Uniform Algebras and Jensen Measures (London Mathematical Society Lecture Note Series)

e Note Seri

Uniform algebras and Jensen measures

CAMBRIDGE ( NIN

PRLS S

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London Mathematical Society Lecture Note Series.

32

Uniform Algebras and Jensen Measures

T. W. GAMELIN Professor of Mathematics

University of California Los Angeles

CAMBRIDGE UNIVERSITY PRESS CAMBRIDGE LONDON

NEW YORK

MELBOURNE

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo

Cambridge University Press The Edinburgh Building, Cambridge C132 8RU, UK

Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521222808

© Cambridge University Press 1978

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1978 Re-issued in this digitally printed version 2008

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Contents Page

Preface

vii

1.

Choquet Theory

2.

Classes of Representing Measures

22

3.

The Algebra

34

4.

The Corona Problem for Riemann Surfaces

46

5.

Subharmonicity with Respect to a Uniform Algebra

54

6.

Algebras of Analytic Functions

83

7.

The Conjugation Operation for Representing

1

R(K)

Measures

107

8.

The Conjugation Operation for Jensen Measures

129

9.

Moduli of Functions in

H

2

(o)

146

List of Notation

156

Index

158

Preface

These notes are based on lectures given in various courses and seminars over past years.

The unifying theme is the no-

tion of subharmonicity with respect to a uniform algebra. Dual to the generalized subharmonic functions are the Jensen measures.

Chapter 1 includes an abstract treatment of Jensen measures, which also includes the standard basic elements of Choquet theory.

It is based on an approach of D.A.Edwards. Chapter 2

shows how the various classes of representing measures fit into the abstract setting, and Chapter 3 deals specifically with the algebra

R(K)

.

In Chapter 4, we present an example due to B.Cole of a Riemann surface ideal space of

R

which fails to be dense in the maximal

H(R)

.

Chapter 5 is based upon recent work of N.Sibony and the author concerning algebras generated by Hartogs series, and the abstract Dirichlet problem for function algebras.

The

abstract development is applied in Chapter 6 to algebras of analytic functions of several complex variables.

Here the

generalized subharmonic functions turn out to be closely related to the plurisubharmonic functions, and the abstract Dirichlet problem turns out to be Bremermann's generalized Dirichlet problem.

Chapters 7 and 8 are devoted to Cole's theory of the conjugation operator in the setting of uniform algebras.

The

problem is to determine which of the classical estimates relating a trigonometric polynomial and its conjugate extend to the abstract setting.

Cole shows that many inequalities

fail to extend to arbitrary representing measures, while

"all" inequalities extend to the context of Jensen measures. In Chapter 9, the problem of characterizing the moduli of the functions in

H2(a)

The discussion is

is considered.

based on Cole's proof of a theorem of Helson, which frees Helson's theorem from the underlying group structure. At the

References are given at the end of each chapter. very end of the notes, there is an index of symbols.

In preparing these notes, I have benefited from mathematical contacts with a number of people.

Let me acknowledge

first and foremost my debt to Brian Cole.

His incisive ideas

and remarkable results form the basis for a sizeable portion of these lecture notes.

Special thanks go to Don Marshall,

for writing up one of the preliminary versions of Chapter 9. I would like to thank Julie Honig for her excellent work typing the penultimate version of the manuscript.

And I

would like to thank the staff at the Cambridge University Press for facilitating the publication of these notes.

T.W.Gamelin

Harcourt Hill Oxford 1978

1

Choquet theory

Here the basic ideas of Choquet theory are developed in a framework suitable for uniform algebras.

The lectures of R.

Phelps[6] provide a very readable account of Choquet theory, as does also the expository paper of G.Choquet and P.A. Their approach has been modified by D.A.Edwards[4],

Meyer[3].

in order to handle Jensen measures and the Jensen-Hartogs inequality for function algebras.

We will follow the develop-

ment of Choquet and Meyer, as amended by Edwards.

R-measures

Let M be a compact space, and let

R

be a family of con-

tinuous functions from M to the extended line will assume always that

R

If

R

R

E--,+-)

and

v,w e R

,

separates the points of

An R-measure for

e M

We

has the following properties.

includes the constant functions.

m E Z+

.

then

(v+w)/m e R

(1.1)

(1.2)

.

M .

(1.3)

is a probability measure

o

on M

such that

w e R

Since

R

(1.4)

.

includes the constants, the estimate (1.4) is

equivalent to the estimate

jwda ? 0

,

for all

w c R

such that

0

.

(1.5)

1

As an example, observe that the point mass always an R-measure for

at

6

is

.

The theory applies to any linear subset

R

of

CR(M)

M

that contains the constants and separates the points of In this case, the fact that

R = -R

measures for a point

are the representing measures

for

,

e M

implies that the R-

that is, the probability measures

a

on M

that

satisfy UM = Judo

all u e R

,

.

In the principal application dealt with by Choquet theory,

M

is a compact convex subset of a locally convex linear

topological vector space, and

is the space of continuous

R

real-valued affine functions on M . probability measure e M ,

$

on M

a

In this case, each

is an R-measure for some

being referred to as the "barycenter" of

a

The main example that will occupy our attention is the

case in which

consists of functions of the form

R

(log IfI)/m , where m

is a positive integer, and

A

longs to an algebra

a

on

M

be-

of continuous complex-valued func-

tions on some compact space measure

f

In this case, a probability

M .

is an R-measure for

e M

if and only

if the Jensen-Hartogs inequality is valid:

log

If

I

log

<_

f e A.

do ,

IfI

J

The R-measures are called Jensen measures. Now fix U

¢ E M , and fix a compact subset

be the set of functions

some

w e R

satisfying

are continuous, then

U

u e CR(X) 0

2

.

such that

of

M .

u > w

If the functions in

Let for R

is simply the algebraic sum of the

positive continuous functions on vanishing at

.

X

X , and the functions in R

On account of (1.2), the cone

is a convex cone.

U

includes the positive functions in

U

Since the restriction of every envelope of functions in X

U

w c R

CR(X)

a probability measure

,

is an R-measure if and only if

.

is a lower

X

to

0 E R,

Since

on

a

for all u E U.

f uda ? 0

In particular, the R-measures can be described by inequalities involving integrals of continuous functions, so that the set

of R-measures on

X

is a convex, weak-star compact set.

Let

X

be a compact subset of

Lemma.

1.1

There exists an R-measure

¢ E M . X

for

a

M , and let

supported on

¢

if and only if WW <_

sup w(x)

w E R

,

(1.6)

.

XE X

If there is an R-measure

Proof.

the inequality

on X , then

for

a

yields (1.6) immediately.

fX wda

Conversely, if (1.6) is valid, then the constant function -1 does not belong to

U , and

is a proper cone in

U

CR(X)

.

By the separation theorem for convex sets, there is a nonzero measure Since

on

T

U

measure.

X

such that

f udT ? 0

for all

includes the positive functions, The measure

U

.

is a positive

T

is then a probability

a = T/T(X)

measure that is nonnegative on

u E U

,

so that

a

is an R-

measure.

The following version of the monotone extension theorem is due in this setting to D.A.Edwards[4].

1.2

Theorem (Edwards' Theorem).

a compact subset of function

Q

from

M .

X

to

Let

E M , and let

X be

For each lower semi-continuous (-co,+cx]

,

the following quantities

are equal:

3

on

w E R, w <_ Q

sup{w(4)

inf{f Qda

X },

(1.7)

is an R-measure for

a

on X}.

p

+°

Here the infimum in (1.8) is declared to be no R-measure for

Let

Proof.

on

4

X

denote the supremum in (1.7) and let

S

w(4) > 0

w < 0

while

,

de-

I

If there are no R-measures for

there is by Lemma 1.1 a function

X ,

if there is

.

note the infimum in (1.8). on

(1.8)

on

X

w

Multiplying

.

w

positive constant, we can arrange that Hence

is arbitrarily large.

w E R

Q

S = +-

,

such that

by a large

on

X , while

S = I

so that

We can assume then that there exist R-measures for X

If

.

w 5 Q

is such an R-measure, and if

a

fwda _< fQda

then

,

w E R

Hence

.

on

satisfies

S 5 I

.

To prove the reverse inequality, suppose first that Let

continuous.

w(4) = 0

b > S

w + b <_ Q

and

long to the cone

Then there is no

.

Consequently

.

such that

does not be-

Q - b

theorem for convex sets, there is a nonzero measure such that

X

is nonnegative on

T

on

a = T/T(X)

U

Since

T ? 0

U

,

more,

.(Q-b)do <_ 0

,

b > S

is arbitrary, we conclude that

so that

is

By the separation

defined earlier.

U

w E R

Q

,

while

f(Q-b)dT :- 0.

Further-

is an R-measure.

fQda <_ b

and

,

I = S

on

T

I <_ b .

Since

in the case

,

at hand.

For the general case, consider a continuous function q <- Q

,

and let

mined by

q

.

Iq = Sq

denote the quantity above deter-

For each such

q

,

choose an R-measure

a q

such that

This choice is possible; since the

= S

fgda q

q

set of R-measures is weak-star compact.

Let

star adherent point of the net

as

Q

.

fies

If

q

increases to

q

is fixed, and the continuous function

q 5 p _< Q

,

then

< fpda

fqda p

4

{oq}q
be a weak-

a

= S

p

<_

P

S

.

p

satis-

Letting

p

increase to

Q , we obtain fQda <_ S

arbitrary,

exists an R-measure

.

fqda <- S

Hence

I = S

such that

a

Since

.

,

is

and moreover there

fQda = I

11

.

Applying Edwards' Theorem to the function

-XE , where

is the characteristic function of a closed subset

XE X

q <_ Q

,

of

E

we obtain the following corollary.

1.3

Suppose there is an R-measure on

Corollary.

Then for any closed subset

sup{a(E)

of

E

X ,

X

an R-measure on

:

a

:

u c -R, u >_ 0

X for

for

is equal to

inf{u(O

The Family

S

on

X, u ? 1

of

M

to

E}

of R-envelope Functions

A lower semi-continuous function X

on

(-oo,+co]

u

from a compact subset

is an R-envelope function on

is the upper envelope on

X

u

if

R

.

The family

is denoted by S

.

In the ab-

of functions in

of R-envelope functions on M

X

stract framework, the R-envelope functions often play the

role that the subharmonic functions play in classical potential theory.

Every R-envelope function on tion to If

X

of a function in

c e ]R, and if

S

X

is evidently the restric-

.

w1,...,wn e R

,

then the function

max(c,wl,...,wn)

is a continuous R-envelope function.

The functions in

S

are simply the upper envelopes of functions of the form (1.9). An elementary compactness argument establishes the following.

5

The continuous R-envelope functions are the

Lemma.

1.4

uniform limits of functions of the form max(c,wl,...,wn) where

is real and

c

wl,...,wn E R

.

has the

S

From the definitions, we see immediately that following properties.

includes the constant functions.

S

(1.10)

If u c S and c> 0, then cu E S. If

u,v E S

If

{ua}

then u+ v E S

,

is any subset of then

S

,

belongs to

sup ua

The following property of

(1.12)

.

S

(1.13)

.

is sufficiently important

S

to merit a separate statement.

Lemma.

1.5

u c S , and if

If

is an increasing continu-

X

ous convex function from an interval containing the range of u

(--,+-]

to

then

,

Xou E S

.

There is only one possible point of discontinuity of

Proof.

an arbitrary increasing convex function S

X(t) = - for

such that

t < S

We must assume that

.

increases to

.

, while

X(t)

tends to

,

X

where

namely, a point x(t) < +as

X(R)

for t

is an upper envelope of a > 0

.

Consequently

is an upper envelope of functions of the form au + b

Xou

where Xou

at + b

,

t >

In this case,

functions of the form

X

a > 0

.

Since each of these belongs to

S

,

so does

0

.

The R-envelope functions are dual, in some sense, to Rmeasures.

6

This duality is exhibited by the following

characterization of the R-envelope functions.

Let

Theorem.

1.6

from M

(--,+-]

to

be a lower semi-continuous function

u

Then

.

u

is an R-envelope function

if and only if

u(4)) <

for all

Judo

4)

(1.14)

and all R-measures

c M

Since (1.14) holds for all

Proof.

on M

a

u e R

for all upper envelopes of functions in u e S

4

.

it also holds

,

R

for

,

hence for all

.

Conversely, suppose that (1.14) is valid for all and all R-measures

on M

a

tinuous function on M

for

4)

Let

.

v < u

such that

$ e M

be any con-

v

According to

.

Edwards' Theorem (Theorem 1.2), there exists for each and each

e > 0

w(4)) > u(¢)-e

,

It follows that

.

functions in

R

a function w E R u

¢ E' M

w < u , while

such that

is an upper envelope of

.

11

From Theorem 1.6 and Fatou's Lemma, we obtain immediately the following.

1.7

Corollary.

{u.}_j_

If

i

bounded above, and if

-1

is a sequence in

u = lim sups

lower semi-continuous, then

u.

S

that is

is bounded and

u E S

There is another simple proof of Lemma 1.5, based on Theorem 1.6 and Jensen's inequality.

Recall that Jensen's

inequality is the estimate

X(Jud(j )

<_

Jxou do

valid whenever

a

(1.15)

,

is a probability measure,

u

is real-

7

valued and

is an increasing convex real-valued function

X

of a real variable.

The validity of (1.15) for simple func-

tions boils down to the convexity of

X

.

To prove Lemma 1.5, one notes that if for

,

X(fudo) <_ fXou do

then Xou E S

Theorem 1.6,

We denote by

tions on M

so that by

,

.

of Continuous R-envelope Functions

SC

The Family

is an R-measure

a

SC

the family of (finite) R-envelope funcAs observed earlier in

that are continuous.

Lemma 1.4, these are the uniform limits of the functions of the form (1.9).

Evidently

SC

is a convex cone that separates points and Consequently

contains the constants.

enjoys the proper-

SC

ties (1.1), (1.2) and (1.3) postulated for we have developed can be applied to

SC

R

.

The theory

in place of

R

Observe though that the SC-measures are precisely the Rmeasures, while the SC-envelope functions coincide with the R-envelope functions.

For many purposes, the family

be replaced by the family

SC

R

,

can

.

An important property enjoyed by by

R

SC

is that of being a semi-lattice.

two functions in

,

but not necessarily The maximum of any

This leads to Sc again belongs to SC the following observation, which plays a crucial role in the .

treatment of maximal measures.

Lemma.

1.8 CR(M)

Proof.

The algebraic difference

is dense in

.

If

v1,v2,w1,w2 E SC

max (v1 - wl , v2 It follows that

8

SC - Sc

_'2 )

,

then

= max (v1 +w

SC - SC

2)

v2 + w1) - w1 - w2

is a lattice.

Since it separates

points and contains the constants, it is dense in

CR(M) (M)

the lattice version of the Stone-Weierstrass Theorem.

11

The R-Dirichlet Problem Let

be a lower semi-continuous function from M

u

(-o,+c]

to

In analogy to the procedure followed by Perron to

.

solve the Dirichlet problem, we define the (lower) solution to the R-Dirichlet problem with data upper envelope

u

on M

u

of the functions in

to be the

dominated by

R

u

on M : sup{w(m)

: w c R , w < u on M }

The supremum defining

functions in S

,

u

could as well be taken over the

or in SC , dominated by

Theorem gives an alternative expression for

inf{Judo

o

:

(1.16)

.

u

.

Edwards'

u

an R-measure on M

}.

for

Some elementary properties of the correspondence

u -; u

are

as follows:

is an R-envelope function

(1.17)

u= u if and only if u e S

(1.18)

cu = cu

(1.19)

u

u+v <_ u+v u <- v

c > 0

if

,

whenever

if a net

,

{u }

(1.20)

(1.21)

u <_ v

of lower semi-continuous functions

increases pointwise on M

creases pointwise on M

u

to to

u

.

,

then

u

a

in(1.22)

9

In applications, we will wish to consider a lower semicontinuous boundary function subset

X

envelope on M on

Again

M .

of

defined only on a compact

u

is defined to be the upper

u

This amounts to declaring

X .

and defined

inf{Judo

where the infimum is declared to be measures for

¢

on

X

an R-measure on

a

:

+-

to be

u

u

M\X

on

From Edwards' Theorem, we obtain

as before.

u

dominated by

S

of the functions in

+00

for

},

(1.23)

if there are no R-

X .

The Choquet Boundary The Choquet boundary of those points

4)

e M

R , denoted by

aR

,

such that the point mass

the only R-measure for

4

consists of at

d

is

4)

.

By Edwards' Theorem, any continuous real-valued function u

on M

u = u

satisfies

on the Choquet boundary.

property characterizes the Choquet boundary.

This

Indeed, suppose

4)

is not a Choquet boundary point, and choose an R-measure

o

for

ing

a x 6 .

such that

4)

Judo < u(4))

Then any

also satisfies

,

u e CR(M)

u(4)) < u(¢)

satisfy-

.

The next lemma shows that the notion of Choquet boundary point is independent, in the appropriate sense, of the com-

pact set on which the functions in

1.9

Lemma.

Let

every w e R

X

be a closed subset of

attains its maximum on

If the point mass at X ,

then

Proof.

x0

such that

x0 e X

X , and let

is the only R--measure for

a

be an R-measure for

a compact neighbourhood of

continuous function on

10

M

belongs to the Choquet boundary of

x0

Let

are defined.

R

X

x0

in

such that

x0 X

,

R

on M . and let

.

Let u

on

x0

E

be

be a

u <_ 0, u(x0) = 0, and

X\E

on

u <_ -1

From (1.23), we obtain

Suppose and

5(4>) = 0

estimates ,

Let

.

supp(a)

,

0 = u(4>) < fudp <_ fudp

so that

are dense in

supp(a)

,

X

show that

is supported on

p

E

for

E

Hence

w(4>) < w(x0)

for all

Since

w(x0) <- fwda

for all

Since

.

E

4>

4>

E supp(o)

4>

has

is an arbitrary

w E R

on

w(x0)

E supp(o)

supp(a)

supp(a)

.

x0

w E R

and all

each w E R

,

is the point mass at

a

on

u = 0

is an R-measure for every point of

ates points,

The

.

x0 , we conclude that the point

compact neighbourhood of

the constant value

4>

a

and since any weak-star limit of R-

an R-measure supported on

x0

(do)

Since such

.

measures is an R-measure, we see that every

mass at

The

.

the closed support of

be an R-measure on

p

u(x0) = 0

u = 0 a.e.

then shows that

belongs to

4>

supp(p)

u <_ 0 on M , while

u(x0) < fudo

estimate

u on M.

Consider the R-envelope function

.

Since

assumes

R

separ-

.

K. de Leeuw[2] have given an example, a por-

E.Bishop and

cupine space, for which the Choquet boundary is not a Borel

This adverse behaviour does not occur when M

set.

is

metrizable.

1.10 aR

Lemma.

M

If

is metrizable, then the Choquet boundary

is a Gd-set.

Let

Proof.

{u.}

dently a point and only if {uj = u

}

4>

=1

be a dense sequence in

CR(X)

.

EviE M belongs to the Choquet boundary if

u.(4>) = u.(4>)

for all

j

.

Since each set

is the intersection of the open sets

{u.-uj <1/n},

j

n >_ 1

,

we see that

aR

is itself a countable intersection

of open sets.

1.11

Theorem.

Each

u e SC

attains its maximum at a

Chouquet boundary point, as does each

w E R

.

11

The latter assertion follows by applying the former

Proof.

to a function of the form u = max(c,w)

, where

is very

c

negative.

The proof of the first assertion is modelled on a standard proof of the Krein-Milman Theorem.

We begin by introducing

an auxiliary notion, corresponding in convexity theory to a face of a convex set.

A closed subset for each point of

is an R-face if every R-measure

is supported by

E

{4}

A singleton

itself an R-face.

E

Evidently M

.

is

is an R-face if and only

is a Choquet boundary point.

if

Let

M

M

of

E

u c SC

E M

Suppose

.

and let

,

mea sure for

satisfies

a

If

.

on

u

is an R-

a

then the estimates u <_ a and a =

q ,

on the support of

u = a

u c SC

which any

be the maximum value of

a

a

Hence the set on

.

attains its maximum is an R-face.

The R-faces evidently form an inductive family, when By Zorn's Lemma, every R-face includes

ordered by inclusion.

Therefore it suffices to show that every

a minimal R-face.

minimal R-face consists of one point. Suppose that

be the subset of

F0

let

E FO

Since

F

is an R-face,

of

R

subset

to

F

F0

Since

.

of

Hence

RIF-face.

is an R-face.

F0 ,

and

and since

v

v

a

Since

single point.

let

M F

for .

.

Furthermore,

is the restriction

RIF

is an RIF-envelope function, the attains its maximum is an

v

is supported on

is constant on SC

where

,

,

attains its maximum,

v

is supported on

a

on which

F

v E SC

be an R-measure on

a

is an RIF-measure for

a

F

on which

F

and let

,

Let

is a minimal R-face.

F

FO

is.minimal,

F F

.

