive
O 4 ~a (s) 4 c,
then
thanks
2.
A
is the n u l l m ~ .
Here
c
is a
with
the
number.
The
proof
We n o w aid of
k
We n o t e
that
~
the e f f e c t
continuous
functions
on
-
a complex-valued
F
combination
on
96
is i m m e d i a t e
turn
to the
as d e f i n e d (s,t)
for e a c h
study
of v e c t o r - v a l u e d
immediately
~ h
s
t ~ F.
(t)
to L e m m a
before
Lemma
is c o n t i n u o u s
Given
functions
an a l l o w e d
on k
3
and
~ x F we
the and
introduce
on
~
constructed
family that HA
of
s ~ h
functions s
(t)
with
domain
from Lemma
3 that
h
s
.
is h a r m o n i c ~
defined
by
HA(s) = A(hs). It
is r e a d i l y
verified
that
HA
is h a r m o n i c .
It f o l l o w s
k ~ HA
is u n i v a l e n t . o IIHAII h a s
We
seek
to c h a r a c t e r i z e
a harmonic
majorant,
~
A
having
satisfying
the p r o p e r t y the c o n d i t i o n s
that given
IIHAIf, resp. in w
Ch. II.
-
The c e n t r a l
Theorem
result
2:(i)
majorant
is the
A necessary
is that
there
following
-
theorem.
and s u f f i c i e n t
exist
97
condition
a non-neqative
IIHAII p o s s e s s
that
Radon measure
a harmonic
o__nn C (P)
H
such
that
(4.4)
llA(f) II 4 p(Ifl) ,
for
f ~ C(F).
Every vector-valued
that
~uU
admits a h a r m o n i c
fyinq
the c o n d i t i o n
(ii) text and
(4.4)
majorant
is
is r e p l a c e d
result
o ~
~
for a
k
on
Q
with
values
(necessarily
in
X
such
unique), satis-
~ o ll...II r e p l a c e s
II...II
in the
finite-valued
(4.5)
I f l ~6~a'
~F
Lebes~ue
measurable
function
on
F
such that
is summable.
linear m a p of non-negative
we u n d e r s t a n d
C(F)
into
C
of
2: The
hs
and
The r e m a i n d e r
which
sufficiency for
(4.5)
takes
Radon measure
each non-negative
for the F . R i e s z - H e r g l o t z
tions
[16].
We shall w a n t
h
with
which
along
of some
a complex-valued
takes
(resp.
representation
be a v e c t o r - v a l u e d
O 4 r I < r 2 4 + ~,
(4.4)
the a v a i l a b i l i t y
the c o u n t e r p a r t
associated
4: Let
of
of the p r o o f p r o c e e d s
given
efficients
by a n o n - n e g a t i v e
on
member
F
of
a continuous C(F)
into a
real.
of T h e o r e m
positivity
Lemma
u
b y the c o n d i t i o n
is a n o n - n e q a t i v e
To be exact,
Proof
HA
holds w h e n
I I A ( f ) ll ~ ~i -
where
function
for some ~ .
The c o r r e s p o n d i n q
(4.4)
harmonic
in
the
lines
theorem
harmonic
X.
is i m m e d i a t e
of the J e n s e n
standard
harmonic
values
(4.5))
Let
inequality.
for n o n - n e g a t i v e
harmonic
facts
Fourier
on
Cn(r)
the
of the p r o o f w h i c h we have
concerning
function
function
on noting
func-
co-
on an annulus.
{r I < rzl denote
< r2},
the
nth F o u r i e r
-
coefficient
of
h(rei8),
Co(r)
1 r 2~ 9 =-2K~O h(rele)e-nlede"
= A o l o g r + B O, Cn(r)
= A rn + B n
A
n
and
B
are elements of
n
-
r I < r < r2, i.e.
Cn(r)
Then
98
X.
r -n
whe n
-n
The c o e f f i c i e n t s
The lemma is an i m m e d i a t e c o n s e q u e n c e
where
n / O,
the c o e f f i c i e n t s
are u n i q u e l y determined.
of the fact that
h
admits a r e p r e s e n -
tation of the form
h(z)
where
A ~ X
and
f
w h i c h take values
in
theoretic
= A loglzl
and
g
X.
+ f(z) + g(z),
are v e c t o r - v a l u e d
analytic
A p r o o f m a y be g i v e n w i t h o u t
a p p a r a t u s w i t h the aid o f the W e i e r s t r a s s
valued continuous
functions
functions
the i n t e r v e n t i o n o f function-
approximation
and the m a x i m u m p r i n c i p l e
on the annulus
theorem
for s u b h a r m o n i c
for v e c t o r -
functions,
cf.
[19, p. 300 - I].
We shall find it c o n v e n i e n t c
small and p o s i t i v e , o n t o
section, w e observe
F.
to introduce
Returning
to the
a m a p of the level set F(c) = {~a(S) ~k
introduced before
L e m m a 2 of this
that
~a
o
~k (z)
-
-w k 2~
loglzl,
~
<
Izl
<
i,
where
Wk = S F k
8nS~a ds,
r e f e r e n c e being m a d e to the inner normal derivative. which we propose
The m a p
~
of
F(c)
onto
is 2H
U1~k~n{(~k(Z),~(sgz))
, Izl = e-~-kk c}.
F u r t h e r w e shall agree that g i v e n a function f o ~
by
fc.
f
with domain
F
= c}
w e shall denote
F
-
To continue,
we consider
a h a r m o n i c majorant.
-
u : ~ ~ X,
U = MllulI.
Let
99
i
harmonic
We i n t r o d u c e
IF
on
for
Ilull
and such that c
has
small and p o s i t i v e
ufC6~a
(c) and
( i I Nc (f) = 2-~
ufC6~a ,
Jr(c) f e C(F).
It is u n d e r s t o o d
that
F(c)
has the s t a n d a r d p o s i t i v e
sensing.
The follow-
ing i n e q u a l i t y holds:
llAc(f) ll,l~c(f) I 4 ~c (Ifl)
With the aid of the W e i e r s t r a s s f o ~ ( e i8)
d
theorem
(trigonometric
and Lemma 4 of this section, w e see that
b y a function and
approximation
F ~ C(F)
which
is such that
(4.6)
x< (maxFlfl)U(a)-
f
lim k (F) c~O c
form)
a p p l i e d to
m a y be a p p r o x i m a t e d u n i f o r m l y and
lim ~c(F) c~O
exist.
For c
small and p o s i t i v e we have
lIAc (f) - kd(f)II 4 llAc(f - F)II + PIAc(F) - hd(F)II + IfAd(F - f)lJ
and a c o r r e s p o n d i n g lira ~c(f) c~O
exist.
inequality
(4.6)
concluded
that
non-negative
inequality
for
IBc(f)
- ~d(f) I.
We denote the former limit b y
holds h
with
A
replacing
is a c o n t i n u o u s
R a d o n m e a s u r e on
Ac
h(f)
linear m a p of
=
Hh, u
=
is
and that
The
now r e a d i l y ~
is a
~ c = {~a (s)
hold:
(4.7)
the second a s s e r t i o n of T h e o r e m
small and p o s i t i v e
X
It
n,
of the first a s s e r t i o n u s i n g the u n i v a l e n c e c
into
gc"
~(f).
and
F.
u
for
C(F)
lim Ac(f) c~O
and the latter b y
and g replacing
We show that the following r e p r e s e n t a t i o n s
w h e n c e we c o n c l u d e
We c o n c l u d e that
> c}
2 (i), and t h e r e u p o n
of k ~ H A . To e s t a b l i s h and t h e r e u p o n
the n e c e s s i t y
(4.7) w e introduce
c s ~ ~c' Gs,
w i t h domain
-
a neighborhood for
~
of
~
with pole
c
Poisson
integral
c s,
1 0 0 -
satisfying
: (I) its r e s t r i c t i o n
(2)
is
harmonic
U(S)
- 2~
Gc s
at
each
point
~
of
is G r e e n ' s
c F . o
The
function
generalized
formula
ij
U 6G: 6~a ,
rlc) a n d the c o r r e s p o n d i n g
one
for
U
hold.
u(s)
a corresponding
to
formula
We shall use
holding
Now
6G c
a (4.8) m a y b e r e w r i t t e n
= xc ~G~
for
(4.8)
s , Oc'
as
(4.9)
~
U.
the fact that
6G c s o ~-1 6G c a tends u n i f o r m l y positive
to
number
h
d
as
s
c
so that
tends 2d
to
O.