Since

separates the point of

It follows that

.

coincides with

F0

v E SC M ,

F

is arbitrary,

reduces to a 0

As a corollary, we obtain a version of Shilov's Theorem on

12

the existence of minimal closed boundaries.

1.12

There exists a smallest closed subset of

Theorem.

namely the closure of the Choquet boundary of each function in

tion in

R

,

on which

attains its maximum.

R

If X is any closed subset of

Proof.

R

R

on which each func-

R

E M

attains its maximum, then by Lemma 1.1 all Therefore

have an R-measure on

X

the closure of

On the other hand, Theorem 1.11 shows

that every

DR

.

.

attains its maximum on

w E R

We will refer to the closure of associated with

includes

X

R

8R

DR

DR

,

hence

0

.

as the Shilov boundary

.

Barriers

In analogy with classical potential theory, we say that an R-envelope function u <_ 0,

rier

u

on M

u u

at

,

any R-measure

is a Choquet boundary point. u

for

a

0

if

a bar-

0

then

deed, the estimates

E M

is a barrier at

and

0 =

show that

Judo

is concentrated at

In-

.

There is also a topological condition that must be satisfied for

to have a barrier.

If

u

then the intersection of the open sets includes only

,

is a barrier at

{u > -1/n}, n ? 1

so that the singleton

Conversely, if

{4}

is any continuous function on v

is a Gd-set.

is a Choquet boundary point which

forms a GS-set, then there is a barrier at v

,

0

M

such that

.

Indeed, if v <_ 0,

a barrier

at

In the setting of classical potential theory, M.V.

Keldysh[51 has proved that any point having a barrier actually has a continuous barrier.

This theorem extends to our

13

In fact we have the following characteriz-

general setting.

ation of Choquet boundary points.

1.13

The following are equivalent, for a point

Theorem.

¢OE M. (i)

belongs to the Choquet boundary of

0

(ii)

.

is any continuous real-valued function on

h

If

R

M , then there exists a continuous R-envelope function u

on M

such that

u <_ h

There exist

(iii)

while

there exists

such that

v E R

.

with the following prop-

a < 0 < B

for any compact subset

erty:

u(¢0) = h4 0)

E

M

of

not containing O ,

v(40) = 0

v <_ a

,

E

on

and v<_S on M. a < 0 < S

Evidently (ii) implies (iii), for any

Proof.

.

Suppose that (iii) is valid, and suppose furthermore that Let

(i) fails.

the point mass where

at 0 .

d0

Write

a = aT + (1-a)60 ,

is a probability measure with no mass at

T

0 < a <_ 1

let

be an R-measure for 0 distinct from

a

Let

.

be a compact set not including

E

be the corresponding function from (iii).

v

0 = v(40) <_ fvda = afvdT 5 aaT(E) + aBT(M\E)

chosen, though, so that most of the mass of by

E

,

diction.

0 ,and

.

T

If

0 ,and Then E

is

is carried

this last sum is negative, and we obtain a contraIt follows that (iii) implies (i).

For the remaining implication, assume that (i) is valid, and let

h E CR(M)

.

Fix real numbers

b

and

c

such that

of

M

not in-

b<minh<_maxh
O , there exists

v and

14

v <_ E

CR(M)

v E R

E

such that c

satisfy

g 5 h(¢0)

on M,

and

on

g <_ b -c

Since

.

w

such that

w c R

E

there exists

is sufficiently small, then

-C

w(40) =

g , while

If

.

e

has the desired proper-

v = w +e

ties.

Now fix

0 < s < 1

and choose a sequence

,

decreases rapidly to zero.

and on

We construct by induction a sequence Let

0 5 j <_ m -1

max

u.(@)

m

's

j J=0

in

R

as

Suppose

5 b on

E

}

m

um c R

we may choose

u

m

u.(¢0) =

e

OSj<_m-l

then does not contain 0 .

and

which

The compact set

.

m

{u.}-

be the constant function

u0

have been chosen so that

u0,...,um-1

E

M }

m m=1 Precise conditions on the e

will be specified momentarily.

s

follows.

{c

By our preliminary observation, um(g0) = h(¢0) ,

so that

um <_ c

m

Now consider the series

u = (1 - s) I s3u.

J

j=0

u.'s

If the

are not uniformly bounded below, we replace u.

max(-y,u.) e SC , when

by

are uniformly bounded, and this insures that the

u 's

the

Then

is a large constant.

y

j

series converges uniformly on M Furthermore,

h(q0)

u .(4)

If

for all

<_

to a function

We must show that

.

j

,

u e SC u <_ h

5 h(4).

then certainly

J

Suppose that u> h(4) E Uk=l Ek

,

and since the

a first index m ? 0

u

(0)

m

<_

c

,

and

u

J

Ek's

for

5 b

0 <-

for

j

.

Then

are increasing, there is

such that

h(O) + cm

Then

for some index

e Em+l

while

j 5 m -1 , j

> m

.

4

Em

while

Substituting

these estimates in the series defining u, and taking into account the appropriate modifications in the case

m = 0

,

15

we obtain

(1 - s)

r

m-1

s] + sm c + b

)

e

j=0

1

j=m+l sJJ

cm) + (1 - s) smc + sm+lb

(1 - sm)

= h(¢) + (1-sIl1)cm + so near

< 1

s

(1 - s)c + sb - min h < 0 and then we choose the

that

1

,

so small that

e 's

m

(1 - siIl)em + sm[(1 - s)c + sb - min h] < 0 we obtain the estimate

m?1

,

,

u(p) <

0

The idea used in the preceding proof stems from Bishop[1,2].

Maximal Measures We define a partial ordering on the probability measures

on M , by declaring u E SC

all

.

w c R

symmetric and transitive.

space generated by p < v

and

fudp 5 fudv

v < p

SC

.

The relation ".e" is evidently

Since by Lemma 1.8 the linear

is dense in

CR(M)

together imply that

,

p = v

the relations .

As an example, observe that the probability measures

on M

such that satisfy

for the point

for

This is equivalent to requiring that for all

fwdp <_ fwdv

to mean that

p < v

E M .

6 < a

a

are precisely the R-measures

Heuristically speaking, the larger a

probability measure is with respect to this ordering, the more it is dispersed towards the boundary.

A probability measure on M

16

is maximal if it is maximal

Since the conditions de-

with respect to the ordering "<".

fining the ordering are weak-star continuous, a weak-star compactness argument shows that the set of probability measures, with the ordering "<", is inductive.

By Zorn's

Lemma, every probability measure is dominated by a maximal In particular, every point

probability measure.

e M

has

a maximal R-measure.

Our aim now is to show that maximal measures are supported We begin by stating a version

"near" the Choquet boundary.

of the Hahn-Banach Theorem that is convenient for our purposes.

1.14

vector space V

Let

Theorem. V

be a sublinear functional on a real

p

that is,

,

is a real-valued function on

p

such that

P(u+v) <_ P(u) + p(v)

p(cu) = cp(u)

u,v E V

,

u E V, c

,

>_ 0

Then there exists a linear functional L <_ p

Furthermore,

.

Proof.

Let

such that

V

L = p

is

p

.

We consider only the uniqueness assertion. be a one-dimensional subspace of

V0

v

by a nonzero vector dominated by <_ p(-v)

on

L

is unique if and only if

L

linear, in which case

.

,

L0

Theorem to extend the interval

on L0

VO to

[-p(-v),p(v)] L

on

V

,

V

L0

LO(v) <_ p(v)

that is, if and only if

Defining first

functional

A linear function

.

if and only if

p

V

generated

,

on and

V0

is

L0(-v)

-p(-v) <_ LO(v) <_ p(v)

.

and then using the Hahn-Banach we see that any real value in is assumed at

dominated by

p

tension is then unique if and only if

v e V , and in this case L = p

.

v

by some linear

The dominated exp(v) = -p(-v)

for all

.

17

1.15

Let

Lemma.

be a probability measure on M , and

v

define a sublinear functional

p(u) = - J

(-u)dv

The functionals

by

u E CR(M)

,

on

L

p

CR(M)

.

dominated by

those arising from probability measures

Suppose first that

Proof.

L(l) = 1

also p(u)

L

<_ 0

,

and

u 5 0

If

.

L(u)

L <_ p

<_ 0

,

satisfying v < p

u

Since

.

-p(-l) = p(1) =1

then -u? 0 ,

is represented by a probability measure

then v < u

(-u)

It follows that

.

are precisely

p

L p

>_ 0

0

Judv = Judv = -p(-u) <_ -L(-u) = L(u) = Judp

,

so that

,

u e SC

If

.

,

,

Hence

.

.

Conversely, suppose that

v < p

Let

.

v,w E SC

It is

.

easy to verify the identity w _v = w +(--V) , and from this

we obtain

-p(v-w) = f(w-)dv <_ J(w-v)dp = J,wdp + f(-v)dp 5

J(w-v)dp = -L(v-w)

v- w are dense in

u E CR(M)

, and

According to Lemma 1.8, such functions

.

CR(M)

L <_ p

Hence

.

-p(u) <_ -L(u)

for all

.

L1

Combining the two preceding lemmas, we arrive easily at the main result of this section.

1.16

A probability measure

Theorem.

if and only if for each GS-set

{u = u}

Proof.

If

u E CR(M)

,

is carried by the

is maximal, then by Lemma 1.15,

unique linear functional on linear function

p

carried on the set

u E CR(X)

{u = u}

.

is the

By the uniqueness

defined in Lemma 1.15.

for all

v

dominated by the sub-

CR(X)

clause of the Hahn-Banach Theorem,

18

v

is maximal

.

v

-p(-u) = Judy

on M

v

v = p .

Since

.

Hence u 5 u ,

Judy = v

is

Conversely, if Judv = Judv

,

and

dominated by

CR(X)

u e CR(X)

for all

v e CR(X)

all

is carried by each set

v

p(v) = fvdv

then

,

for

is the unique linear functional on

v p

Hence

.

{u = u}

By Lemma 1.15,

.

is maximal.

v

11

As a simple application of Theorem 1.16, we observe that every maximal measure ary.

Indeed, suppose that Choose

Shilov boundary.

while

is supported by the Shilov bound-

v

does not belong to the

p e M

such that

u c CR(M)

on the Shilov boundary.

u = 0

Then

1.16 implies that the closed support of the set

{u <_ 0}

Now each of the sets aR

,

and

u <_ 0

v

p

Theorem

.

is a G6-set that includes

{u = u}

is the intersection of these sets.

3R

.

is contained in

and hence does not include

,

u(p) > 0

The ques-

tion arises as to whether the maximal measures are precisely those probability measures carried by hope fails spectacularly.

.

This fleeting

The examples of Bishop and de

Leeuw cited earlier show that Moreover, even if

aR

need not be a Borel set.

aR

is a Borel set, there may exist maxi-

DR

mal R-measures with no mass on

3R.

.

In the positive direction, Bishop and de Leeuw have shown that a maximal measure has zero mass on any Baire set that is disjoint from

aR

.

From this, they obtain easily their

generalization of Choquet's Theorem to the nonmetrizable case.

In our setting, their result asserts that for each

¢ e M , there is a probability measure generated. by

aR

u e S

In the case that DR

sets of the form

on the a-ring

and the Baire sets, such that

and

1.10 shows that

p

M

p(aR) = 1

.

is metrizable, the proof of Lemma

is the intersection of a sequence of

{u = u}

.

From Theorem 1.16 we deduce

that the maximal measures are the probability measures carried by

aR

.

In particular, we have the following version

19

of Choquet's Theorem.

M

Suppose that

Theorem.

1.17

Then every

is metrizable.

$ c M

has an R-measure that is carried by the Choquet bound-

ary

R.

Examples

M

For the most trivial example, we take on the real line, and we take

[a,R]

c + at

functions of the form

to be the set of

R

where

,

to be an interval

is real and

c

a >- 0.

In this case, the R-envelope functions are the continuous convex increasing functions from

[a,R]

(-,+-]

to

{g}

Choquet boundary consists of the singleton

M

More generally, let

is real and

where

c

in

are convex.

S

which

be any compact subset of

be the set of functions of the form

R

let

s

<-

t

a.

0

1

,

<_

s.

<_

tJ .

1 5 j

,

<_ n

IRn, and

u(t) = c

+En

a.t. J

j=1 J

The functions

.

Rn with the ordering for

If we provide

means

j

The

.

.

<_ n

then the func-

,

J

tions in

S

are increasing.

The Choquet boundary consists

of those extreme points of the closed convex hull of which are maximal with respect to this ordering of

E IRn

Next let us return to the principal example for convexity

theory, in which M

is a compact convex subset of a locally

convex real linear topological vector space

V

,

and

the family of continuous affine functions on M . the uniform limits on where

c

V

.

4

c + L

,

is a continuous linear

L

if and only if

is the barycenter of

is

These are

Recall that a probability measure

is an R-measure for

that is,

of functions of the form

is a real constant and

functional on

M

M

R

a

.

a

a

on

represents

The functions in SC

turn out to be the continuous convex real-valued functions on

M , and the Choquet boundary of

set of extreme points of Choquet's Theorem:

20

If

M .

M

R

coincides with the

Theorem 1.17 specializes to

is metrizable, then the set of

,

extreme points of

M

is a GS-set, and every

4 e M

is the

barycenter of a probability measure on the set of extreme points of

M .

References 1.

Bishop, E.

A minimal boundary for function algebras,

Pac. J. Math. 9 (1959), 629-642. 2.

Bishop, E. and de Leeuw, K.

The representation of linear

functionals by measures on sets of extreme points, Ann. Inst. Fourier (Grenoble) 9 (1959), 305-331. 3.

Choquet, G. and Meyer, P.A.

Existence et unicite des

representations intdgrals dans les convexes compacts quelconques, Ann. Inst. Fourier (Grenoble) 13 (1963), 139-154. 4.

Edwards, D.A.

Choquet boundary theory for certain spaces

of lower semicontinuous functions, in Function Algebras,

F. Birtel (ed.), Scott, Foresman and Co., 1966, pp.30030 9 . 5.

Keldysh, M.V.

On the solubility and stability of

Dirichlet's problem, Uspekhi Mat. Nauk USSR 8 (1941), 171-231. 6.

Phelps, R.R.

Lectures on Choquet's Theorem, Van Nostrand

Mathematical Studies No.7, Van Nostrand, 1966.

21

2 Classes of representing measures

A

Throughout these notes, we will denote by X

gebra on a compact space closed subalgebra of

A

will be denoted by

In other words,

.

X

is a

The maximal ideal space of

.

MA , and

algebra of functions on

A

containing the constants and

C(X)

separating the points of

a uniform al-

A

will be regarded as an

MA .

Associated with A , there are three natural choices for the space

R

of the preceding chapter.

in turn these three choices.

We will consider

The R-measures become respect-

ively the representing measures, the Arens-Singer measures, and the Jensen measures.

Representing Measures For this example, we take

R

real parts of functions in A . a probability measure

a

to be the space Since

on MA

Re(A)

of

is a linear space,

R

is an R-measure for

if and only if

u E Re(A)

.

E MA

This is

equivalent to

f W = J fdo

,

fEA

A probability measure

a

(2.1)

.

on

MA

that satisfies (2.1) is

said to be a representing measure for ary of

Re(A)

4

.

The Choquet bound-

is the set of points for which the point mass

is the only representing measure.

The Edwards Theorem (Theorem 1.2) specializes immediately to the following useful theorem.

2.1

22

Theorem.

If

4 E MA , and if

Q

is a lower semi-

continuous function from

X

to

u E Re(A), u < Q

(-co,+co] then

on

X },

(2.2)

is equal to

inf{1 Qda

a a representing measure for

:

on X} .

(2.3)

As an application of Theorem 1.11, we prove the existence

A

of a "Shilov boundary" for

There is a smallest closed subset

Theorem.

2.2

.

MA

of

E

such that every function in A attains its maximum modulus on

E

.

Let

Proof.

Re(A)

be the closure of the Choquet boundary of

E

By Theorem 1.11, every

.

mum modulus on

E

,

and consequently every

its maximum modulus on

Let

u c Re(A)

E

attains its maximum modulus on

f E A

f E A

attains

.

be another closed subset of

F

attains its maxi-

MA F

.

such that every Fix

E MA

.

Then

If(s) I

f(x) I,

sup

<_

feA

xE F

Hence there is a measure

a

4

on

u E Re(A)

u

By Theorem 1.11, F D E

such that

F

11011 = 1

is a probability measure, so that

representing measure for

and

on

Applying (2.1) to the function

while (2.1) holds. find that

a

F

F

.

a

1

,

we

,

is a

Since

attains its maximum on F

includes the Choquet boundary of

Re(A),

.

The subset

E

of Theorem 2.2 is called the Shilov boundary

23

of A

,

and it is denoted by

aA

Evidently

.

9A c X

.

Theorems 2.1 and 2.2 depend only on the linear structure of

A

of

C(X)

They are valid for any closed separating subspace

.

containing the constants.

Now we turn to a

characterization of the Choquet boundary points of

A

as the generalized peak points of on the algebraic structure of

A

Re(A)

This result depends

.

.

We begin by defining several concepts related to peak sets and peak points.

For more background and details,

see [4].

A closed subset function

f E A

MA\E

on

E

set

such that

The function

.

MA

of

E

is a peak set if there is a on

f = 1 f

E

while

,

fl

< 1

E. A closed

is said to peak on

is a.generalized peak set if it is an intersection It is easy to show that a generalized peak set

of peak sets.

is a peak set if and only if it is a GS-set .

A point

is a peak point for

¢ E MA

A

if the singleton

is a peak set, and it is a generalized peak point if is a generalized peak set.

A

algebra

on

(i)

cluding

exists MA

¢0

,

,

0 < a < 1

For each compact subset there is

g c A

gl <_a on

and

E

MA

of

not in-

satisfying

1

E .

For each strictly positive f E A

with the follow-

c ? 1

and

Re(A)

such that

h E CR(MA)

h(@0)

while

,

IfI

there <_ h

on

.

Proof.

Suppose that (iii) is valid.

then the function

24

and 0 E MA .

There exist

ing property.

(iv)

,

belongs to the Choquet boundary of

0

(iii)

II
MA

is a generalized peak point.

0

(ii)

II

The following are equivalent, for a uniform

Theorem.

2.3

u = Re(g) -1

in

If

Re(A)

g

is as in (iii), satisfies

u(40) = 0 ,

u <_

c

and

,

a -1

u <_

on

E

Consequently

.

the condition (iii) of Theorem 1.13 is met, and Choquet boundary point for

Re(A)

is a

0

Of course, it is easy

.

to show directly from (iii) that the point mass at the only representing measure for

is

0

In any event, (iii)

0 .

implies (ii).

By taking powers of peaking functions, one sees that (i) implies (iii), with

c = 1

and

a

arbitrarily small.

Since

(iv) obviously implies (i), it remains to prove that (ii) We will mimic the proof of the corresponding

implies (iv).

implication of Theorem 1.13.

Suppose then that (ii) is valid.

Choose

and

b

such

c

that

0 < b < min h <_ max h < c

.

We claim that for any compact subset

O , there exists and

Ifl

<_ b

f E A

on

E

If

not including <_ c'

IfI

Indeed, using Theorem 2.1 as in the

.

v(40) = log h(40) , g

MA

of

such that

proof of Theorem 1.13, we find that

E

v = Re(g)

v < log c

,

is normalized so that

in

and

such

Re(A)

on E.

v < log b

is real, then

f = eg

has the desired properties.

Now we proceed as in the proof of Theorem 1.13, fixing and a sequence

0 < s < 1 zero.

{em}m=1

decreasing rapidly to

We construct by induction a sequence

{f

.}J=1

in

A

J

as follows.

Let

fo,.... fm-1

so that

f0 = h(q0) f

be constant.

.4 0) =

0 <_ j

Having chosen < _ m-1

,

set

J

max

Em

and choose If M

I

< b

fm E A on

E

.

M

If.(4)I > h(j)+cm} so that

fm(¢0) =

IfmI < c

,

and

The series

25

f = (1-s)

s3f

I

j=0

converges uniformly, so that

f E A , and

The estimates in the proof of Theorem 1.13, with

re-

u. J

placed by

If .I

show that

,

Ifl 5 h

providing

s

and

J

fem}

are chosen properly.

Arens-Singer Measures As the second application, we take

to be the set of

R

functions

jm log fi : m E Z+, f e where

(2.4)

,

denotes the group of invertible elements of

A-1

Evidently

includes the constant functions, and

R

arates the points of

log

Al

If-1 I

= -log

MA

fl

.

A

.

sep-

R

From the identities

,

m2 ml

1 log If1I + m log If2I ml

=

2

we see that

R

In particular,

f1 f2

m m log 12

is a linear space over the rational numbers. satisfies (1.1) through (1.3), and the

R

theory developed in Chapter 1 applies to Since

R = -R

,

any R-measure

a

.

on

MA

for

a

on

MA

satis-

u

fies

the probability measures

for

R

log IfIda = log

f c A-1

such that

.