< ~a(S)
O < c 6 2d.
assured point
that
Gc s
We n o t e
m a y be t a k e n
of its d o m a i n
restricted
that w i t h
and t h a t
save Gc
s
To see this,
is so taken.
s ~ ~
given we
the a i d of S c h w a r z i a n
that
O
N o w the
~ (t)>-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~
to h a v e d o m a i n
provided
for
and
F(c)
when
(4.10)
O.
property,
Consequently
concerning
(4.10)
that
6G~/6G~ follows
(4.11)
GCs
tends u n i f o r m l y
tends u n i f o r m l y with
to
to
6~s/6~a
the a i d of the e q u a l i t y
~s
in in
- 101 -
./ ,<.
6G c s o ~ -1 6G c
Given
fl,f2
r C(F)
I
dG c s ~ 4 -i i I ~, C \ <.Ga iI
-h
= S
/
we see that
llAc(fl ) - k(f2)II ,( liAc(fl - f2 ) Jl + llAc(f2 ) - A(f2)li
4
Taking
into account
that of
f2'
this
inequality,
we see that
u = HA .
(maxlfl-f21)U(a)+ F
with
(4.10)
ilAc(f2) - A(f2)II.
playing
The c o r r e s p o n d i n g
the r o l e of
formula
for
fl
U
and
hs
is c o n c l u d e d
the same a r g u m e n t .
The p r o o f of T h e o r e m
A minimal that
p
harmonic
Radon measure
on
function C(F)
is a n o n - n e g a t i v e
The non-negative from
of p.
is the R a d o n m e a s u r e
negative
- N
property
2(i)
A.
We i n t r o d u c e
on
is n o w c o m p l e t e .
Suppose
obtained ~
P = Hv,
Radon measure
Radon measure
C
on
C (F)
(f)
i /F = 2H
IIHA.II p o s s e s s e s
in the a b o v e proof,
satisfying
satisfying
c
that
b y the u n i c i t y
p
and t h a t
P >/ ]fHA]i. W i t h we have
a harmonic
~
P
is a n o n -
the n o n - n e g a t i v e
Hv_ p = Hv - HB
>, O,
of the r e p r e s e n t a t i o n
m a y be o b t a i n e d
by a simple
~a"
(c) Now
INc(f ) - ~)c(f) f ~< (maxifl)[H
ifH, 6~.a ],
(a)
(c)
whence H ~-N
limit p r o c e s s
by
UHAII fc6
majorant,
by
-
f ~ C(F) , pointwise
and the r i g h t on
(~)
to c o n s i d e r
linear m a p of
Proof H
(4.4).
C(F)
for all
of T h e o r e m
= MIIHAIi
to
O
We r e m a r k
as
c
tends
to
O.
that this c o n d i t i o n
the case w h e r e
is a fixed u n i v a l e n t
satisfied
tends
-
Hence
tends
c
to
C(F).
The c o n d i t i o n It suffices
side
102
sequence
into
1
(C).
X = l~(C).
of p o i n t s
of
But there
does
m a y be g e n u i n e l y
Suppose
F.
Then
that A
not exist
A(f)
=
restrictive.
(f(tk))
is, of course, a
~
such
that
where
a continuous (4.4)
is
f ~ C(P).
2(ii).
When
is q u a s i b o u n d e d
~ o flHAII has a h a r m o n i c
as is
M(~
o IIHAII) = M ( ~
majorant,
o H ).
then b y Th.
The m e a s u r e
~
2, Ch. II, is g i v e n b y
~I-~fcS~ .
~(f)
Indee,
the n o n - n e g a t i v e
H~ = H .
Further
function H*
of
serves
M(~ for
5. B o u n d a r y
Radon measure
~ o ~
for
K4p(n,X),
f = HA
for a c o n t i n u o u s
Ch.,for
some n o n - n e g a t i v e
with characterizing
g i v e n b y the r i g h t - h a n d
and is equal
p.p.
It is n o w e a s y to e s t a b l i s h
in the n e c e s s i t y
theory
da
v
is summable
o H ). a
i = 2--~3F ~
(ii)
F
satisfies
to the Fatou
of T h e o r e m
boundary
2 on n o t i n g
that
statement.
I ~ p 4 + ~-
linear m a p
A
of
Radon measure
~
the c o n t i n u o u s
on
side
If
C(F) on
linear m a p s
f ~ ~p(n,X), into
X,
I 4 p ~ + ~,
which
satisfies
C(F).
We are c o n c e r n e d
A
C(F)
of
into
X
then (4.4),
in this
this
section
such that
HA ~ S p ( n , X ) . Necessary is bounded.
Throughout
that
14 p
that
MflHAIIP
condition
conditions.
< + ~.
on
this
B y Th.
When section
with
(I)
~(x)
We o b t a i n
the aid of Th.
we
see t h a t
it is u n d e r s t o o d
2, Ch. II, w i t h
is q u a s i - b o u n d e d . A
p = + ~,
2(ii),
liA(f) fl 4
that
= exp(px)
MIIHAll H and
is b o u n d e d
= MIIHAII.
Suppose
u = logllHA11
from these o b s e r v a t i o n s
a n d so H* now
we c o n c l u d e
a first n e c e s s a r y
this Ch.:
Iflu6~_, du
(5.1)
-
f ~ C(F),
where
a
is a n o n - n e g a t i v e
i03
-
member
of
L
(F). P
A second siderations. ~.
With
necessary
Let
~
license
is o b t a i n e d
be an a n a l y t i c
the n o t a t i o n
notational
condition
abelian
of the p r e c e d i n g
in w r i t i n g
the C a u c h y
differential
section
in
~,~~
w/6~a,
theorem
on some
force w i t h
open
u = HA,
and
limit
con-
set c o n t a i n i n g allowing
a slight
tend
O
we h a v e
/ ,
from
-i
2H
H~ ~
6~a = 0
6~ a
(c)
for s m a l l
positive
c
b y the C a u c h y
integral
theorem.
On
letting
c
to
we
obtain / k I~
(II)
~ ,I= O
(5.2)
\ ja/ for e a c h
admitted
We now thereby
~.
show
a theorem
has
IIHAIIP
There showing
that
analytic.
1
at the of
The
conclude
a harmonic
majorant.
to be
shown
linear
time)
with
and the
and hence
analyticity
of
that HA
analyticity
(I) a n d
(II) , t h e n
then
H A ~ %(~,X).
functions.
H A is b o u n d e d
is a n a l y t i c
on
We a c c o m p l i s h
the
function
locally
in t e r m s
of a uniformizer
function
composed
with
(being
is c o n s t a n t
b y the
is
as the
conjugation, analytic
( + ~;
this b y
1 o HA
HA
function
X
I < p
~.
1
latter
on
and when
functional
is c o n s t a n t
We o b t a i n
sum
we c o n c l u d e
and antianalytic arbitrariness
follows.
of
1 o HA
(II) , r e p l a c i n g
admits
HA
function the
and
p = ~,
an a n a l y t i c latter
(I)
for v e c t o r - v a l u e d
that
t h e n on r e p r e s e n t i n g function
that ~
type
that when
To s h o w the satisfies
satisfies
(I)
composed
i.
A
for e a c h b o u n d e d
For
same
from
remains
o f an a n a l y t i c that
if
of Cauchy-Read
It is c l e a r then
that
A
a representation
we proceed
, ~ being of the
as
follows.
appropriately
form
Let
changed.
~
= 1 o k. From
(I) w e
Then
-
f E C(F)
where
the fact that
~
[
= ~-~
f~6~a,
is a complex-valued member of
MIH~I
is quasibounded
of quasibounded harmonic
and so
functions on
condition of Cauchy-Read. taken together
-
f
i
,~(f)
1 0 4
Hence
~.
H~
L(F).
Re H ~
Using
This may be concluded
and
Im H ~
from
are both the difference
(II) we see that
~
satisfies the
is analytic.
The sufficiency of
There remains
to be considered
(I) and
(II)
follows.
Continuous boundary characterizing
AIF
is determined by
for
AIF
function. A
a continuaus map of
~
into
X,
is obvious by the maximum principle
which yields max~llAll = maxFflAfl. ]
Suppose that
B = AfF
the problem of
analytic
in
for subharmonic
for some allowed
~. [That A functions A.
Then we
have (III)
i
(5.3)
Be = 0
_.v
for allowed
w
as we see from the Cauchy integral theorem.
Suppose that ~.
We introduce
B
is a continuous map of
F
into
X
satisfying
(III)
for allowed
k by
A(f)
= i
j
fB6 ~a, F
f ~ C(F). Hence
The conditions
HA u B
(I) and
is an allowed
A
(II) are fulfilled. satisfying
HA u B
is continuous
on
Q.
AIF = B.
To sum upj we have Theorem 3=
Let A be a continuous
condition that H A ~ ~ ( e , X )
linear map of C(F)
is that A satisfy
(I) and
into X. A necessary and sufficient (II).
Let B be a continuous map of F into X. A necessary and sufficien~ conditio~ that there exist a continuous map A o f ~ into X, analytic B satisfy
(III).
in ~, satisfyinq AIF = B is that
-
1 0 5
-
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