(2.5)

J

Such measures are called Arens-Singer measures. There is another way to view Arens-Singer measures. is easy to check that the functional

26

It

L(log Igi) = log Ig(4))I

g E A-1 ,

,

extends to a continuous linear functional on the closed linear span of

log IA-1I

The Arens-Singer measures corre-

.

spond to the positive extensions of this functional from the

log JA11

linear span of

By applying (2.5) to

CR(MA)

to

.

g E A

eg , where

is arbitrary, we

find that

gEA,

Re(g)da =

J

so that every Arens-Singer measure for measure for

This can also be deduced by observing that

.

4)

is a representing

4>

the family (2.4) defining the Arens-Singer measures for includes the family measures for

4

Re(A)

4>

defining the representing

.

The Arens-Singer boundary points are defined to be the points for which the point mass is the only Arens-Singer These points are now characterized by Theorem 1.13.

measure.

What is the Shilov boundary associated with this choice of

Since

R ?

point for

A

R

includes

Re(A)

,

every generalized peak

is an Arens-Singer boundary point.

other hand, every function in maximum on the Shilov boundary

evidently assumes its

R 3A

the Shilov boundary associated with In particular, every supported on

$ e MA

On the

of R

A

.

It follows that

coincides with

DA

has an Arens-Singer measure

3A

There is one class of examples for which every representing measure is an Arens-Singer measure.

If

MA

is simply con-

nected, in the sense that the Cech cohomology group = {O}

,

H 1 (MA;Z)

then according to the Arens-Royden Theorem, every in-

vertible function in the family

R

coincides with

A

is an exponential.

In this case,

of (2.4), defining the Arens-Singer measures, Re(A)

.

27

Jensen Measures The third choice for

R

leading to the most important

,

application for our purposes, is the family of functions

{m log IfI

:

f E A, M E Z+}

(2.6)

These functions are continuous, from MA line

[-co,+0o)

to the extended

Again (1.1), (1.2) and (1.3) are easy to

.

establish, so that the theory developed in Chapter 1 applies. It is convenient to note that we could as well work with the

cone of functions of the form

c log IfI

,

where

c > 0

and

f e A .

In this case, a probability measure measure for

log

If

a

on

MA

is an R-

if and only if

<_ I

1

log

I f I do ,

fEA.

(2.7)

Such measures are called Jensen measures, and the inequality (2.7) will be referred to as the Jensen-Hartogs inequality. Every Jensen measure is an Arens-Singer measure.

This

follows from the observation that the family (2.6) leading to the Jensen measures includes the family (2.4) leading to the Arens-Singer measures.

One can also obtain (2.5) di-

rectly by applying the Jensen-Hartogs inequality to

1/f

f

and

.

The Choquet boundary associated with the family (2.6) is called the Jensen boundary of

These are the points for

A .

which the point mass is the only Jensen measure. boundary includes the Arens-Singer boundary.

The Jensen

Since every

function in the family (2.6) assumes its maximum on

aA

the Shilov boundary associated with the family (2.6) co-

incides with the Shilov boundary

aA

of

A .

The R-envelope functions associated with the family (2.6) will be referred to as the log-envelope functions.

28

Thus a

function w

from a compact subset

is a log-envelope function on

w

and

w

if

E

MA

of

E

(-W,+-]

to

is bounded below,

is an upper envelope of functions of the form , where

c log Ifl

c > 0

f E A

and

.

The Jensen-Hartogs

inequality remains valid for log-envelope functions. The R-Dirichlet problem associated with this family of functions will be referred to as the A-Dirichlet problem. Thus the solution to the A-Dirichlet problem with boundary data

on the compact subset

u

envelope where

MA

of

E

of all functions of the form

u

c > 0

and

f E A

is the upper c log

c log Ifl

satisfy

< u

fl

on

,

E

We will return to study log-envelope functions and the ADirichlet problem in some detail in Chapter 5. By Lemma 1.5, the composition of an increasing convex function

with any log-envelope function is again a log-

X

In particular,

envelope function.

X(log

f

X(log If(x)I)da(x)

for any Jensen measure

for

a

,

f E A

,

(2.8)

This estimate follows

.

directly from the Jensen-Hartogs inequality and Jensen's X(t) = ept

In the special case

inequality.

,

where

p > 0

is fixed, (2.8) leads to

If(s) I

5

[JlfxIPdox]

1/p

fEA, p>0

,

Hence the evaluation functional at LP-metric, for all

p > 0

.

(2.9)

is continuous in the

.

The estimate (2.9) actually characterizes Jensen measures among the probability measures on

X

.

Indeed, the limit

relation

f(x)Ipda(x)]I/p =

lim

efloglf(x)Ida(x)

If

29

shows that (2.9) becomes the Jensen-Hartogs inequality (2.7) as

decreases to zero.

p

For future reference, we state the specialization of Theorem 1.13 to the case at hand.

The following are equivalent, for a point

Theorem.

2.4

X0 E X .

X

on

,

(i)

x0

is a Jensen boundary point.

(ii)

If

h

is any continuous real-valued function on

then there exists a continuous log-envelope function X

such that

u <_ h ,

There exist

(iii)

exist on

E

and

,

E

c > 0

and

c log IfI

u(x0) = h(x0)

a < 0 < S

for any compact subset f E A

while

5 S

of

.

with the following property:

not containing

X

such that on

u

f(x0) = 1 ,

x0

,

there

c log If I <_ a

X .

An Example For the simplest nontrivial example, we take boundary

aA

disc algebra A

of

of the open unit disc A(A)

case,

A

, and

8A

is dense in

CR(@A)

.

0

aA

,

In this Z

and this represent-

ing measure is necessarily a Jensen measure. 28

.

Hence each point in

has a unique representing measure on

resenting measure

to be the

The maximal ideal space

.

coincides with the closed unit disc

Re(A)

to be the

consisting of the analytic functions on

that extend continuously to A(A)

A

X

For the rep-

for the origin, the inequality (2.7)

becomes the usual inequality reflecting the subharmonicity

of log

fl ,

log If(0) I

<_

J

log If(ei6) I20

,

f E A(A)

(2.10)

.

In turn, (2.10) serves to generate inequalities of the form

X(log If(0)I) < J X(log If(ei8)I)28

30

,

f E A(A)

,

(2.11)

valid whenever

is an increasing convex function.

x

There is a converse assertion of sorts, that shows that the Jensen-Hartogs inequality (2.10) is essentially the best It asserts that if

inequality that can be expected.

x

is

a real-valued Borel function of a real variable such that (2.11) is valid, then

is an increasing convex function.

x

Let us prove this assertion. Suppose that (2.11) is valid. such that

f e A(A)

Substituting this so that

If1 = eb f

convexity of

x

0 < c < 1

t

.

Let

u harmonically to A,

log If(0) I

f(0) = ea

x(a) < x(b)

To establish the

f e H (A)

.

= J

Let

s,t e ]R,

be a real-valued function on

u

let

and set

u ,

there exists

, while

s

on a set of mass

on the complementary set of mass

function of

,

first observe that the inequality (2.11)

,

that assumes the value

value

8A

in (2.11), we obtain

is readily established for all

8A

on

a < b

is an increasing function.

x

and let

If

*u

c

1 - c

,

.

and the Extend

be the conjugate harmonic

f = exp(u+i*u)

.

Then

ue = cs + (1-c)t

and (2.11) becomes

x(cs+ (1-c)t) <_ cx(s) + (1-c)x(t) Hence

x

is convex, and our assertion is established.

Nomenclature We close with some remarks on the history of Jensen measures and the Jensen-Hartogs inequality (2.7).

In 1899, J.L.W.V.Jensen published a paper in the Acta Mathematica "Sur un nouvel et important thdorhme de la thdorie des fonctions".

The important theorem that Jensen

refers to is now known as the Jensen formula, that

31

rli...16 I

2u

= loglf(0)I + log

lf (re

fo

where

f

fall " 'lanl

is meromorphic on the closed disc of radius a1,...,an

with zeros

and poles

1,.... Sm

.

r

,

Jensen applied

his formula to entire functions, in order to relate the growth rate and the distribution of zeros.

From the Jensen

formula one readily obtains the inequality (2.10) in the case that

is analytic.

f

However, the estimate (2.10)

does not appear explicitly in the paper. Apparently Jensen was not the first to write down the

J.Hadamard[5,p.501 asserts that he had

Jensen formula.

discovered the Jensen formula already in 1888.

He did not

publish the result, though, as he was waiting to be able to derive some significant consequences from the formula. In their fundamental paper in this area, R.Arens and I.M.Singer[1] refer to (2.10) as Szego's inequality, citing the paper [101 in which G.Szego establishes the integrability of

for

log Ifs

f E H2(d8)

.

Again while (2.10) is immedi-

ately obtainable from Szegb's work, it does not appear explicitly in the paper.

More recently, K.Hoffman[7], and also V.S.Vladimirov[ll], refer to (2.7) and (2.10) as Jensen's inequality.

The no-

menclature is justifiable, but it leads to confusion since the name has already been attached to the inequality (1.15), proved by Jensen[9] in 1906. S.Bochner and W.T.Martin[3] refer to the estimate (2.10) as the Jensen-Hartogs inequality, in view of the seminal research of F.Hartogs[6] which hinged upon the subharmonicity of

log Ifl

.

We have adopted the terminology of Bochner

and Martin.

The term "Jensen measure" is due to Bishop[21, who established the existence of Jensen measures in general by roughly the same argument we have used in Lemma 1.1.

32

References 1.

Arens, R. and Singer, I.M.

Function values as boundary

integrals, P.A.M.S. 5 (1954), 735-745. 2.

Holomorphic completions, analytic continu-

Bishop, E.

ations, and the interpolation of seminorms, Ann. of Math. 78 (1963), 468-500. 3.

Bochner, S. and Martin, W.T.

Several Complex Variables,

Princeton University Press, Princeton, 1948. 4.

Gamelin, T.W.

5.

Hadamard, J. The Psychology of Invention in the Math-

Uniform Algebras, Prentice-Hall, 1969.

ematical Field, Princeton University Press, 1949. 6.

Zur Theorie der analytischen Funktionen

Hartogs, F.

mehrer unabhangiger Veranderlichen, insbesondere caber

die Darstellung derselben durch Reihen, welche nach Potenzen einer Veranderlichen fortschreiten, Math. Ann. 62 (1906), 1-88. 7.

Hoffman, K.

Analytic functions and logmodular Banach

algebras, Acta Math. 108 (1962), 271-317. 8.

Jensen, J.L.W.V.

Sur un nouvel et important thdorbme

de la thdorie des fonctions, Acta Math. 22 (1899), 359364. 9.

Jensen, J.L.W.V.

Sur les fonctions convexes et les

indgalitds entre les valeurs moyennes, Acta Math. 30 (1906), 175-193. 10.

Szego, G.

Uber die Randwerte einer analytischen Funk-

tion, Math. Ann. 84 (1921), 232-244. 11.

Vladimirov, V.S.

Methods of the Theory of Functions of

Several Complex variables, M.I.T. Press, Cambridge, Mass., 1966.

33

3 The algebra R(K)

Let

K

let

R(K)

be a compact subset of the complex plane denote the uniform closure in K

rational functions with poles off

uniform algebra on coincides with

and

,

of the

C(K)

Then

.

C

is a

R(K)

K , and the maximal ideal space of

R(K)

We wish to specialize the discussion of

K .

the various classes of representing measures introduced earlier to the algebra

mation on

R(K)

R(K)

.

Pertinent background infor-

is contained in [2].

Recall that the Cauchy transformv K

ported on

J

d-(Z) z -

,

E

1/z

the integral defining

,

Since

and the locally integrable

v

sup-

v

is defined by

providing the integral converges. ution of

of a measure

function

converges absolutely on a

set of full Lebesgue measure.

off the closed support of

is the convol-

v

(dxdy)

v

Furthermore, ,

and

is analytic

v(o') = 0

It is easy to approximate a rational function with poles off

K

by linear combinations of functions of the form

z 3 1/(Z-C), functions

C E C\K .

1/(z-C) ,

Consequently the linear span of the

C E L\K , is dense in

this statement is the fact that a measure thogonal to

R(K)

if and only if

v = 0

R(K) v

on

Dual to

.

K

on

O\K .

is orThis

points to the important role played by the Cauchy transform in rational approximation theory.

Related to the Cauchy transform of a measure is the logarithmic potential of negative

34

Vv

v

.

v

on

K

We will work with the

of the logarithmic potential, defined by

VVW = J log lz-Cldv(z) The kernel

cEC

,

.

(3.1)

appearing in the convolution integral

log lzl

is locally integrable

(dxdy)

so that the integral con-

,

verges absolutely on a set of full Lebesgue measure. function

V

is harmonic off the closed support of

v

- .

tends to

If

v

v

,

and

(3.2)

Vv(C) = v(K) log lkl + o(l)

as

The

is positive, then

V

v

is (upper

semicontinuous and) subharmonic. The potential v

is related to the Cauchy transform of

Vv

by the formula

(3.3)

which is to be interpreted in the sense of distributions.

We will only use (3.3) on

(C\K

, where the formula is easy

to establish by differentiating by hand.

Indeed, locally

one can express

log k-zl = 2

log(Z-z)1

,

from which one obtains

a log lC-zl = 12 C-z

,

C z z.

(3.4)

Differentiating the expression (3.1) for

Vv

,

and substi-

tuting (3.4), one obtains (3,.3).

3.1

Let

Theorem.

measure on (i)

(ii)

K . v

p E K , and let

v

be a probability

Then the following are equivalent:

is a representing measure for

p

C E C\K ,

35

C\K

of

Proof.

.

From (3.3) and (3.4) we have

[Vv(0 -

L1P

2

Vv(4) - logy-pl

Since

is constant on each component

logIC-pi

(iii)

is real, the function is constant

just as soon as its c-derivative vanishes.

Hence (ii) and

(iii) are equivalent.

The identity (ii) means precisely that on the functions

E C\K

z -}

v

represents

p

As observed

.

earlier, the linear span of these functions is dense in R(K)

,

so that (i) and (ii) are equivalent.

11

It is easy to establish directly the equivalence of (i)

and (iii) in Theorem 3.1, without passing through the Cauchy transform.

and that

Suppose for instance that C0 E C\K .

R(K)

represents

near O ,

For each fixed

a single-valued determination of belongs to

v

log[(z-r)/(z-r0)]

p

,

there is that

For this determination we have

.

J log\Z_ jdv(z) 0

= log(P_c CO \

1

Taking real parts, we obtain

VvW - Vv(C0) = log1p-d - loglp-c0I

This shows that ponent of

K\C

V

loglp-cl v

is constant on each com-

.

Observe that the constant value of the unbounded component of (3.2), which shows that

36

C\K

V W - log y -pI

is zero.

on

This follows from

lim VvW - logy-PI = 0 whenever

has unit mass.

v

Now we turn to the Arens-Singer measures on point

K

for a

Recall that these are defined to be the

p e K .

probability measures on

K

that represent the functional

"evaluation at p" on the linear span of

loglR(K)-11

,

that is, the linear span of the logarithms of the moduli of the invertible elements of

Denote by

R(K)

.

the closure in

H(K)

functions harmonic in a neighbourhood of

3.2

incides with

H(K)

K .

logjR(K)-11

K

co-

.

Any function in

uniformly on f

K .

The closed linear span of

Lemma.

Proof.

of the family of

CR(K)

loglR(K)-11

can be approximated

by a function of the form

loglfl

, where

is a rational function with neither poles nor zeros on In particular,

belongs to

loglfl

logjR(K)-ll

closed linear span of Suppose points

is included in

zl,...,zm e C\K

and constants

H(K) K .

cl,...,cm

.

Choose

such that

cj loglz-zjl

has a single-valued harmonic conjugate hood of

so that the

,

is harmonic in a neighbourhood of

u

w = u -

H(K)

Then

K .

so do the functions E cj loglz-zjl

f = exp(w + i*w) z-zj ,

1

<_

*w

in a neighbour-

belongs to

j 5 m .

Hence

belongs to the linear span of

R(K)_l , and

u = loglfl + loglR(K)-1l

.

0

3.3

Let

Theorem.

measure on (1)

K . v

p e K , and let

v

be a probability

The following are equivalent:

is an Arens-Singer measure for

p

37

(ii)

represents

v

for every function

p

on

H(K)

u(p) = Judy

harmonic in a neighbourhood of

u

(iii) Vv(c) = logy-PI for all

r E T\K

K

.

The equivalence of (i) and (ii) follows immediately

Proof.

from the preceding lemma.

That (ii) implies (iii) is immedi-

ate from the definition of

V

V

Suppose that (iii) is valid.

differentiable function on that

that is

,

C

Let

be an infinitely

u

with compact support, such

is harmonic in a neighbourhood of

u

Then

K .

u

can be represented as the logarithmic potential of its Laplacian,

u(0 = - I

1

2Tr

Since

over

u

(Au)(z) loglz-Cldxdy

is harmonic near

K , this integral can be taken

6\K , and we obtain

I I t\K (Au)(z)LJ logI

dxdy

2Tr

J

L

1

(Au)(z) loglz-pl dxdy

= - 2m

This establishes (ii), and the proof is complete.

Let

Finally we turn to Jensen measures.

measure on

K

for

p

.

Since

v

J

log e-zldv(z)

z + z-C , we obtain

,

C E

CL

.

(3.5)

is an Arens-Singer measure, the identity (iii) of

Theorem 3.3 becomes

38

be a Jensen

Applying the Jensen-Hartogs in-

equality (2.7) to the functions

logy-PI <_

v

logy-pl = J log y-zldv(z)

E C\K

,

(3.6)

.

It turns out that these properties characterize Jensen measures.

Let

Theorem.

3.4

K

measure on

p E K , and let

be a probability

v

Then the following are equivalent:

.

(i)

v

is a Jensen measure for

(ii)

v

satisfies (3.5) and (3.6);

(iii)

u(p) <_ Judy

whenever

u

p

is a real-valued function K

that is subharmonic in a neighbourhood of

.

We have already observed that (i) implies (ii).

Proof.

Suppose that (ii) is valid, and let a neighbourhood of

K .

be subharmonic in

u

By the F.Riesz Theorem,

u

can be

expressed in the form

u(z) = v(z) +

where

T

log lz-OdT(C)

zEK,

,

is a positive measure supported on a compact neigh-

bourhood of K

f

K , and

v

is harmonic in a neighbourhood of

Using Fubini's Theorem, we obtain

.

J

u(z)dv(z) = J v(z)dv(z) +

JJ

log

v(p) + J logy-pldT(C) = u(p)

Hence (ii) implies (iii). Finally, suppose that (iii) is valid. rational function with poles off

harmonic in a neighbourhood of

K .

Let

Then

f

be a

loglfl

is sub-

K , so that

39

loglf(P)l

Since such for all

<_

loglfldv

J

.

are dense in

f

f E R(K)

and

,

v

R(K)

, we obtain the inequality

is a Jensen measure for

p

.

0

The equivalence of (i) and (iii) in Theorem 3.4 is dual to the following statement.

The continuous log-envelope functions on, K

Theorem.

3.5

with respect to

K

are the uniform limits on

R(K)

of the

functions that are continuous and subharmonic in a neighbourhood of

K

.

Any continuous log-envelope function

Proof.

K

on

u

is a

uniform limit of functions of the form max(c1 loglfll,..., cm loglfm1)

cl,...,cm > 0

, where

rational functions with poles off

and

fl,---,f

m

are

In particular,

K .

u

is a uniform limit of functions that are continuous and sub-

harmonic in a neighbourhood of Conversely, if

K .

is a uniform limit of such functions,

u

u(p) <- fudv

then from (iii) of Theorem 3.4 we obtain

p e K

all 1.6,

and all Jensen measures

v

for

p

.

for

By Theorem

is a log-envelope function.

u

0

Now consider the R(K)-Dirichlet problem with boundary data

u E CR(8K)

.

In view of Theorem 3.5, the solution of

the problem is the upper envelope

harmonic functions on

v <_ u

8K

.

is harmonic on

u

is continuous on u

K

in a neighbourhood of

v

K°

.

such that

K

Under certain regularity conditions, and coincides with

u

on

aK ,

so

is the solution to the classical Dirichlet problem.

This accounts for the terminology.

40

of all continuous sub-

The classical proof of Perron shows that

u

that

u

The characterization of Theorems 3.3 and 3.4 yield the following theorem of A.Debiard and B.Gaveau[ll.

measure on and only if

p c K , and let

Let

Theorem.

3.6

3K

Then

.

is a Jensen measure for

v

is an Arens-Singer measure for

v

Suppose that

Proof.

that is carried by

loglz-pl = Vv(z)

3K

,

To show that

<_ V

v

v

(z)

,

p

if

p .

is an Arens-Singer measure for

v

p

Then

.

z c O\K

.

From the upper semi-continuity of

loglz-pl

be a probability

v

Vv

,

we obtain

z c aK .

(3.7)

is a Jensen measure, it suffices to show

that this estimate holds also on

K°

We will appeal to the following maximum principle for the logarithmic potential

u

of a positive measure

supported on a closed set

E

(cf. [5, pp.53-541):

v

lim sup u(z) = lim sup u(z)

E D z}C

z->C

valid for any point of

E

If

E E

that is not an isolated point

.

p E K° , we easily conclude the proof.

maximum principle above to

u = -Vv

,

Applying the

we obtain from (3.7)

the estimate

log IC-pi 5 lim inf Vv(z)

,

C E 3K

z-*

Since

loglz-pl

is subharmonic, while

V

v

is harmonic on

41

we obtain the estimate (3.7) for

K°

z E K°

The proof is slightly more complicated if K

trary point of

loglz-(I < 0

belong to some neighbourhood of

C

is an arbi-

p

K

Assume for convenience that

.

tained in a small disc, so that and

as required.

,

K .

is con-

whenever

z

Define

ue(z) = 1k-pIke

Then

u

v

loglz-pl

K , so that from (3.7) we obtain

near

>_ V e

<_ u

(z)

z c aK .

,

e

Furthermore, K°

u

is subharmonic, and

e

is harmonic on

u

Applying the maximum principle again, we obtain

.

loglz-pl 5 lim inf ue(c) C-;z

z c BK,

for all

neighbourhood of

z z p p

.

.

Also

loglz-pI < uE(z)

We conclude from the usual maximum

principle for harmonic functions that on

K°

Vv(z)

3.7

Letting

.

on

K°

,

Corollary.

c

loglz-pi 5 ue(z) loglz-pl <

tend to zero, we obtain

as required.

0

The Jensen boundary for

with the Arens-Singer boundary for

R(K)

coincides with the Choquet boundary for

Proof.

in a

coincides

R(K) ,

and this in turn

H(K)

.

The first statement follows from Theorem 3.6, Lemma

1.9, and the fact that

aK

is a closed boundary for

R(K)

The second statement follows immediately from Theorem 3.3. 11

The requirement in Theorem 3.6 that the measure situated on

42

aK

is essential, even in the case that

v

be

K

is

the closed unit disc

4

representing measure

a

z = 1/2

Since

.

for

with positive mass at (say)

0

is connected, any representing measure

CAE

However, the Jensen-

is an Arens-Singer measure.

R(E)

for

Indeed, it is easy to construct a

.

Hartogs inequality shows that a Jensen measure for have positive mass at any point other than

0

cannot

0

so that

,

a

is not a Jensen measure.

The following striking observation is due also to Debiard and Gaveau.

Suppose that

Theorem.

3.8

Jensen boundary of

R(K)

p e K

Then for each integer

.

the differentiation functional

f - f(m)(p)

to a continuous linear functional on

For

Proof.

,

k >_ 1

,

let

Ek

R(K)

p

,

logarithmic capacity, and let

y(Ek)

,

extends

.

be the intersection of

{2-m-1 < Iz-pl < 2-m}

and the annulus

m ? 0

defined on

that extend to be analytic near

f e R(K)

those

does not belong to the

let

Q\K

cap(Ek)

be its

be its analytic capacity.

According to the Wiener criterion (cf. [3]), the series M

k/log[l/cap(Ek)]

I

k=1

converges. to

0

.

lim k->O

Let

In particular, the terms of the series converge

Since

y(Ek) <_ cap(Ek)

k

log[l/y(Ek)]

M be large.

,

we obtain

=0 Then

k/log[l/y(Ek)]

1/M

for

k

large,

so that (Ek) <

for

k

-Mk

large.

It follows that

43

2mk(Ek)

(3.8)

G

k=1 converges for each fixed integer

m ? 1

According to the

.

Melnikov-Hallstrom Theorem (cf. [41), the convergence of the series (3.8) is necessary and sufficient for the differentiation functional R(K)

to extend continuously to

f - f(m)(p)

This concludes the proof.

.

[1

K

It is easy to find an example of a set 0 e aK

has a unique Jensen measure, while

a unique representing measure for

R(K)

such that does not have

0

Since such an ex-

.

ample is required for the next chapter, we provide some details (cf. [31).

K

Let

be the roadrunner set obtained from the closed

unit disc by excising the open discs 1

<_

<_

j

.

{Iz-3 'I

One checks that the measure

2d 1Z

<

8-j)

on

,

aK

has

finite variation, and it is a complex representing measure for

0

.

It follows [2, p.331 that

0

measure distinct from the point mass.

has a representing On the other hand,

the functions

f.(z) = 8/(z-3) K , while

are bounded in modulus by one on tends to

+o

continuous,

Theorem 3.8.

.

0

If!(O)I = (9/8)j

Since the differentiation functional is not belongs to the Jensen boundary for

R(K)

,

by

Of course, one can check directly that the

Wiener series diverges, so that

0

is a regular boundary

point for the Dirichlet problem.

References 1.

Debiard, A. and Gaveau, B.

Potential fin et algebras de

fonctions analytiques I, J. Functional Analysis 16 (1974), 289-304.

44

2.

Gamelin, T.W.

3.

Gamelin, T.W. and Rossi, H.

Uniform Algebras, Prentice-Hall, 1969. Jensen measures and algebras

of analytic functions, in Function Algebras, F. Birtel (ed.), Scott, Foresman and Co., 1966, pp.15-35. 4.

Hallstrom, A.

On bounded point derivations and analytic

capacity, J. Functional Analysis 4 (1969), 153-165. 5.

Tsuji, M.

Potential Theory in Modern Function Theory,

Maruzen, Tokyo, 1959.

45

4 The corona problem for Riemann surfaces

This chapter is based on some unpublished work of B.Cole, that has been circulating since about 1970.

Cole constructed

an open Riemann surface for which the corona problem has a The key idea of the construction is related

negative answer.

to Jensen measures.

Let

be a set with an analytic structure, and suppose

D

that the algebra

of bounded analytic functions on

H (D)

separates the points of

D

We can then regard

.

subset of the maximal ideal space identifying at

z

.

of

M(D)

D

D

as a

H(D) , by

with the ideal of functions that vanish

z e D

The corona problem asks whether

D

is dense in

M(D)

The corona problem can be rephrased in terms of the functions in

HA(D)

.

The density of

D

in

M(D)

is equivalent

to the validity of the following condition:

Whenever

fl,...,fn e Hw(D)

and

6 > 0

If11 + ... + Ifnl ? 6 > 0

g1, ...,gn e

D

on

,

satisfy

(4.1)

there exist

such that f1g1 + ... + fngn = 1.

H°°(D)

We will also consider the following stronger property that D

might or might not enjoy:

For all stants e HA(D)

6 > 0 C(n,6)

n e Z+ , there exist con-

and

such that whenever

satisfy

if .I

<_

1

,

1

<_

j

fl,...,fn <_ n ,

and

J

If1I +... + Ifnl

g1, ...,gn E and

46

H°°(D)

6

,

then there are

satisfying f1g1 + "' + fngn = 1 '

(4.2)

1 <_ C(n,6)

I g

1 < j

,

<_ n

J

We will focus our attention on the property (4.2).

For more

information and references, see [1].

According to the corona theorem of L.Carleson, the open unit disc that

A

is dense in

A

M(A)

.

has the property (4.2).

Carleson actually proves In [1] it is shown that

any finitely connected planar domain with constants

C

m

has property (4.2),

depending only on the number

(n,6)

boundary components of

D

D

The corona problem for planar

.

domains can be reinterpreted as asking whether the are bounded as

m -} -

.

m of

Cm(n,6)

If so, then all planar domains have

property (4.2), and the same constants will serve for all such domains.

If not, then it is possible to construct a

planar domain

D

that fails to be dense in

M(D)

.

Our purpose here is to prove Cole's Theorem to the effect that the analogous constants for finite bordered Riemann surfaces fail to be bounded.

Let

Theorem.

4.1

0 < 6 < 1

,

and let

exists a finite bordered Riemann surface fl,f2 E H (R)

If11

<_

1

,

,

.

Then there

together with

R ,

such that

1f21 < 1

If11 + If21 ' 6 if

M > 0

(4.3)

;

(4.4)

;

g1,g2 E Hw(R)

satisfy

f1g1 +f 2g2 =

1

,

(4.5)

then

max(11g1IIR,1Ig2IIR) > M Proof.

.

We begin with the roadrunner set

the close of the preceding chapter.

Choquet boundary of

H(K)

K

constructed at

For this example, the

coincides with

8K

.

It is an

47

elementary fact (cf. reference [31 of Chapter 3) that every function in

CR(3K)

is then the restriction to

function in

H(K)

On the other hand,

.

is constructed so

K

that there is a representing measure v

of a

3K

for

with

0 E 3K

respect to the algebra R(K) , such that v has no mass at 0.

{Izl
Denote the open disc we can choose

and

c > 0

p > 0

by

4p

.

Since v({O}) =0

so small that

Me + Mv(4 ) < 1

(4.6)

P

ul and u2

Choose functions

K, u1 < log e

on K\0

max(ul,u2) > log 6

H(K)

such that

, where

f1

This yields a neighbourhood

U

ul =

and f2 K , and

of

K

on

, and

By Lemma 3.2, we can assume that

.

analytic and nonzero in a neighbourhood of an integer.

ul <0

u2 <0 on K, u2 (0) < log e

, P

u2 = N log If2I

and

log Ifll

in

are

N

is

which is

bounded by a finite number of closed analytic curves, and functions

fl,f2

analytic on

If1I,If2I < 1

fl,f2

(4.7)

Iflll/N + If2I1/N > 6

U\0

on

that satisfy

U

on

do not vanish on

Iflll/N < e

U

U

7

(4.8)

U

(4.9)

and

(4.10)

on

,

P

If2(0)I1/N < e

(4.11)

.

Consider the set of points z3 e U ,

N

zl

= f1(z3)

,

and

(zl,z2,z3) E C3 N

z2

= f2(z3)

(4.8), the coordinate projection

.

that satisfy

On account of

onto the third coordi713

nate plane is an unramified N2-sheeted covering of this surface over

48

U

.

Let

R

be a component of this surface, so

,

that

is an unramified covering map of

Tr3

F1 and F2

fine analytic functions F .(z) = z.

,

j = 1,2

J

and (4.9) we obtain on

R

FN =f

Then

.

J

IF1l < 1 ,

°Tr3

U

.

De-

by setting

R

on

onto

R

and from (4.7)

,

j

and

IF21 < 1

IF1I + IF2I > 6

.

Suppose there exist

G1,G2 E HE(R)

F1G1 + F2G2 = 1

IG2I < M and

on

R

IG1l< M

such that

,

We will complete the

.

proof by showing that this leads to a contradiction. H = F1G1

Set

IHI < MIF1I

H

U , that is, define

on

values of lytic on

H

on the fiber

.

h(z3) Tr

Let

h

Ih-11 5 MY

IHI <M

be the trace of

to be the average of the

31(z3) n R

.

Then

h

is ana-

1/N

(4.12)

211/N

(4.13)

From (4.7), (4.10) and (4.12) we see that Ihl 5 Me

R,

and

U ,

< MY I

Ihl

is analytic on

H

I1-HI <MIF2I

and

,

Then

.

off the p-neighbourhood of

Ih(0) I = IfhdvI

<_

0

.

Ihl 5 M , while

Hence

Ifo hdvl + I,fK\A hdvl < Mv(Op) +Me

.

p

p

On the other hand, from (4.11) and (4.13) we have

lh(0) - 11

<_ Me

.

These two estimates for

Theorem (Cole).

4.2

R

h(0)

contradict (4.6).

There exists an open Riemann surface

that is not dense in the maximal ideal space

H_ (R)

Proof.

M(R)

of

.

Fix

0 < 6 < 1

.

According to Theorem 4.1, there is

49

a sequence

of finite bordered Riemann surfaces,

{Rm}m=1

together with analytic functions

F

and G

m

extend analytically across the border of

IF

IF

m

m

IGm

I

+ IG

I

if

m,

then

m

<_

on

on

R m

I I $m l

m

that

such that

(4.14)

m

(4.15)

;

satisfy

E

m

Rm ,

R

R

I

? 6

I

1

on

m

mFm +

=1

G

m m

,

(4.16)

I c+ I I m I I.? m.

We claim that there is an ambient (connected) Riemann containing the

R

surface

F,G E H (R)

joint subsets, and functions

FI

<_

FI +

2

,

IGI

<-

2

IGI ? 6/2 > 0

IF-FmI < 1/m

and

on

and their borders as dis-

Rm's

(4.17)

R ;

on

(4.18)

R ,

IG-GmI < 1/m

Suppose for the moment that

R ,

constructed, and that 4, e H (R)

F + iG = 1

On

R

m

4 F

m m

we then obtain

,

where

m = /[l

50

on

F

Rm

and

(4.19)

.

G

have been

satisfy

(4.20)

.

+ p mGm = 1

such that

,

F) + (Gm G)1

,

m = p/[1 +4(F MF) + (Gm G)] Since the norms of

$

and i

m

.

on

m

m

- , we obtain a contradiction to (4.16) for large

tends to

We conclude that the equation (4.20) is not solvable in

m .

HE(R)

consequently

,

is not dense in

R

R, F

to construct

and

G

M(R)

R 's

Let

as follows.

m

tained from

each m ? 1

m Km

Tm

Let

be a finite bordered surface ob-

S m

Then

.

is obtained by attaching, for

S

one of the boundary components of

,

of those of

containing the

S

by attaching an annulus to each boundary

R

component of

are disjoint.

Let

E

DRm+l

Tm's

Then

.

for each m >_ R

m

Ift

1

E

<_

1

and

on

Igl S 1

on

E

F

m

be functions analytic on

G

IG-gl < min(6/4,1/(2m))

on Rm u T m

and

,

their

G

on and

such that

E g

with

G

m

.

on

F

m

E

IF-fl < min(6/4,1/(2m))

Such

R 's

Tm's

is a closed subset of

f and g

coincides with

f

and such that

and

F

,

,

Ift + Igi ? 6 Let

to

is connected.

Define continuous functions

on

Sm

DR

We assume that the

.

be the union of the

E

boundaries, and the , and

to one

be a simple analytic arc that starts at

and terminates at

,

Sm

by means of a rectangular strip.

Sm+l

passes directly through the rectangle joining Sm+l

It remains

.

.

First we construct an ambient surface

S

m

are bounded as

R

RE

S

such that

m u Tm ,

can be constructed by means of a simple

51

iteration argument, together with Bishop's Theorem asserting that if S

K

is a compact subset of an open Riemann surface S\K

such that

tion on

K

is connected, then every continuous func-

K

holomorphic on the interior of

proximated uniformly on K The existence of

F

and

can be ap-

by functions holomorphic on

S

also follows from a version of

G

Arakelyan's Theorem for Riemann surfaces due to S.Scheinberg [2].

In any event, with

the open surface

R

and

F

in hand, we take for

G

a connected open neighbourhood of

E

small enough so that (4.17), (4.18) and (4.19) are valid. 11

Cole's example can be modified to provide an example of a

bounded domain of holomorphy in

03

the corona problem is negative.

Indeed, by embedding the

surface

Rm

in a polydisc in

obtains a domain

D

m

in

03

for which the answer to

and fattening it, one

,

with smooth strictly pseudo-

03

Fm

convex boundary, and functions

and

Gm

in

H (Dm)

that satisfy (4.14), (4.15) and (4.16) relative to translating and dilating the Dm's

have pairwise disjoint closures, while the

verge to

0

It can be shown that the

.

D

m

By

.

Dm's , we can arrange that the

D 's

m

Dm's

con-

can be joined

by smooth narrow tubes so as to obtain a domain

D

in

03

with the following properties: (i)

Every point of

pseudoconvex boundary point for (ii)

D

is a smooth strictly

(2D)\{0} D

.

is holomorphically convex, and in fact

D

is

a Runge domain. (iii)

D

is not dense in

M(D)

An example of a holomorphically convex domain that is not dense in

M(D)

cently by N.Sibony[3].

D

in

02

has been constructed more re-

Meanwhile, it is unknown whether the

corona problem has an affirmative solution for the unit ball

or the unit polydisc in

52

02

References 1.

Gamelin, T.

Localization of the corona problem, Pac. J.

Math. 34 (1970), 73-81. 2.

Scheinberg, S.

Uniform approximation by functions ana-

lytic on a Riemann surface, Annals of Math., to appear. 3.

Sibony, N.

Prolongement analytique des fonctions holo-

morphes born6es, C. R. Acad. Sci. Paris, t. 275 (1972), 973-976.

53

5 Subharmonicity with respect to a uniform algebra

Let A MA

be a uniform algebra on a compact space X, and let

denote the maximal ideal space of A.

In this chapter,

we continue the line of investigation begun in Chapters 1 and 2.

We will introduce and treat various classes of

"quasi-subharmonic" functions.

The lower semi-continuous,

quasi-subharmonic functions will be the log-envelope functions introduced in Chapter 2.

The upper semi-continuous,

quasi-subharmonic functions will be called simply "subharThe subharmonic functions in this context correspond

monic".

to the subharmonic functions on an open subset of

0

or to

,

the plurisubharmonic functions on an open subset of

Tn

The main theorems of this chapter are Theorems 5.9 and 5.10, asserting that a locally subharmonic function is subharmonic, while a bounded, locally log-envelope function is a log-envelope function.

Our exposition will be based on

work of the author and N.Sibony[3,41.

Quasi-subharmonic Functions Let

[-,+o]

u .

be a Borel function from a subset We say that

Judo

for all

u

is quasi-subharmonic on

S

implicitly that the negative part of

for

u , min(u,0)

,

is inte-

for those

a

satisfying

is quasi-subharmonic on

u

S

if and only if

is quasi-subharmonic on each compact subset of

A function

u

from

S

to

[--,+-)

S

.

is subharmonic if

is upper semi-continuous and quasi-subharmonic.

54

a

It is understood

.

grable with respect to the Jensen measures

u

to

if

S

and all Jensen measures

¢ e S

supported on a compact subset of

e S

MA

of

S

u

A subharmonic

function on S

S

is bounded above on each compact subset of

.

Recall from Chapter 2 that a lower semi-continuous function from a closed subset

E

MA

of

(-,+-]

to

is a log-

envelope function if it is the upper envelope of functions

of the form

c loglfl

c > 0

, where

f e A

and

Some el-

.

ementary properties of log-envelope functions are collected in (1.10) through (1.13).

Lemma.

5.1

Let u be a lower semi-continuous function from

a compact subset

E

of

MA

to

(-co,+-]

log-envelope function if and only if

Then

.

u

is a

is quasi-subharmonic.

This is a special case of Theorem 1.6, in which

Proof.

is the family of functions f E A

u

c > 0

, where

c loglfj

R

and

.

In particular, a continuous function on

is subharmonic

E

if and only if it is a log-envelope function.

A compactness

argument yields easily the following further characterization, which is a special case of Lemma 1.4.

5.2

Lemma.

The following are equivalent, for a continuous,

real-valued function

u

on a compact subset

(i)

u

is subharmonic on

(ii)

u

is a log-envelope function on

(iii)

u

is a uniform limit on

E

form

of

MA

.

E

E

of functions of the where

positive real numbers, and

E

fl,...Ifm e A

cl,.... cm

are

.

In order to characterize the subharmonic functions, we

recall some facts from Chapters 1 and 2 concerning the ADirichlet problem. Suppose that

u

is a lower semi-continuous function, de-

fined on a compact subset

E

of

MA , with values in

55

(-c,+o]

.

The solution to the A-Dirichlet problem with

boundary data

on

u

is defined to be the upper envelope

E

MA

of all log-envelope functions on

u

u on E

by

The solution

.

that are dominated

can be expressed in the form

u

c>O,fEA, clogIfI
uW = sup{c

(5.1) According to Edwards' Theorem,

can be expressed in terms

u

of Jensen measures,

u0) = inf{ J udo

a

:

a Jensen measure on E for

}

.

(5.2) Some elementary properties of the correspondence

are

u -> u

listed in (1.17) through (1.22).

5.3

Let

Lemma.

be an upper semi-continuous function

u

from a compact subset subharmonic on

MA

of

E

to

if and only if

E

[-co,+-)

u

Then

.

is

is the pointwise limit

u

of a decreasing net of continuous subharmonic functions on E

.

Suppose that

Proof.

creasing net E E

pose a

,

,

{u

and let

v E CR(E)

u

is the pointwise limit of a de-

}

of subharmonic functions on

a

be a Jensen measure on

a

u < v

satisfies

so that

Then

.

u (4) <_ fu do a

mum over such

fvda

a

then

For this, choose If

.

a

Sup-

.

for large

Taking the infi-

.

v

c > 0

on

E

It

.

u < w

such that

such that E

u < w- e. for

,

Taking the infimum over such

.

-e

E

satisfying

is a Jensen measure on

Judo <_ Jwda -e

a , we obtain

56

ua < v

is subharmonic on

a continuous subharmonic function

E E

for

E

u

suffices to find, for each w c CR(E)

Let

Let

.

v , we obtain

u

u < v < w .

E

.

By Edwards' Theorem, there

c > 0

while

u(4) < c loglf(4)l

and

in a neighbourhood of

The latter inequality persists

.

.

on E,

c loglfl < w

such that

f E A

exist

Covering

of such neighbourhoods, and letting

by a finite number

E

be the

cJ loglfi I

corresponding functions, we find that

v = max cJ loglfi I

is a continuous subharmonic function that satisfies u < v < w .

Let U

to

N

has a compact neighbourhood N

A function

.

uIN

such that

U

is subhar-

The locally log-envelope functions and the

.

locally quasi-subharmonic functions on larly.

from

u

is locally subharmonic if each point of

[--,+o)

monic on

MA

be an open subset of

U

are defined simi-

U

In order to study these classes of functions, we

take a detour through algebras generated by Hartogs series.

This will lead to a further characterization of log-envelope functions.

Hartogs Series A Hartogs series is a series of the form

E f

.

, where

J

C

is a complex variable, and the

ft's

depend on other

We are interested in the case that the

parameters.

f.'s j

belong to the algebra Let (-co,+co]

u .

A .

be a lower semi-continuous function from X Define a subset

x E X,

Y =

ICI

Y

of

X x C

<_ e-u(x)}

The lower semi-continuity of

u

to

by

(5.3)

.

guarantees that

Y

is com-

pact.

A

We regard

as a subalgebra of

way, and we define

erated by A algebra of

B

C(Y)

to be the uniform algebra on

and the coordinate function B

in the obvious

.

Y

gen-

A dense sub-

is formed by the Hartogs polynomials

57

N

F (x, 0) _

fj WC 3

(X' O e Y ,

,

(5.4)

j =0 where

f0,...,fN e A

Since each

.

i e MB

is determined by

its restriction to

A

and its value on the coordinate func-

tion

i

has the form

any such

,

N

j=0

(F)

_ flA

where

MB

sequently

CO = (O , and

is as in (5.4).

F

Con-

'

can be regarded as a compact subset of

MA X C . There is another way to regard

MB

Let

.

the algebra of Hartogs polynomials on

denote

A e P

MA x 0

that is, the

,

algebra of functions of the form N

fj

F =

,

f09 ...5fN e A

(5.5)

.

j =0 A ® P-convex hull

The

t

of a compact subset

consists by definition of those the evaluation functional

of

E

(¢,C0) e MA x Q

F -

MA x 0

such that

is continuous with

respect to the norm of uniform convergence on

E

.

This oc-

curs if and only if

I

Then

E

<_ IIFIIE ,

F E A ®P .

is easily seen to be the maximal ideal space of the

uniform closure in

C(E)

coincides with the

A ®P-convex hull

of

Since the A-convex hull of

hull of

X x {0}

A 0 P

X

is

In particular,

.

Y

Y

of

the

M A ,

MB

.

A O P-convex

coincides with MA x {0} , and MA x {0} c MB

Now the algebra of Hartogs polynomials is invariant under rotations in the c-coordinate, and so is

Y

,

so that

is invariant under rotations in the c-coordinate.

MB

Further-

more, the Hartogs polynomials depend analytically on the

58

r

variable, so that the maximum modulus principle shows that includes the entire disc

MB

{(4,c)

:

just as

IfI s r}

Itl = r}

soon as it contains the boundary circle It follows that

MB

has the form

ICI <

MB =

is some function from MA

R

is compact,

How can

[0,+o)

to

Since MB

.

is bounded and upper semi-continuous.

R

be captured from the function

R

turns out to be reasonably simple:

The answer

u ?

R = exp(-u)

We state

.

the result in full.

Let

Theorem.

5.4

from

X

to

be a lower semi-continuous function

u

(-°°,+-]

,

let

be the uniform closure in

where

-logI 0I

< 60)

integer

m

.

ICI

<_

e-50)}

.

M

.

kol > exp(-u(cf))

m loglfl < u

that is, that

,

f E A

and a positive

X , while

on

The first estimate shows that

X , so that

shows that

lCmfl < 1

c0mf(4)1 > 1

.

on

Hence

Y

-logIC0l

lfl exp(-mu)

The second estimate

.

does not belong to

0

.

It suffices now to show that Since that

(5.6)

,

Then there exist

such that

<-1 loglf(4)l

MB

u

Suppose that

on

of

is the solution to the A-Dirichlet problem with

u

boundary function

< 1

of the algebra A ® P

C(Y)

4 E MA ,

Proof.

B

Then

Hartogs polynomials.

MB =

be as in (5.3), and let

Y

C0 x 0

,

MB

includes

MA x {0}

kol < exp(-u(¢))

,

Ikol

we may assume

+o

so that in particular

Suppose first that

whenever

E MB

.

Let

F

be a

59

Hartogs polynomial of the form (5.5), and suppose that IFI <_ 1

Y

on

For fixed

.

is bounded in modulus by

the polynomial

xE X ,

on the disc

1

ICI

The Cauchy estimates for the coefficients

f

<- exp(-u(x))

then take

.(x) j

the form

If.(x)I 5 exp(ju(x))

,

logIf .l <_ ju

It follows that

so that

MA

on

logIfjl<- ju on X

Consequently

.

J

exp(Ju($))

and

,

N I

I

J =0 N

ko eu(OIJ

< C

j =0 1/[l - ICOI

0) ]

e

Since this estimate is independent of functional

O,c0)

belongs to

MB

,

Since

whenever

the evaluation

extends continuously to

F -x

<

N ,

B

.

Hence

this for all O satisfying is compact,

MB

Icol = exp(-u(4))

MB

IkOI

also includes 11

.

As a corollary, we obtain a characterization of logenvelope functions.

5.5

Corollary.

from MA

to

Let

(--,+-]

w .

be a lower semi-continuous function Then

w

is A-subharmonic if and

only if the set

E MA ,

ICI <

e-wWI

is an A ®P-convex subset of Proof.

given by

60

MA x Q .

Applying Theorem 5.4 in the case

that the

(5.7)

X = MA

,

we find

A ®P-convex hull of the set defined by (5.7) is

e-w(0 }

E MA

<_

Iii

,

.

Consequently the set defined by (5.7) is

w = w

only if

A ®P-convex if and

.

Algebras Generated by Hartogs-Laurent Series There are analogous results for algebras generated by Hartogs-Laurent polynomials of the form

Ff. 3 N

,

(5.8)

j =-N

where the

belong to

fits

A

Let

.

semi-continuous functions such that compact subset

v S -u , and define a

by

X x C

of

Y

be lower

v

and

u

e-u(x)}

Y = {(x,0

Let

B

ev(x)

:

I-

1 k i

`

.

be the uniform closure in

form (5.8), so that

B

of functions of the

C(Y)

is generated by

and

A ,

Since every complex-valued homomorphism of mined by its action on space

MB

B

of

MA x C

subset of

A

and on

,

.

is deter-

the maximal ideal

It consists of all pairs

Hartogs-Laurent polynomials

F

F -> F(@,C) ,

,

such

defined on

extends continuously to

is described directly in terms of

MB

.

can again be identified with a compact

that the evaluation functional

The space

r

B

1/r

u

and

B. v

as follows.

Theorem.

5.6

M B = { ( $ ,

where

u

and

Let

)

u ,

: E MA ,

v

v, Y ev(e)

and

<

B

be as above.

I kI < e-uW }

Then

,

(5.9)

are the solutions to the A-Dirichlet

problem associated with

u

and

v

respectively.

61

As in the preceding proof, it is easy to check that

Proof.

is included in the set described by (5.9).

MB

Suppose that

el (,W

Let

<

satisfies

l 0I <

(5.10)

be a Hartogs-Laurent polynomial of the form (5.8),

F

and suppose that f

e-uW

IFI

on Y

5 1

The coefficient functions

.

are given by

.

J

J-ldc

f.() = 2 i

F(x,C)C

J

x E X .

,

(5.11)

II=c

J

Suppose that

? 0

j

Taking

.

are led to the estimate

c = exp(-u(x))

in (5.11), we

exp(ju(x))

If.(x)I

loglf .I

or

,

J

5 ju

on

Hence

X .

exp(ju(q))

loglfj1 5 ju

MA , and

on

Ifj(ol

<_

This yields the estimate

.

N

f . ($);g j =O

On the other hand, suppose that in (5.11), we obtain

on

X .

Hence

if

Setting

j <0 .

.(x)I_exp(-jv(x)) J

logIfj 1 5 -jv

on

c = exp(v(x))

loglf .I _ -jv J

or

,

I fj (0 I

M A , and

<_

This yields the estimate 1/[l - IkOI

J

j=-N

e

(5.13)

.

7

Combining (5.12) and (5.13), we obtain a bound for

which is independent of

.

cludes all closed, <_

MB

(4,C0)

B

that satisfy (5.10).

includes all

For the general case, let

.

Hence

MB

Since

MB

inis

that satisfy

providing that

proves the theorem whenever

62

It follows that the functional

extends continuously to

F -;

Ifl

N

v(4) < -u(4)

v < -u c > 0

on ,

.

This

MA .

define

ve = v-c

,

set

Ye = {(x,0 and let

ev(x)-e

x E X,

:

ICI

<

be the uniform closure on

Be

Laurent polynomials.

Since the

<

e-'(x) }

Ye

,

of the Hartogs-

decrease to

Y

Y

as

e

e

decreases to

0

also

,

decreases to

MB

MB

as

de-

e

e

creases to

0

Now observe that since

.

u+ve s uu+ve <_ -e

Hence

.

on

ve < -u

u+ve <- -c

,

MA , and the special

case of the theorem already established applies to Since

also

BE

.

= v-e , we obtain

v e

M =

$ E MA ,

ev($)-e

< d

e-uW}

E:

Letting

decrease to zero, we obtain (5.9).

c

Since

u+v <_ 0

,

also

u+v < 0

,

and

v < -u

on

MA

.

The slices of the set (5.9) corresponding to fixed values of are therefore either circles or annuli, never empty.

case the slice is an annulus, the functions in analytically on

depend

B

inside the annulus.

C

Consider now the Laurent series of a function

for a fixed

In

F

in

B

E MA

The coefficient

is given by

f J

fj W = where

c

J-ldC

1

2Tri

ICI=c

F($,C)C

,

is any fixed number in the range F

If

c

<_

is the Hartogs-Laurent polynomial given

by (5.8), then the coefficients

of

f.

F

occurring in

J

(5.8) coincide with those defined above.

The integral formula

makes it clear that the coefficient functions tinuously. on

F

,

in the norm of

B

.

fj

depend con-

It follows that the

63

coefficient function

A

belongs to

fj

F E B

for each

.

One consequence of this observation is that a Laurent poly-

g.WC3 =-N

nomial

belongs to

EN

coefficient functions B

belongs to

gj

is a proper subalgebra of

a proper subalgebra of

only when each of the

B

.

In particular,

A

just as soon as

C(Y)

C(X)

A

is

.

As an illustration of this class of algebras, we mention an example due to R.Basener, of a uniform algebra B x C(MB)

that B

for

, while every point of

MB

B

such

is a peak point

The first such example had been discovered by B.

.

Cole[l, p.2551,

and Cole's construction also depends cru-

cially on the notion of Jensen measure.

For Basener's example, we start with a compact subset of the open unit disc in every point of

K

such that

C

R(K) x C(K)

K

, while

has a unique Jensen measure (namely, the

point mass) with respect to the algebra

R(K)

.

For such a

K , one can take J.Wermer's example [511 of a compact subset

K

of

such that

C

has no continuous point deri-

R(K)

(Recall that by Theorem 3.8, point derivations of

vations.

all orders exist at any point for which the point mass is not a unique Jensen measure.)

One can also take

McKissick's celebrated Swiss cheese. to be the algebra

R(K)

A

K

to be

In any event, we take

in the preceding discussion, and

we define

u(z) = -v(z) = -log/,---, z12

The set

Y

,

z cK

.

of the preceding theorem becomes the compact sub-

set

z E K, 0 <_ 6 < 2Tr}

Y= of

C2

.

Since Jensen measures are unique, we have

u= -v , and the preceding theorem shows that

64

MB

u = -4 =

coincides

with

Y

According to our earlier remarks,

.

On the other hand, note that (2

so that every point of

,

B s C(MB)

.

lies on the unit sphere of

Y

is a peak point for

MB

B

In fact, the peaking functions can be taken to be a linear combination of

and

z

C

.

Localization Principle for Subharmonic Functions Before beginning, we introduce two pieces of notation that will be useful. If

is a closed subset of

E

the uniform closure in

MA , then

can be identified

AE

.

By Rossi's local maximum

.

modulus principle, the Shilov boundary of

(8E) u (X n E)

will denote

of the restriction algebra

C(E)

The maximal ideal space of AIE with the A-convex hull E of E

in

AE

AE

is included

.

For the second piece of notation, we will denote by the upper semi-continuous regularization of a function on a set

S

,

u* u

defined by

u*(y) = lim sup u(s)

S3s- y

for

belonging to the closure of

y

S

.

The following lemma provides the key to the localization of subharmonicity.

5.7

Lemma.

from

X

to

and let v<_ u

Proof.

v

on

Let

u

(--o,+co]

be a lower semi-continuous function .

Let

E

be a compact subset of

be a quasi-subharmonic function on X n E° , while

v <_ u

on

8E

,

then

Define a lower semi-continuous function

E

.

MA

If

v
w

on

(X n E°) u 8E by

65

u(q)

E X n E°

,

WO)

E

be the solution to the A-Dirichlet problem with

w

Let

w on

boundary data

w5u on Let

(XnE°) u

DE

.

First we show that

E .

be defined as in (5.3), so that the maximal ideal

Y

of the uniform closure

MB

space

8E .

Let

is given by (5.7).

H

:

B

MB - MA

A ® P

of

in

C(Y)

be the natural pro-

jection, and define

Q = [Y n II-1(E°)] u II-1(DE)

Note that

includes the boundary of

Q

includes the intersection of ary of

B

.

II-1(E)

II-1(E)

,

Q also

and

with the Shilov bound-

By Rossi's local maximum modulus principle,

.

Q

includes the Shilov boundary of the restriction algebra BII-1(E)

cludes (-u(4)))

In other words, the

.

(E)

II

A ®P-convex hull of

In particular, if

.

belongs to the

E E

,

Q

then

A ®P-convex hull of

inexp

Q

Now observe by inspection that

E (XnE°) u 8E, ICI

Q=

By Theorem 5.4, applied with place of the

Y

w

and

c E, and

We obtain then

v <_ w on

is a Jensen measure on

5 fvda 5 fwda all such

66

a

of

in place of u ,

.

A ,

in

Q

lying in

any

satisfies Q

e-w0)}

ICI

5

coincides with the AO P-

exp(-u(4))

for

<_

on E.

w <_ u

By hypothesis, a

Q

AE ®P-convex hull of

convex hull.

and

in place of

AE ® P-convex hull of

Now the

AE

<

.

fwda

(XnE°) u 8E

.

(XnE°) u BE

If

for

E E ,

then

By Edwards' Theorem, the infimum over

coincides with

Hence we

obtain

v(4)

w

<_

on

a

E

Q

.

The next lemma is a version of the maximum principle.

5.8

Let

Lemma.

every point of u

MA

is bounded above. then

X

v <_ u

v < u

If

on

on

Let

.

to

X n U , while

U

that

on

v* <_ u

U

Note first that the Jensen boundary is included in

Proof.

so that

X ,

A

be a locally quasi-subharmonic function on

v

,

such that

is a Jensen boundary point for

3U

be a lower semi-continuous function from

Let

3U

be an open subset of

U

is defined on

u

3U s X

.

For the purposes of obtaining a contradiction, suppose that

a=

E U}

Since

is strictly positive. finite.

Define

E

E U

:

Since points of 3U

The hypothesis

.

does not reach out to of

U

is bounded above,

a = u

are Jensen boundary points, v* <- u 3U

on

so that

,

then implies that

3U E

on E

is a compact subset

.

Let o belong to the Shilov boundary of pothesis, there is a compact neighbourhood

which

is

a

-a(01 = a)

lim sup y 3U

v

v

Choose

is quasi-subharmonic. g c A

sup{Ig(1)I

:

AE N

.

of

We may assume that

By hy0

on N c U

such that

4 E E} = 2

,

67

while

IgI

< 1/2

on

E\N°

AE

This choice is possible, since the Shilov boundary of meets E

N°

The choice of

.

show that for

c > 0

c

and

a

sufficiently large,

c(v-u-a) + loglgl < 0

We choose

and the definition of

g

on

aN

so large that also

-ca + log I I g l < 0 l

the norm being that of

with the hypothesis

A

X n U

on

v <_ u

c(v-a) + loglgl < cu

The latter estimate, together

.

on

,

shows that

x n N ,

whilst the former estimate shows that

c(v-a) + loglgl < cu

Now

c(v-a) + loglgl

8N

.

is quasi-subharmonic on

Lemma 5.7, we find that

c(v-u-a) + loglgl

on

<_ 0

Now choose 1 E E n N°

c(v-a) + loglgl

on

N

<_ cu

N

.

on

N

or

,

(5.14)

.

such that

Applying

lg(41)I = 2

.

Then

lim sup [c(v(y) -u(y) -a) +loglg(y) l] = log 2 and this contradicts (5.14).

68

0

This concludes the uphill stretch.

Now we can coast

through the localization theorems and their corollaries.

Theorem.

5.9

from

MA

Let

[-",+-)

to

such that

point of

MA\U

Let

u

a

MA

Judo

be an open subset of

v <_ u

u > v u

q E MA

v 5 u

Then

.

v

is

Let

.

MA\U , since

on

on the Jensen boundary.

On

U

we ob-

Judo <

Hence

from Lemma 5.8.

v

Then

.

Taking the infimum over such

.

MA

U , while every

be a Jensen measure for

coincides with

tain

U

is a Jensen boundary point.

satisfy

U E CR(MA)

Let

.

is locally subharmonic on

v

subharmonic on

Proof.

be an upper semi-continuous function

v

u

,

we obtain

is subharmonic.

The exact analogue of Theorem 5.9 for log-envelope functions is false.

Suppose for instance that each point of

is a Jensen boundary point for

Then the characteristic function of

empty.

semi-continuous function on envelope function on MA

on

U = MA\X

A , while

.

U

MA

X

is not

is a lower

U

that is a locally log-

but is not a log-envelope function

,

However, there is a slightly more complicated ver-

sion of the localization theorem that is valid for logenvelope functions.

5.10

Theorem.

each point of

Let

MA\U

U

be an open subset of

is a Jensen boundary point.

a bounded, lower semi-continuous function on u

is a locally log-envelope function on

continuous at each point of envelope function on

Proof.

MA

MA

MA\U

.

Then

MA

such that Let

such that

U , while u

be

u

u

is

is a log-

.

Since each point of

MA\U

is a Jensen boundary point,

69

u = u

MA\U

on

clude that

u <_

Now apply Lemma 5.8 to

.

U

on

u

Hence

.

v = u , to con-

u = u , and

is a log-

u

envelope function.

Corollary.

5.11

11

A locally subharmonic function on

MA

is

A bounded, locally log-envelope function on

subharmonic.

is a log-envelope function.

MA

Applying the preceding corollary to the algebra where

is an A-convex subset of

E

AE

MA , we obtain the

following.

5.12 E

Corollary.

Let

be an open subset of

U

be a compact, A-convex subset of

subharmonic function on

U

U

.

MA

Then any locally

is subharmonic on

E

.

locally bounded, locally log-envelope function on log-envelope function on

E

and let

,

Any U

is a

.

Try to prove the following theorem from the definitions:

5.13

Theorem.

Let

U

be an open subset of

MA

.

The

pointwise limit of a decreasing net of locally subharmonic functions on

U

is locally subharmonic.

The pointwise limit

of an increasing net of bounded, locally log-envelope functions on

Let

Proof. {ua}

is a locally log-envelope function.

U

u

be the pointwise limit of the decreasing net

of locally subharmonic functions on

Choose a compact, A-convex neighbourhood in

U

By Corollary 5.12, each

.

u

a

U N

.

of

Let p

p e U

.

contained

is subharmonic on

N

.

The proof of Lemma 5.3 shows that a pointwise limit of a decreasing net of subharmonic functions is again subharmonic. Hence

u

monic on

is subharmonic on U

N

,

and

u

is locally subhar-

.

The proof of the second statement is similar, once one observes that the pointwise limit of an increasing net of log-

70

envelope functions is a log-envelope function.

Suppose that every point of

5.14 Theorem.

boundary point.

Furthermore, the restrictions to

is subharmonic.

MA\X

of the subharmonic functions on MA\X

subharmonic functions on

Let

E

that are bounded above.

MA\X

has a Jensen measure supported on

p E E

does not meet

MA\X

Hence

X , and

are precisely the

MA

be a compact subset of

E

is a Jensen

X

Then any locally subharmonic function on

MA\X

Proof.

11

E

Then any

.

.

Consequently

is a compact subset of

t

MA\X

is A-convex, and the first statement of the

theorem follows easily from Corollary 5.12.

MA\X

The restriction to MA

Conversely, if

is subharmonic and bounded above.

a subharmonic function on u*

of a subharmonic function on

MA\X

u*

is

that is bounded above, then

is an upper semi-continuous extension of

Furthermore,

u

u

to

MA

is subharmonic, by Theorem 5.9.

Localization of the Jensen Boundary Let x0

X

MA

in

that if

AIN

,

x

N

E X , and let

0

Rossi's theorem on local peak points asserts

.

0

then

be a compact neighbourhood of

is a peak point for the restriction algebra is a peak point for

xo

A .

However, it remains

x0

is a peak point for

an open problem to determine whether

A

under the weaker hypothesis that

the closed restriction algebra

AN

x

0

be a peak point for

.

The problem can be reformulated in terms of representing measures.

Suppose that

x0

such that the point mass at

has a compact neighbourhood x 0

measure for

x

0

on

resenting measure for

N

is the only representing

Is the point mass a unique rep-

.

x

N

?

0

The analogous problem for Jensen boundary points can be solved on the basis of the localization theorem.

71

x0 E X x0

in

of

AN

such that

MA

Then

.

Let

u(x0) = 1

that c log

off

N


Then

e > 0

A

v = c log Ifl

is continuous, v

v

on

N

u ,

and

is any Jensen measure on

Judo >_ Jvda ? v(x0) >_ 1-c

obtain

Judo = 1

,

and

.

Since

MA

x0

f E AN

and

,

v

such

v = 0

vanishes

Clearly

.

x0

for

,

then

is arbitrary, we

is supported on the set

a

u

v

is subharmonic.

v

> 0

c

On account of the freedom of choice of mass at

N

By Theorem 5.9,

is locally subharmonic. a

x0

Define a function

.

off a compact subset of the interior of

If

Since the

.

and

c > 0

belongs to

f

A

N , while c log i f (x0) '> 1- e

on

by setting

.

Let

.

there exist

AN ,

We may assume that MA

MA\N

on

is the unique Jensen measure for

x0

Ifi

of

0 5 u <_ 1, while

satisfy

u = 0

and

,

N

belongs to the Jensen boundary of

x0

with respect to

on

and let

X ,

belongs to the Jensen boundary

x0

U E CR(MA)

point mass at

v

be a uniform algebra on

Suppose there is a compact neighbourhood

.

Proof.

A

Let

5.15 Theorem.

,

a

{u= 11

is the point 11

.

One corollary to the preceding theorem is that if

is

$

not a Jensen boundary point, then any subharmonic function u

on a neighbourhood of

¢

satisfies

lim sup u(y)

{a

Indeed, on the basis of Theorem 5.15, there is a net

of Jensen measures for aa's

tend to

{0

sup u(y)

.

Judo a

as

has no mass at

then implies that

u

at

.

¢

.

u(4) <- lim

The reverse inequality is the upper semi-

continuity of

72

such that the supports of the

¢

, while each

The estimate

}

a

Supports of Jensen Measures As another immediate application of the localization theorem, we mention the following result.

Let

5.16 Theorem.

every point of

U

be an open subset of

is a,Jensen boundary point.

DU

Jensen measures for points of

Proof. u

on a

u = 1

Define

on

U

.

u = 0

and

By Theorem 5.9,

u

u

shows that

a

Then all

on

MA\U

U

.

Then

.

is locally subharmonic

is subharmonic on

is a Jensen measure for a point

1 = u(4) 5 Judo

such that

are supported on

U

is upper semi-continuous, and MA\8U

MA

MA

If

.

e U , then the estimate

assigns full mass to

U

.

11

It is proved in [41 that the Jensen measures for supported on a compact subset of

U

E U

are weak-star dense in

the set of all Jensen measures for

.

Theorem 5.16 is interesting, in part, because the corresponding assertion for representing measures is false.

To

see this, consider the "string of beads" example of A.M.Davie

and J.Garnett (cf.[2,p.1221), with a disconnected Gleason In this example,

part.

K

plex plane, every point of and

K°

is a compact subset of the com8K

is a peak point for

R(K)

,

consists of two connected components which together

form a single Gleason part for component of

K°

R(K)

.

Then points of each

have representing measures with positive

mass on the other component of

K°

Bremermann Functions We wish now to introduce a class of functions, the "Bremermann functions", which appear as solutions to a generalized Dirichlet problem. defined locally.

The class of functions will be

In one complex variable, these functions

73

will correspond to the harmonic functions.

In several com-

plex variables, they will correspond to the plurisubharmonic functions whose complex Hessian matrices have determinant zero.

Let

be an open subset of

U

continuous function each

N , where

on

.

A lower semi-

is a Bremermann function if

U

has a compact neighbourhood

q E U

u = v

on

u

MA\ X

v = ulaN

N c U

such that

This latter condition is

.

equivalent to the identity

inf

uda

e N

,

(5.15)

,

faN

a

where the infimum is taken over all Jensen measures supported on

Note that the hypothesis

8N .

for

a

U c MA\X

guarantees, by Rossi's local maximum modulus principle, that every

assumes its maximum modulus over

f E A

Consequently each

E N

N

on

has a Jensen measure on

8N

8N ,

so

that the infimum in (5.15) is well-defined.

Note that every Bremermann function is a locally logenvelope function.

The next two results serve to generate a number of Bremermann functions.

5.17 Lemma.

Let

E

be a compact subset of

be a lower semi-continuous function from Then

Proof.

E MA\(XUE)

Let

neighbourhood of

4

and let

,

disjoint from

log-envelope function on

N

f E A

< u

satisfy

c log Ifi

on

c log Ifl

w = v

on

74

< v

on

MA\N , while

8N

be any compact

N

X u E

.

aN

MA

(--,+o]

to

Then

.

Suppose that

.

tinuous subharmonic function on while

E

MA\(XuE)

is a Bremermann function on

u

MA , and let

.

c > 0

Let

w

on

w = max(v, c logIfI)

and

be a con-

v

such that

Define

is a

u

v < u MA on

,

so that N .

Then

u

w

is continuous, and w

w

By Theorem 5.10, on

E

also

,

w
is a log-envelope function. Since

on N.

Hence

clog jfj

u

5.18 Lemma.

c > 0

d e ]R, g e A

,

U ,

Bremermann function on

In particular,

U

.

then

are Bremermann functions on g e A

and all invertible

Proof.

If

a

cu + d logigi

floglglda

validity of (5.14) for ,

whenever

so that u

is.

is a

f e A

.

is any Jensen measure for

obtain

u

and

logjgI

for all

MA\X

,

ing the Jensen-Hartogs inequality to both

loglgl

on N.

is invertible, and

is a Bremermann function on

Re(f)

<_ u

w _< u

is a Bremermann function.

It follows that

If

is locally a log-envelope function.

.

g

then by applyand

1/g , we

It follows easily that the

implies its validity for

u

cu + d loglgl

cu + d

is a Bremermann function

The final statement of the theorem follows

from the first statement, upon setting

g = exp(f)

11

.

The maximum of two Bremermann functions need not be a Bremermann function, not even in the complex plane.

Worse

yet, the sum of two Bremermann functions need not be a Bremermann function. of Chapter 6.

To see this, we anticipate some results

It turns out that

Bremermann functions on

G2 ,

Iz1I2

and

Iz2I2

since they are plurisubharmonic

and their complex Hessian matrices are degenerate. Izlj2

+

Iz2I2

that

+

However,

is not a Bremermann function, since its com-

plex Hessian matrix is strictly positive. lzll2

are

Iz2I2

One can also see

is not a Bremermann function by observ-

ing, from the expression (5.14), that a Bremermann function cannot attain a strict local minimum. The main result on Bremermann functions is the following version of the maximum principle.

The proof proceeds along

the same lines as that of the maximum principle in Lemma 5.8.

75

Let

5.19 Theorem.

be an open subset of

U

be a Bremermann function on v

u

Let

that is bounded

U

If

lim sup v(z) 5 lim inf u(z)

all

,

U3z -

U3z +> then

Let

.

that is bounded below.

U

be a quasi-subharmonic function on

above.

MA\X

v 5 u

U

on

E

8U

.

Towards obtaining a contradiction, we assume that

Proof.

a = sup{v(q)-u(4)

E U}

:

is strictly positive.

Since

below,

Set

is finite.

a

E =

U

is bounded above and

v

u

lim sup [v(y)-u(y)] = a}

:

y -}$ Evidently

E

Choose

4

algebra

to

8N

Let

.

u = g

N

By shrinking

is the restriction of

g

contained in

0

N

subharmonic on

large,

be a compact neighbourhood of N , where

on

neighbourhood of ,

N

g E A 1

.

The proof of Lemma 5.17 shows that any compact

.

Choose

U

to be a Shilov boundary point for the

E E

0

AE

such that u

is a compact subset of

has the same property.

N

then, we can assume that

v

is quasi-

.

so that

for some

Igl < 1/2 E E n N0

c(v-u-a) + loglgl < 0

on

.

aN

on

E\N°

,

For

c > 0

.

Letting

while sufficiently c = 1/c

we obtain

v-a+c loglgl If

76

E N , and if

u

o

on

aN .

is a Jensen measure on

8N

for

then

a + e loglg(oI 5

fry - a + e loglgl]da

Jud° Taking the infimum over such

a

, we obtain

a + e loglg($)I 5 u($)

Hence

5 a - e logIgo1)I

a = lim sup

Igo1)I > 1

This contradicts

.

According to the definition, a Bremermann function assumes

locally the form w

for appropriate

The next theorem

w .

shows that often it assumes this form globally.

Let

5.20 Theorem. E°

is disjoint from X

on

E

such that

u = w

on

aE

be a log-envelope function

u

<_

u

on

and

c > 0

E°

f c A

let

u

Let

v = c logIfl

Hence w 5 u

.

.

Since

U

u

be an open subset of

U

,

then

,

we obtain is itself

.

be a bounded Bremermann function on

any compact, A-convex subset of

where

Then

.

c loglfl 5 u

satisfy

a log-envelope function, we obtain w = u

5.21 Corollary.

E°

w = uIaE

Applying Theorem 5.19 to

.

c loglfl

Let

.

such that

is a Bremermann function on

, where

Suppose

Proof.

on

E

u

MA

be a compact subset of

E

MA\X , and U

.

u = w

If

E

on

E

is

w = uIaE

77

This follows immediately from Corollary 5.12 and

Proof.

Theorem 5.20.

The pointwise limit of an increasing net of

5.22 Theorem.

bounded Bremermann functions is a Bremermann function.

Suppose that

Proof.

pointwise to

{u

U

on

u

is the increasing net, converging

}

a

Let

.

e U

pact, A-convex neighbourhood of ua = wa

N

on

we obtain

wa = uaIDN

where

,

u =

be a com-

N

and let

.

By Corollary 5.21,

.

Passing to the limit,

w = uIM .

N , where

on

,

The Generalized Dirichlet Problem Fix an open subset

U

MA\X

of

real-valued function on

8U

Let

.

h

The generalized Dirichlet

.

problem is to find a Bremermann function on the boundary values

on

h

be a bounded,

8U

that attains

U

We aim to study the sol-

.

ution to the problem given by the classical Perron process.

Define a subsolution to be a continuous, locally subharmonic function

on

u

U

such that

u* <_ h

on

8U

The

.

upper envelope of all subsolutions will be referred to as the solution of the generalized Dirichlet problem with boundary function

h

and be denoted by

,

locally log-envelope function on ditions on

and on

U

h

h

h

on

h

If

If

h ? b

on

8U

is a subsolution, so that

78

h

is a

We wish to give con-

.

8U

.

actually

h

First we observe

is a bounded function on

is a bounded Bremermann function on

w* <_ c

Evidently

is a Bremermann function.

5.23 Lemma.

Proof.

.

that guarantee that

attains the boundary values that

U

h

on

8U

,

U

8U

,

then

.

then the constant function h >_ b

.

h

If

for any subsolution w .

h <_ c

on

b

8U ,

By the maximum

then

principle (Theorem 5.19), and

w <_ c

U

on

.

Hence

h <-

c

is bounded.

h

p c U , and let

Let

bourhood of

p

Let

.

be a compact, A-convex neigh-

N

To show that

v = hIDN .

Bremermann function, it suffices to show that By Corollary 5.12, so that

h <_ v

such that

c loglfl < h

y c DN

function

u

y

on

DN

such that

U

on

Choose c> 0 while

,

f c A

and

c logIf(q)I > v(q)-c

u* <_ h

on

DU , while

This estimate persists in a neighbour-

.

Covering

.

N

there is a continuous, locally subharmonic

,

u(y) > c loglf(y)I hood of

on N.

v = h

We must obtain the reverse inequality.

.

For each

is a

is a log-envelope function on

h

q E N, and let e> 0.

Fix

h

DN

by a finite number of such

neighbourhoods, and taking the maximum of the corresponding u , we obtain a continuous, locally subharmonic

functions function

w

w > c loglfl

on

that

w0 =w on

Then

wo

on

DU

.

such that

U

on

DN

w* <_ h

on

DU

Define a function w

.

U\N , while

0

, while

U

on

w o = max(w,cloglfl)

is continuous and locally subharmonic, and

w

Consequently

c loglf(q)I <- h(q) v(q) <_ h(q)

.

In particular,

0

Letting

so

on

N

w o*

<_ h

v(q) -e <

tend to zero, we obtain

e

0

.

The theory of barriers can be,adapted to give conditions under which the boundary values of For simplicity, we assume that ric

d

U

h

coincide with

h

is metrizable, with met-

.

A barrier at a point subharmonic function

lim

u(z) = 0

u

C E DU on

U

is a continuous, locally such that

,

U_-4z --

while for each satisfying

d

> 0

d(z,C) ? S

,

the supremum of

u(z)

is strictly negative.

for

z e U

The point

79

is a regular boundary point of

c e 8U

a barrier at

C

From Theorem 1.13, it follows that any

.

boundary point of

U

.

is a regular boundary

5.24 Theorem.

Suppose that

point.

is a bounded function on

h

If

lim sup h(p) <_ lim sup

8U

lim inf h(p) < lim inf

U3p--

,

(5.16)

h(q)

.

(5.17)

8U3q->

is continuous on

aU

continuously at

Let

8U

U

,

for

.

satisfies

lim inf h(z) ? a }

and

.

Let

c > 0

h <_

8U

v* <_ h on 8U This proves

.

in a neighbourhood

6

M be an upper bound for

large,

on

M + cu* <_ 6

subsolution.

Using the estimate

the estimate

w* <_

on

S

J

,

supremum over subsolutions lim sup h(z)

<_

S

.

on .

w* <_ M on

we obtain

w ,

h

(8U)\J

By the maximum principle, w + cu 5 S

and

in

C

z

Suppose next that 8U

C

v = a +cu

(5.17).

in

h

assumes the boundary values

h

in a neighbourhood of

large,

on

is regular, and if

8U

.

h >- a

c > 0

v <_ h

then

,

be a barrier at

u

Suppose that

Hence

then

,

h(q)

In particular, if each point of

Then for

8U

8U3q-

U3p->

Proof.

is a regular

A

that is a Jensen boundary point for

C E 3U

h

if there exists

U

Let

we obtain

U

.

Then

.

w

(2U)\J

w* + cu* <_

on

of

J 8U

6

be any and

,

on

DU.

Taking the

h + cu 5 6

on

U

This proves (5.16).

In our definition, we have specified that subsolutions be continuous on

U

.

However, we might just as well consider

the upper envelope of all locally subharmonic functions on

80

U

such that

w* <_ h

on

aU

.

w

This upper envelope might

be strictly greater than

h

The following theorem gives

.

conditions under which this upper envelope coincides with h

Let

5.25 Theorem.

every point of

be an open subset of

U

is regular.

8U

MA\X

Suppose that

bounded, lower semi-continuous function on a locally subharmonic function on 8U

,

then w 5 h on

u = h

while

on

8U

By Lemma 5.23,

u

on

U

8U

is a

w

is

w* <_ h

on

If

.

such that

u = h

so that

on

is a Bremermann function on

u

U ,

u is lower semi-continu-

By (5.17),

.

h

.

Define a function

Proof.

ous.

U

U

such that

U

Since subharmonic functions are bounded above on compacta,

w

and since

above on

U

is bounded above near

8U

w

,

From Theorem 5.19 we obtain

.

is bounded on

w <_ u

U 11

In general, the continuity of tinuity of

h

even if

,

8U

h

does not imply the conConsider for ex-

is regular.

ample the algebra of analytic functions on two analytic discs {Izl 5 1}

w = 0

and

For

.

{lwl

l}

<_

,

z =

attached at their origins

we take the boundaries of the two discs.

X

The set of Jensen measures for the origin consists of the interval joining the measure on

{1wl = 1}

d6/27

on

{lzl = 1}

and

Each other point has a unique Jensen

.

measure, either a point mass or a Poisson kernel.

boundary function

Owl = 1} {0 <

lwl

,

<_

,

is zero on

h

then 1}

is zero on

h

so that

h

Conditions under which following theorem.

Let

{1zI 5 1}

and and

is discontinuous at

1

on on

1

z = w = 0

.

is continuous are given in the

We will see in Chapter 6 that the hy-

potheses are met in

5.26 Theorem.

OzI = 11

If the

0n

U

be an open subset of

MA\X

.

Suppose

81

that if on

U

is any bounded, locally log-envelope function

u

h E CR(8U)

is locally subharmonic on

u*

then

,

is such that

continuously at

8U

.

(h)*

By Theorem 5.25,

continuous on

U

If

.

attains the boundary values

h

then

By hypothesis,

Proof.

on

8U ,

U

is continuous on

h

U

is subharmonic, and (h)* 5 h

on

U

.

h

.

(h)* = h

Hence

h

is

0

.

References 1.

Browder, A.

Introduction to Function Algebras, W.A.

Benjamin, Inc., 1969. 2.

Gamelin, T.W.

Uniform algebras on plane sets, in Ap-

proximation Theory, G.G. Lorentz (ed.), Academic Press, New York, 1974, pp.101-149. 3.

Gamelin, T.W.

Uniform algebras spanned by Hartogs

series, Pacific J. Math. 62 (1976), 401-417. 4.

Gamelin, T.W. and Sibony, N.

Subharmonicity for uniform

algebras, to appear. 5.

Wermer, J.

Bounded point derivations on certain Banach

algebras, J. Functional Analysis 1 (1967), 28-36.

82

6 Algebras of analytic functions

In this chapter, we aim to study the abstract notion of subharmonicity in the context of algebras generated by analytic 0n functions on compact subsets of The key to the appli.

cation of the abstract theory is a theorem of H. Bremermann,

asserting that the abstract subharmonic functions essentially coincide with the plurisubharmonic functions.

We begin with

a review of plurisubharmonic functions.

Plurisubharmonic Functions Recall that a function to

from an open subset

u

is plurisubharmonic if

[-co,+-)

ous, and the restriction of

u

D

is upper semi-continu-

u

to the intersection of

and any complex line is subharmonic.

0n

of

D

Good references for

plurisubharmonic functions are the monographs of L. Hormander [7] and V.S. Vladimirov[9].

The plurisubharmonic functions on if

and

and

u

cu

D

are plurisubharmonic and

v

are plurisubharmonic.

form a convex cone; c > 0

,

then

The cone of plurisubharmonic

functions includes the cone of functions of the form Ifj

, where

c ? 0

and

f

u +v

is analytic on

D

.

c log

The maximum

of a finite number of plurisubharmonic functions is plurisubharmonic.

The composition of a convex increasing function

and a plurisubharmonic function is again plurisubharmonic.

The complex Hessian matrix of a smooth function

u

is the

hermitian matrix function n

2

Hu(z) =

8 u

(6.1)

(z)

8z.j

k

j,k=l

83

A smooth function H (z) ? 0

if

is plurisubharmonic on

u

z c D

for all

u

strictly plurisubharmonic on positive definite at each

.

if and only

D

A smooth function if the matrix

D

z E D

u

is

Hu(z)

is

.

A decreasing limit of plurisubharmonic functions is plurisubharmonic.

on

is an arbitrary plurisubharmonic function

u

If

then there is a sequence of domains

D ,

increasing

{D .} J

D

to

and infinitely differentiable plurisubharmonic func-

,

tions

defined on

u.

D. .

,

such that the sequence

creases pointwise to

u

{u.}_

Suppose that

J J=1

subharmonic functions on limit

u

.

Since

j

.

is an increasing sequence of pluriD , which has a finite pointwise

need not be upper semi-continuous,

u

is not necessarily plurisubharmonic. semi-continuous regularization u* = u

and

Suppose

A

in

u*

of

is plurisubharmonic,

u

almost everywhere with respect to volume measure.

A

is a uniform algebra, and that the functions

are analytic on some open subset

D

Cn

of

f.'s belong to A, is plurisubharmonic on

function on

u

However, the upper

any function of the form max(loglf1l,...,loglfm1) the

de-

{u .}

J

D

D

,

.

that is subharmonic with respect to

Then

.

where

Since any A

can

be expressed on compacta as a decreasing limit of such functions, the A-subharmonic functions are plurisubharmonic on D

.

Later we will consider the problem of determining which

plurisubharmonic functions arise from A-subharmonic functions.

Plurisubharrnonic Barriers and Pseudoconvex Boundary Points Let

D

barrier at

be a bounded domain in C E

aD

lim v(z) = 0 D3 z- C

84

.

A plurisubharmonic

is a plurisubharmonic function

such that

while for each

Cn

r > 0

,

v

on

D

v(z) < 0

sup

zED,lz-CI>r

For example, if f

on

at

D

C

forms a plurisubharmonic barrier

Ifl- 1

then

,

is a peak point for an analytic function

C

.

Note that the notion of plurisubharmonic barrier is local.

If a plurisubharmonic function near

defined in the part of

v ,

has the properties above, then for

,

iently small, the function defined to be and

away from

-d

A point

C

near

max(v,-d)

is a plurisubharmonic barrier at

is a smooth point

CE aD

suffic-

> 0

6

containing

of

B

such that

dp # 0 , while BnD = {p <0}

C

a strictly pseudoconvex boundary point of Hessian matrix associated with

p

p

The point

.

C C

.

if there is an

DD

and a smooth function

open ball

D

D

on

B is

C

if the complex

as in (6.1) determines a

positive definite bilinear form on the complex tangent space az.2

to

at

91)

nonzero

C

,

that is if

a E Cn

E

that satisfy

E az

for all

0

J az k

(p)a. = 0

This defi-

.

J

nition depends neither on the smooth defining function nor on the analytic system of coordinates at p

by

etp - 1

for

t

C

p

Replacing

.

large, we can always arrange that the

complex Hessian matrix of a defining function at a strictly pseudoconvex boundary point is positive definite.

6.1 Lemma.

point of

Proof.

If

D

,

Choose

C

then there is a plurisubharmonic barrier at

B

Hessian matrix of p

is a smooth, strictly pseudoconvex boundary

and p

C

as above, so that the complex

p

is positive definite on

B

.

Expanding

in a Taylor series, we find an analytic quadratic poly-

nomial

f

such that

p(z) = Re(f(z)) + C aZ.aZ J

k 85

.

Since the Hessian of exists

near

z

C

there

,

such that

c > 0

clz-Cl 2

Re(f(z)) <_ p(z) -

for

is positive definite at

p

(6.2)

It follows that

.

Re(f)

is a barrier at

at least locally, and hence there is a global barrier

,

at

E

.

There is a converse to Lemma 6.1.

6.2 Lemma.

Let

be a smooth boundary point of

7

D

Sup-

.

pose there is a smooth strictly plurisubharmonic function on a ball and

containing

B

v(C) = 0

point of

Proof.

D

.

Then

C

v 5 0

such that

5

v

B n D,

on

is a strictly pseudoconvex boundary

.

Making an analytic affine

We follow H.Rossi[81.

change of variables, we can assume that is defined by a function

= 0

E

and that

D

of the form

p

P(xI + iy1) ...,xn + iyn) = x1 + o(Izl)

We must show that

,

8z az

.

for nonzero

0

A e do

k

al = 0

satisfying

Making a linear analytic change of the

z2,...,zn , we can arrange also that

variables = 0

.

2

2

... An

Near

x2 0

,

8D

by the variables ...,yn)

2

2(0)+a2(o) >0

(0)

az2 Di2

(6.3)

3y2

forms a manifold, which is coordinatized (yl,x2,y2,...,xn,yn)

by the x1-coordinate on

Let

X1 = X1(y1,x2)

.

8D , regarded as a function

of the remaining variables, so that

86

a3

It suffices then to show that

.

p(XI(yl,x21.... yn),yV x2,...,yn) = 0

near

For a function of

to

i

DD

0

,

.

r

denote by

the restriction yl,x2,...,

as a function of the coordinates

,

so that

yn r

(yl,x2,...,yn) = (Xl(yl,x21.... yn)'yVx2,...,yn)

.

From the chain rule we obtain

W =n+

ate,

ax2

axl ax2

2

ax2 r

a

a

=

2

ax2

aXl

DX1 ax

a2

2

2ax2 ax ax 2 1

2

ax

i IX

a2X1

ad

a2V

2 ax

2

1 ax2

ax2

ax 2 \2

X

2 ax

2

a

1

C

axl

Evaluating at

0

2 r

and noting that

,

2 a

2 (0) =

2(0) + X (o) ax2

ax2

1

Applying (6.4) to ax

p

ax

,

we obtain

a2X 21(0) ax2

(6.4)

and noting that

,

(0) = 0 2

while

pr = 0

(0) = 1 , we obtain 1

a2x

2

0 = a 2(o) +

(6.5)

1(0) ax2

ax2

Applying (6.4) to the plurisubharmonic function

u

,

and now

using (6.5), we obtain 2 r D

ax2 Since

2

2

2 (0) =

2(0) a

ax2 u

ax

(o) 2(0)

1

ax2

attains its maximum over

locally, we have

a22ur

(0) 5 0

,

aD

at

0

,

at least

and

ax2

87

a 22P (0)

2

2(0) 5 ax (0)

ax2

ax2

1

2

There is a similar estimate for

a 2(0)

mates, we obtain

9y2 2

2

2

.

Adding the esti-

2 a

2(0) < aXl(o)

2(o) + 2(0)

a

ax2

ax 2(0) + 2

y2

ay2

2

The term on the left is

4 az28z2 , which is strictly posi-

tive on account of the strict plurisubharmonicity of Since

>_ 0

ax (0)

u

, we obtain (6.3), as required.

1

SS-sets

K

A compact subset

of

0n

K

is an SS-set if

is the

limit of a decreasing sequence of open sets, each component of which is a domain of holomorphy.

We denote by

0(K)

the

algebra of functions that are analytic in a neighbourhood of K , and by

the uniform closure of

H(K)

It is the algebra

H(K)

0(K)

in

C(K)

that we are interested in.

Any compact subset of the complex plane is an SS-set, and coincides with the usual algebra R(K) Cn Any compact polynomially convex subset of is an SS-set, in this case

H(K)

.

and in this case the algebra gebra

P(K)

generated by the polynomials in

6.3 Theorem.

space of

Proof.

coincides with the al-

H(K)

H(K)

If

K

zl'...,zn

.

is an SS-set, then the maximal ideal

coincides with

K

.

This follows readily from the fact that every nonzero

complex-valued homomorphism of the algebra of analytic functions on a domain of holomorphy is the evaluation homomorphism at some point of the domain.

88

K

Now fix a compact subset

and

Cn

and a smooth function

C

.

Let

be a smooth

c

K , so that there is an open ball

boundary point of taining

of

B n K = {p =0}

on

p

B

dp x 0

with

If the complex Hessian

.

is not

positive (indefinite) on the complex tangent space to at

C

then there is a ball

,

containing

B'

con-

B

c

2K

such that

all functions satisfying the tangential Cauchy-Riemann equa-

tion on {p =0} extend to be analytic in B' n {p >0} (cf. In particular, all functions in

[6, Theorem 2.6.131).

extend holomorphically to spectrum of

H(K)

.

B'

,

so that

We conclude that if

K

0(K)

cannot be the

K

is an Sd-set,

then the Hessian of any smooth defining function is positive on the complex tangent space at any smooth point of the smooth points of

8K

and let

,

Let

K

sisting of the smooth boundary points. of

and the Shilov boundary of

N

adherence in

Proof.

for

K

Then the intersection

H(K)

coincides with the

be a smooth strictly pseudoconvex boundary

r

K

point of

con-

8K

of the strictly pseudoconvex boundary points.

N

Let

be an Se-subset of

be the relatively open subset of

N

There is then a smooth defining function

.

so

are pseudoconvex.

6.4 Theorem (Rossi's Theorem). (n

8K ,

such that

near

H (C)

p

is positive definite.

As

P

earlier, there is a quadratic analytic polynomial Then

that (6.2) is valid. tion at

c

r

H(K) 0(K)

in

,

(cf. [51). H(K)

Conversely, suppose that H(K)

.

Let

In particular,

e

.

is a

C E N

e > 0

C

belongs

.

belongs to the Shilov

be small, so that

a defining function on the open ball with radius

C

and in fact there is a function peak-

to the Shilov boundary of

boundary of

is a local peaking func-

By Rossi's local peak point theorem,

.

peak point for ing at

exp(f)

such

f

Be

has

2K

centered at

C

We must find a strictly pseudoconvex

89

K within

boundary point of Choose

such that

f e 0(K)

K\Be

small on

For

.

BE

.

lifli = 1

, while

is

Ifl

sufficiently small, the func-

d > 0

tion dizl2

u(z) =

+ loglf(z)l

We can assume that

at some point 1 e Be n BK

K

attains its maximum over

f(C1) z 0

,

so that

u

is smooth and

strictly plurisubharmonic in a neighbourhood of Lemma 6.2,

is strictly pseudoconvex at

BK

.

Cl

El

By

.

.

CI

There is a geometric argument which makes it plausible that the Shilov boundary should fail to meet an open set on on which the complex Hessian is degenerate.

BK

It turns

out that "most" of such points lie on analytic varieties in BK

.

More precisely, if

Q

is a relatively open subset of

consisting of smooth boundary points, such that the com-

BK

plex Hessian

H (C)

for a defining function

has constant

p

P

rank

r

on the complex tangent spaces

each point of

the space of vectors in

M

T

r

T.'s

.

Indeed, let

annihilated by

are complex subspaces of

checks that the

T

HP(C)

there passes a manifold

Since the

M

M

in

Q

r

.

C E M

e Q

coincides with

M

MC

.

is a complex

For details, see [4].

Bremermann's Theorem Our next task is to prove a theorem of Bremermann

90

so that

such that the tangent

are complex subspaces,

analytic manifold.

,

be

form an involutive distribution.

at each point

Mc's

M

of dimension

Frobenius' Theorem then shows that through each

space to

then

,

has a neighbourhood that is fibered by

Q

analytic varieties of dimension

the

at

T

C e Q

One

H(K)-subharmonicity.

relating plurisubharmonicity and

The

proof we will give is, in broad outline, the same as that of We begin by establishing several lemmas.

Bremermann[2].

6.5 Lemma. v

let

the set

Let

be plurisubharmonic on

(-°,+-]

and

Then each component of

.

i

+-

that tends to

U

on

be a convex increasing function from

X

,

is pseudoconvex, there exists a plurisub-

U

harmonic function Let

U

0n

is a domain of holomorphy.

{v <0}

Since

Proof.

be a domain of holomorphy in

U

such that

x(O) = +-

harmonic function on at the boundary of

{v <0} {v <0}

Then

.

,

.

to

is a plurisub-

X°v

+-

tends to

{v<0}

Since

.

8U

(-o,0]

4 + X°v

and

at

has a plurisub-

harmonic exhaustion function, each component is a domain of holomorphy.

6.6 Lemma. let

Let

D

be a domain of holomorphy in

be plurisubharmonic on

u

Then

Proof.

E

,

and

Define

.

e-u(z) )

Cn+l

D* _

D

tn

:

z E D,

Icl

<

(6.6)

.

is a domain of homomorphy.

D*

Apply the preceding lemma to the plurisubharmonic

function v(z, C) = u(z) + logk

l

on

DxC

. 11

6.7 Lemma. u

Let

D

be a .domain of holomorphy in

be plurisubharmonic on

D

,

and define

D*

&n

,

let

as in (6.6).

Then there is an analytic function

F(z,C) _

j

fj(z)CI

,

z E D,

ICI

<

e-u(z)

(6.7)

91

on

such that for each fixed

D*

convergence

{Km}

Let

Proof.

convex subsets of

z E D

,

that increases to

D*

Km

choose

E D*\Km

holomorphic on

D* .

{z = z0}

slice

,

neighbourhood of

(6.8)

be a sequence of compact holomorphically

be the projection of

Icf(z,0l 5 1/2

the radius of

,

of the series (6.7) is given by

RF(z)

RF(z) = e-u(z)

z E D

into

D

0

ve 0

Choose

.

C0f(z0)?0) = 1

z0

m

f

, while on

Since f is not identically the variety

E

z0 E Em

For fixed

.

such that

such that

D* , and let

Km

on the

1

projects onto a

{Cf = 1}

Covering Em by a finite collection

.

of such neighbourhoods, we obtain a finite collection fl,...,fk

Km ,

on

1/2

exists = 1

.

D*

of analytic functions on 1 5 j

<_ k

(z0,C0) E D*

such that z0 e Em ,

, while for each

and an index

j

lcfjl

such that

<

there

C0f.(z0,c0)

Define k

F

[1 - (cf.)N]

II

m

j =1

where the integer Km .

on

N

is chosen so large that

Then for each

has a zero in the disc

I1-Fml

z0 E Em , the function flCl

e-u(z)}, while

<

< 1/2m

Fm(z0,0

F(z0,0) =1.

The product

F(z, ) =

II

Fm(z,

m=1

then converges normally on function disc

F(z0,C)

.

For each

z0 E D ,

the

has an infinite sequence of zeros in the

< e-u(z)} , while

{ICI

D*

F(z010) = 1

.

It follows that

the radius of convergence of the Hartogs series expansion (6.7) of

F

is given by (6.8).

0

92

6.8 Theorem (Bremermann's Theorem). of

Cn

and let

,

K

Let

be an Se-subset

be a function that is defined, continu

u

ous and plurisubharmonic in a neighbourhood of is subharmonic with respect to

approximated uniformly on

Proof.

that is,

,

D

fined on the set

modulus by

M

and

<_

ICI

D*

u as

D*

on

F

such that

M > 0

z E K,

such that

Define

.

in (6.6) and choose a holomorphic function r > 0

and

K .

D D K

Choose a domain of holomorphy

is continuous and plurisubharmonic on

Choose

u

can be

u

cl,.... cm > 0

, where

are analytic in a neighbourhood of

Lemma 6.7.

Then

.

K by functions of the form

max(cI loglfll,...,cm logifmI) f1,...,fm

H(K)

K

as in is de-

F

and bounded

r}

in

The Cauchy estimates for the coef-

there.

ficients of the power series (6.7) then become

z c K, j? 0.

if . WY <- M ,

It follows that the functions

u .(z) = 1 loglf .(z)I j

J

z E K

,

J

are bounded above on

K , uniformly in

j

.

Since

u(z) = -log RF(z) = lim sup u.(z)

j.>

J

we conclude from Corollary 1.7 that

u

is H(K)-subharmonic. 11

6.9 Corollary.

Let

open subset of

MA

of

Cn

on

U

A

be a uniform algebra.

Let

that is homeomorphic to an open subset

in such a way that the functions in

A

are analytic

and give local coordinates at each point of

the functions on

be an

U

U

U

.

Then

that are locally subharmonic with

93

respect to

A

are the plurisubharmonic functions on

monic functions on Let

z e D

are plurisubharmonic on

D

Since

.

A

coordinate functions

A

by functions in closed ball ,

B

.

z

on which each of the

z

We can take the neighbourhood to be a z

.

If

is dense in

is plurisubharmonic

u

then by Bremermann's Theorem,

A

.

is approximable uniformly

z1,...,zn

centered at

B

monic, and since on

.

D

gives local coordinates at

there is a compact neighbourhood of

D

.

We have already observed that the locally A-subhar-

Proof.

on

U

H(B)

u

is H(B)-subhar-

,

u

is A-subharmonic

Hence plurisubharmonic functions are locally A-

subharmonic.

Note that we have used Bremermann's Theorem only for closed balls.

It would be of interest to find an elementary

proof of Bremermann's Theorem for this case, which does not depend on the characterization of domains of holomorphy in terms of pseudoconvexity.

Subharmonicity with Respect to H(K)

Fix a compact S-subset

K

of

Cn

We wish to combine

.

Bremermann's Theorem with the localization theorem of Chapter 5,,to obtain conditions under which a given function is

H(K)-subharmonic, that is, subharmonic with respect to the algebra

H(K)

6.10 Theorem.

functions on

We begin with the following.

.

Let K°

K be an SS-subset of

94

Then the

K°

In view of Corollary 6.9, it suffices to show that

a locally H(K)-subharmonic function on monic.

.

that are H(K)-subharmonic are precisely

the plurisubharmonic functions on

Proof.

Cn

K°

is H(K)-subhar-

This follows immediately from Corollary 5.11, once

K°

we show that Let

is H(K)-convex.

be a compact subset of

E

K°

Let

.

be a

{Dk}

sequence of domains of holomorphy that decreases to Ek

let

from Ek aDk

,

E

Then the distance

.

coincides with the distance from

aDk

to

E

to

by one of the characterizing properties of domains of

holomorphy.

aK

to

,

E c Ek , we find in the limit that the

Since

distance from E

O(Dk)-convex hull of

by the

K , and

E

and

coincides with the distance from

aK

to

is a compact subset of

E

K°

.

11

Let

6.11 Theorem.

point of

be an SS-subset of

K

0n

is a Jensen boundary point for

aK

If every

.

H(K)

then

,

the H(K)-subharmonic functions are precisely the upper semiK

continuous functions from

to

that are pluri-

[-co,+-)

K°

subharmonic on

This follows immediately from Theorem 6.10 and

Proof.

Theorem 5.14.

0

Now suppose that u

is a boundary point of

C

is an upper semi-continuous function defined on a neigh-

bourhood of

in

C

K .

Under what conditions is

locally H(K)-subharmonic function near condition is that B

K , and that

containing

C

u .

u

a

The most obvious

?

extend to be plurisubharmonic in a ball Then

is H(B)-subharmonic, and hence

u

H(B n K)-subharmonic.

Another condition is that that

u

be continuous near

harmonic on the part of cumstances hold, let

C

and that

,

near

K°

K

at

Suppose these cir-

.

C

.

v

B

centered at

C

be the

u(z) =

Then

is plurisubharmonic in a neighbourhood of

for some fixed small ball

aK

be plurisub-

u

be small, and let

e > 0

outer unit normal vector to u(z-ev)

be a smooth point of

C

.

B n K

Consequently

95

u

is locally H(K)-subharmonic on

e

verges uniformly to

u

also

,

B n K , by Theorem 5.14.

on

u

B n K .

Since

cone is locally H(K)-subharmonic u

This leads us to the following

theorem.

Let

6.12 Theorem. that

K

functions on K

such

Then the continuous H(K)-subharmonic

is smooth.

8K

Cn

be a compact Ss-subset of

K

are precisely the continuous functions on

that are plurisubharmonic on the interior of

K .

The previous considerations show that any continuous

Proof.

K

function on

that is plurisubharmonic on

By Theorem 5.9, such a function is H(K)-

H(K)-subharmonic.

The reverse implication is clear.

subharmonic.

Now let

is locally

K°

K be a compact Sa-set with smooth boundary.

It has already been seen that the strictly pseudoconvex

K

boundary points of

are peak points for

may also be peak points for

H(K)

pseudoconvex boundary points.

ball with the bulge, {1z112 + function

+ Iz21

Izll

- 1

H(K)

There

.

which are not strictly

Consider for instance the 1z2I4

C'2 <_

1}

,

in

The

.

is a defining function for

the boundary, and its complex Hessian is diagonal, with entries

z2 = 0}

,

4Iz212

and

1

.

On the circle

{1z11 = 1

the complex tangent space is spanned by

,

,

so

8z 2

that the Hessian matrix annihilates the complex tangent space, and points on the circle are not strictly pseudoconvex boundary points.

a peak point for (a,0)

, where

H(K)

However, each point of the circle is ,

and in fact

(l+az1)/2

peaks at

Al d= 1

One problem that is currently unsolved is the following. Suppose that

8K

is smooth, and that each point of

8K

with one exception, is a strictly pseudoconvex boundary

96

Is the exceptional point necessarily a peak point

point.

H(K)

for

It turns out that the exceptional point is

?

necessarily a Jensen boundary point for

H(K)

In fact,

.

much more is true.

6.13 Theorem.

K

Let

be a compact Se-subset of

smooth boundary, and let

T

with

0n

be the set of points on

3K

that are not strictly pseudoconvex boundary points.

every Jensen measure for a point of

ary point for

H(K)

is supported by

T

In particular, any isolated point of

Then T

is a Jensen bound-

T

.

The second assertion follows from the first and the

Proof.

localization theorem for the Jensen boundary.

(See the re-

marks after Theorem 5.15.) Let

E T

be a Jensen measure for

a

,

a strictly pseudoconvex boundary point for

and let K

By perturb-

.

ing a strictly pseudoconvex defining function for ,

we may obtain a compact set

and an open ball J\B = K\B

,

E E J°

,

,

such that

and each point of

pseudoconvex boundary point of

J

near

2K

with smooth boundary,

J

centered at

B

be

E

J D K ,

is a strictly

B n 8J

In these circumstances,

.

K.Diederich and J.E.Fornaess[3] have constructed a continuous plurisubharmonic exhaustion function for tinuous function on

J0

6.12, a

,

p

and

p

p

on

J

such that

Hence

is evidently supported on the set

ticular, the closed support of moreover Since

p = 0

is plurisubharmonic on

is H(K)-subharmonic.

J

J°

,

that is, a conon

8J

By Theorem

0 = p(C) < fpda {p =0) n K .

K°

does not belong to the closed support of

C E (8K)\T

is arbitrary,

a

is supported on

and

,

In par-

is disjoint from

a

p < 0

,

,

and

a

T

97

Bremermann Functions

Let

be an open subset of

D

ous function

B

B c D

,

from D

to

if each point of

D

tion on

u

,

such that

functions of the form

.

A lower semi-continu-

(-co,+-]

is a Bremermann func-

D

0n

is included in an open ball

is the upper envelope of all uIB f is an c loglfl , where c > 0 ,

analytic polynomial, and

c log1f1 < u

bounded, this occurs if an only if

on

8B

D

If

.

is

is a Bremermann func-

u

tion with respect to the uniform algebra spanned by the analytic polynomials on any ball or polydisc containing The locally log-envelope functions on

D

D

are defined

similarly, to be the lower semi-continuous functions from D

to

that are locally the upper envelopes of

(--,+-]

functions of the form

c log(fI

an analytic polynomial.

one sees that if D

A

, where

c > 0

and

is

f

As in the proof of Corollary 6.9,

is any uniform algebra that contains

as an open subset of its maximal ideal space, such that

the functions in

A

are analytic on

D

and give local co-

ordinates, then the locally log-envelope functions on

D

(as just defined) coincide with the locally log-envelope functions on

D

with respect to

functions on

D

(as defined above) coincide with the

Bremermann functions for

A

on

A ,

D

.

and the Bremermann

In particular, the

class of locally log-envelope functions and the class of Bremermann functions are invariant under analytic changes of variables.

Return now to the definition of Bremermann function. There are various ways to restate the condition on

The condition is simply that, uIB ution

v

uIB

coincides with the sol-

of the A-Dirichlet problem with boundary data

v = ulaB , where

A

is the algebra of analytic polynomials.

From the maximum principle (Theorem 5.19), we see that uIB

is also the upper envelope of all plurisubharmonic

functions

98

w

on

B

such that

w* <_ u

on

8B

.

The condition on

uIB

Let

polynomial hulls.

n+l

Y = {(z,C) e

can also be restated in terms of

z c

:

Then the condition

u = v

DB,

ICI < e-u(z) }

on

B

v = uIDB , holds

, where

if and only if the polynomial convex hull of

{(z,1) E

n+l (t

:

z e B,

Icl

<_ e-u(z) }

(6.9)

.

Y

is given by

(6.10)

.

This follows immediately from Theorem 5.4, since the polynomial convex hull of

Y

coincides with the maximal ideal

space of the uniform algebra on

generated by the poly-

Y

nomials.

In the case

n = 1

,

u = v , where

the condition

v =

, means simply that uIB is the Poisson integral of ulDB its boundary values on DB It follows that the Bremermann .

functions on a domain in the complex plane are simply the harmonic functions.

In particular, the Bremermann functions of one complex We will show presently that

variable are real analytic.

this result fails miserably for Bremermann functions of several complex variables.

Nevertheless, the following

characterization of the smooth Bremermann functions is available.

A smooth function

6.14 Theorem.

function if and only if complex Hessian matrix of

u

on

D

is a Bremermann

is plurisubharmonic, and the

u u

is singular at each point of

D .

Let

B

be an open ball with closure included in

D ,

and let

K

be the set defined by (6.7).

DK

lying over

Proof.

B

The piece of

is smooth, with defining function

99

p(z,0 = u(z) + log ICI

.

H

The complex Hessian matrix

of

can be expressed

p

p

simply in terms of that of

Hu(z)

0

0

0

u

:

Hp(z, Recall that the complex tangent space consists of vectors in

T

orthogonal to

Ctn+l

(z,c) E aK

at

(z'O

dp

Since

d p .

has a nonzero component in the c-direction, it is easy to see that the rank of

Hp(z,C)

the rank of

In particular,

Hu(z)

.

pseudoconvex point of

T(z

on

is a strictly

(z,C)

if and only if

BK

coincides with

O

Hu(z)

is positive

definite.

Now suppose that From the definition, Hu(z)

>_ 0

on

B

.

is a smooth Bremermann function.

u

is plurisubharmonic, so that

u

is the polynomial hull of the set Since the Shilov boundary of 8K

point of

3K , by Rossi's Theorem.

lying over

Since

u

Hu(z)

,

no

(This assertion is also

We conclude that

H (z) u

is

is a smooth plurisubharmonic

u

is singular at each point of

is plurisubharmonic,

spect to the algebra on nomials.

Y

.

Conversely, suppose that function, and that

is included in

can be a strictly pseudoconvex

B

easy to establish directly.) z c B

defined by (6.10)

defined by (6.9).

Y

P(K)

point of

singular at each

K

Moreover, the set

By Theorem 5.4,

B

is subharmonic with re-

u

generated by the analytic poly-

B

K

coincides with the maximal

ideal space of the algebra generated by the analytic polynomials on

K .

Now

Hu

is singular at each point

is singular, so that also (z,C) E 8K

lying over

Rossi's Theorem, the Shilov boundary of in the set

Y

defined by (6.9).

100

Y

,

and

.

By

is included

P(K)

It follows that

incides with the polynomial hull of

B

Hp(z,O

u

K is a

co-

1

Bremermann function.

The following theorem provides us with a fairly wide class of Bremermann functions.

Let

6.15 Theorem.

and let Ck

.

be an analytic map of

F

function on

D'

Then

function.

,

into a domain

w

woF

is also smooth and plurisubharmonic,

at each point of

k

in

D'

is a smooth plurisubharmonic

and the rank of the complex Hessian matrix of most

1 5 k < n

let

is a Bremermann function on D.

woF

then

,

Suppose first that

Proof.

D

Cn

is any locally bounded, locally log-envelope

w

If

be a domain in

D

D

.

woF

By Theorem 6.14,

is at

woF

is a

Bremermann function. For the general case, one expresses

locally as an

w

w

increasing limit of smooth plurisubharmonic functions Each

a

is a Bremermann function, and by Theorem 5.22,

w of a

the increasing limit of Bremermann functions is a Bremermann function, so that

woF

is a Bremermann function. 11

Now let Define

X

on

u

be a convex function of a real variable. 0n

by

u(z1,...,zn) = X(xl)

,

zI = XI +iyI

.

Regarded as a function of the complex variable subharmonic. ment,

u

choosing

,

X

is

By Theorem 6.15, or by a simple direct argu-

is a Bremermann function on X

zI

Cn

for

n ? 2

.

By

to be continuous but not smooth, we obtain in

this manner a continuous Bremermann function that is not smooth.

In like manner, we can find a Bremermann function

with any specified degree of smoothness, which fails to have

101

a higher degree of smoothness. Nothing could be easier than producing a discontinuous Define

Bremermann function. while

w = 1

C\{O}

on

w

Near

.

so that

on 0

0

Izl

u(zl,.... zn) = w(zI)

Hence

.

Bremermann function on

Cn

,

is the increasing

a

limit of the subharmonic functions to

w

,

w(O) = 0

,

as

decreases

a

is a discontinuous

n ? 2

One may construct more ill-behaved examples, for which the upper semi-continuous regularization is not continuous,

Let

as follows.

aJ +0

,

let

rapidly, let

e. +0

be large, define the subharmonic function

M > 0

v(z) = Ee

.

log

J

Iz-ai I

Then

and set

,

w

w = max(v,-M)

on

w

is lower semi-continuous, and

to be quasi-subharmonic, so that envelope function.

For

n ? 2

,

w

w(O) = -M

,

.

is easily seen

is a locally log-

the function

u(zl,...,zn)

is a Bremermann function that is discontinuous at

= w(z1)

zl = 0

C\{O}

and

,

is also discontinuous at

u*

zl = 0

.

Bremermann's Generalized Dirichlet Problem Let

D

be a bounded domain in

bounded, real-valued function on

Cn aD

,

.

and let

be a

h

The solution to the

generalized Dirichlet problem defined in Chapter 5 is the upper envelope tions

u

on

of the continuous plurisubharmonic func-

h D

satisfying

u* _< h

to consider also the upper envelope monic functions

u

on

D

on h

satisfying

upper semi-continuous regularization of harmonic function on

D

8D

.

It is natural

of all plurisubharu* s h h

on

8D

is a plurisub-

that is Bremermann's solution to

the generalized Dirichlet problem with boundary data Evidently incide.

h = h

h s h

.

Often the functions

h

and

h

h

.

co-

For instance, it follows from Theorem 5.25 that whenever

h

is lower semi-continuous and there is a

continuous plurisubharmonic barrier at each point of

Another condition guaranteeing that

102

The

.

h = h

8D

.

is given in

The abstract ana-

the following theorem of J.B.Walsh[10].

logue of Walsh`s theorem, Theorem 5.26, does not quite suffice to yield the Walsh theorem in the case at hand.

Let

6.16 Theorem. 8D

E

h

Then

Let

e > 0

D

lie in a (26)-neighbourhood of h(z)-h(w)I < c

then

and define of

v

on

y

.

DD

v

= h

y

.

z,w E D

Iz-wl < 6

such that

y E Cn

so that

D

and if

,

C

h = h

so that if

> 0

6

Fix

tends to

In particular,

.

Choose

.

z E D

as

is continuous on

Proof.

Suppose that for each

.

tends to

h(z)

,

h E CR(BD)

Jyl

,

< 6

on a 6-neighbourhood

8D , while

vy(z) = max{h(z),h(z+y) -e}

Since the term h(z)

elsewhere.

dominates when

the boundary of the 6-neighbourhood of

lies on

z

8D , we see that

v y

is plurisubharmonic.

v

Since

= h

near

8D

, while

h

y

tends to 5 h

v

h

at

D

on

DD

,

the definition of

Consequently

.

h

shows that

h(w) -s <_ h(z)

whenever

Y

z,w c D

satisfy

tinuous on

D

Iz-wl

< 6

.

It follows that

h

is con-

.

If we specialize to S6-sets with regular boundaries, we obtain the following.

Let

6.17 Theorem.

that every point of H(K) on

Let

.

8K

.

Then

continuous on

Proof.

u

K be a compact Se-subset of 8K

0n

such

is a Jensen boundary point for

be a bounded, lower semi-continuous function u = u = u 8K ,

then

on u

K°

.

Furthermore, if

is continuous on

From the definitions, we have

u

is

K .

u <_ u <_ u

on

K°

103

Let

be any plurisubharmonic function on

v

on

v* <_ u

3K°

Since every point of

.

boundary point, u = u and furthermore

K°

.

on

K°

.

Hence

.

is a Jensen

aK

on

v* <_ u

is a Bremermann function on

u

BK°

K°

by

,

By the maximum principle (Theorem 5.19),

Lemma 5.17. on

aK

on

such that

K°

Passing to the upper envelope, we obtain

v <-

u

u

u _<

This proves the first assertion of the theorem.

The second assertion now follows from Theorem 6.16, or from Theorem 5.26.

[1

Now let us focus on the open unit ball n ? 2

B

'n

in

for

,

The existence of discontinuous Bremermann functions

.

shows that

need not be continuous on

u

discontinuous on continuous on ties on

B

aB

BB ,

B

when

u

is

It turns out, though, that if

.

is

has certain regularity proper-

u

then

u

E.Bedford and B.A.Taylor[l] have shown in this

.

case that the first partial derivatives of

u

Lipschitz continuous on compact subsets of,

B

lar, the second partial derivatives of

u

exist and are In particu-

.

exist almost

everywhere with respect to volume measure.

On the other hand, the second partial derivatives of need not exist everywhere on analytic function on function

w

8B

B

,

even though

u

u

is a real-

To see this, consider first the

.

of one complex variable, defined by (4r2-1)2

r >_ 1/2

,

w(rei6) = 0

Then > 0

w

is continuous.

for

r > 1/2

,

w

u(z) = (4z1z1 - 1)2 Since

u ? 0

,

also

for fixed values of

104

551/2 .

,

Since

w ? 0 , while

Aw = 32[8r2-1]

is seen to be subharmonic.

zE

,

u ? 0 zl

,

.

Define

aB

The maximum principle, applied

shows that

u(z)

<_

(41z112 -1)2

u(z) = 0

Since

c B

1z11 = 1/2

for

Hence

z11 5 1/2

for

z

,

,

u(z1,O,...,O) = 0

also

u(z1,0,...,0) <_ w(zl)

is a subsolution corresponding to

w(z1)

u(z1,O,...,O) = w(z1)

In particular,

Iz11 s 1

,

u

,

and since

we obtain

.

is not twice differentiable.

u

,

This example

is taken from [5].

We remark in closing that the Bremermann functions can be thought of as generalized solutions to a special case of the These equations are studied

complex Monge-Ampere equations.

by E.Bedford and B.A.Taylor in [1], where a minimum principle is obtained that overlaps with the maximum principle of The Dirichlet problem associated with the com-

Theorem 5.19.

plex Monge-Ampbre equations is to find a plurisubharmonic function

u

on

ary values on

such that

D 8D

, while

takes on prescribed bound-

u

det(H (z)) u

is prescribed on

D

The Bremermann functions are those solutions for which det(H (z)) = 0 U

on

D

References 1.

Bedford, E. and Taylor, B.A.

The Dirichlet problem for

a complex Monge-Ampbre equation, Inventiones Math. 37 (1976), 1-44. 2.

Bremermann, H.

On a generalized Dirichlet problem for

plurisubharmonic functions and pseudo-convex domains. Characterization of Shilov boundaries, Trans. A.M.S. 91 (1959), 246-276. 3.

Diederich, K. and Fornaess, J.E., Pseudoconvex domains: bounded strictly plurisubharmonic exhaustion functions, Inventiones Math. 39 (1977), 129-141.

105

4.

Local complex foliation of real submani-

Freeman, M.

folds, Math. Annalen 209 (1974), 1-30. 5.

Gamelin, T.W. and Sibony, N.

Subharmonicity for uniform

algebras, to appear. 6.

Hakim, M. and Sibony, N. de

A(D)

Frontibre de Silov et spectre

pour des domaines faiblement pseudoconvexes,

C. R. Acad. Sc. Paris 281 (1975), 959-962. 7.

Hormander, L.

Complex Analysis in Several Variables,

North Holland/American Elsevier, New York, 1973. 8.

Rossi, H.

Holomorphically convex sets in several com-

plex variables, Ann. Math. 74 (1961), 470-493. 9.

Vladimirov, V.S.

Methods of the Theory of Functions of

Several complex variables, M.I.T. Press, Cambridge, Mass., 1966. 10.

Continuity of envelopes of plurisubharmonic

Walsh, J.B.

functions, J. Math. Mech. 18 (1968), 143-148.

Added in proof:

In the development of the notion of sub-

harmonicity with respect to an algebra of functions, the following reference also plays a key role. 11.

Rickart, C.E.

Plurisubharmonic functions and convexity

properties for general function algebras, Trans. A.M.S. 169 (1972), 1-24.

106

7 The conjugation operation for representing measures

There are a number of classical inequalities involving a trigonometric polynomial *u

and its conjugate polynomial

u

The prototypal result, due to M.Riesz, asserts that

.

there exist constants

such that

c

p

*u(eie)IPde

cP

<_

1

Iu(ei8)IPdO

1
,

J

Another important inequality is due to A.Zygmund:

I*uldO

Jul log+IuldO + B.

A l

(7.2)

J

Both of these inequalities can be expressed in the form

J H(u,*u)dO ? 0

where

(7.3)

,

is a real-valued function on

H

C

Our aim is to

.

study inequalities of the form (7.3), in a function algebra We begin by introducing the conjugation operator.

setting.

The Conjugation Operator Let Let

be a uniform algebra on

A

X ,

be a representing measure on

a

u e Re(A)

u+ i*u e A ement of

,

let

*u

Re(A)

0 ,

then

(*u-v) 2($) = J

X

for

denote an element of

and

.

If

v

e MA

and let

Re(A)

.

For

such that

is another such el-

*u -v e A , and

(*u-v)2da

,

107

v = *u a.e.

so that

mined uniquely by

(do)

It follows that

.

and

u

is deter-

at least on a set of full

,

measure for each representing measure for is the conjugate function of

*u

*u

u

,

.

The function u ;*u

and the operator

is the conjugation operator. vanishes at

u +i*u -u(4)

Since

,

we have

0 = Re J (u+i*u-u(0)2do u2do -

2

(*u)2 - J

do

J

we are led to the identity

Judo =

Substituting

(*u)2do + uW 2 =

I

u2do

(7.4)

.

J

In particular,

J

1u12

2 (*u) do <_

do

so that the conjugation operator is bounded with respect to the norm of

L2(a)

.

More generally, the M.Riesz inequality is valid in this general setting, providing

7.1 Theorem. let

p

Let

a

p

is an even integer.

be a representing measure for

be an even integer.

Then there is a constant

and

,

c

P

such that

I*ulPdo

cP 1 Iul da

,

u c Re(A)

.

(7.5)

J

Proof.

Write

p = 2m , where

m

is a positive integer.

The polynomial s2m -2(-l)mRe(l +is) 2m has leading term 2m Hence the polynomial is bounded above on the real _s .

108

axis, say

s2m

- 2(-l)mRe(l +is) 2m

Substituting

y2m <_

Let

<

cx2m + 2(-l)mRe(x +iy) 2m , .

Substituting

and integrating with respect to

If

(*u) 2mdo <_ c f u2mdo

m

m

.

s = y/x , we are led to the estimate

u E Re(A)

f

-- < s < -

c ,

x,y E 7R and

u a

.

for

*u

x

and

y

we obtain

,

+ 2 (-l)mRe j (u + i*u) 2mda

.

(7.6)

is odd, we obtain the estimate (7.5) immediately.

If

is even, we have 2m

(u+ i*u)2mdo =

udo\l

u2mdo

<

.

(7.7)

Substituting (7.7) into (7.6), we obtain the estimate (7.5). 11

In the case that

is the disc algebra, one can invoke

A

the M.Riesz Interpolation Theorem, to conclude that the estimate (7.1) is valid for

2

<_ p <

timate (7.1) is also valid for

2

,

and the M.Riesz

Moreover, it is easy to obtain bounds

Theorem is proved. for the constants

1 < p

By duality, the es-

appearing in (7.1).

c

p

Cole's Theorem It turns out that the M.Riesz Theorem simply fails in the general setting, unless

p

is an even integer.

That will

be a consequence of the following theorem, due to B.Cole.

7.2 Theorem.

Let

D

be a domain in the complex plane, let

109

z0 E D

on

D

H

and let

,

be a continuous real-valued function

Then the following are equivalent:

.

There exists an analytic function

(i)

Re (F)

on

F

such that

D

<_ H

and

F(z0) ? 0

A

If

(ii)

.

is a uniform algebra,

representing measure for

J

and

,

a

is a

then

(Hof)da ? 0

Proof.

f(MA) c D

Suppose (i) is valid. Since

(ii).

satisfying

f E A

for all

Let

f(q) = z0

and

A,4,a and

be as in

f

is analytic in a neighbourhood of

F

A

belongs to

Fof

,

E MA

f(MA)

and

(Fof)da =

F(z0)

>_ 0

.

J

Since

J

(Hof)da

(Fof)do <_

,

J

we obtain (ii).

Observe that the proof of the M.Riesz The-

orem given earlier is essentially a special case of this proof, in which

F

is the polynomial

Suppose now that (ii) is valid.

gebra

110

,

and

R(Kn)

uKn = D

.

{Kn}n_1

Let

creasing sequence of compact subsets of z0 E K0

2(-l)mz2m

.

be an in-

such that

D

Applying the hypothesis to the al-

and the coordinate function

z

, we find that

inf{J Hda

represents

a

:

R(K)} ? 0

on

z0

.

From Theorem 2.1 we conclude that the supremum of over all negative.

Hence there is a function

K

neighbourhood of > -1/n

fn(z0

n - -

as

Re(f) < H

satisfying

f c R(Kn)

.

Re(f

such that

n

Any limit

F

f

n

on

Kn , is non-

analytic in a

n

< H

)

Re(f(z0))

K

on

n

,

and {fn}

of the normal family

has the properties listed in (i). 11

7.3 Corollary.

on the plane.

Let

H

be a continuous real-valued function

Then the following are equivalent:

There is an entire function

(i)

Re(F)

on

F

C

such that

<_ H

and

F(0) (ii)

If

>_ 0 .

A

is a uniform algebra,

representing measure for

,

cp

e MA , and

is a

a

then

J H(u,*u)da ? 0

for all

u e Re(A)

satisfying

0

The M.Riesz Estimate

As a first application, we show that the version of the M.Riesz Theorem given in Theorem 7.1 fails in the case that p

is not an even integer.

7.4 Theorem.

even integer.

Suppose that

1

<_ p <_ -

,

and

p

Then there is a uniform algebra

is not an A

,

and a

ill

representing measure

u - *u

gation operator

LP(a)

c E MA ,

for

a

such that the conju-

is not bounded in the norm of

.

Suppose that there is a constant

Proof.

f*ulpdo <_ c

for all

c > 0

such that

updo

J

(7.8)

satisfying

u E Re(A)

0

.

According to

Corollary 7.3, there exists an entire function

F

Re(F) <_ clxlp - lylp

such that

(7.9)

and

F(0) ? 0

For fixed

(7.10)

.

t > 0

the function

,

Ft(z) = F(tz)/tp also satisfies (7.9) and (7.10). form a normal family.

F(z) = a0 + a z + a I

t

Ft

is

amtm p

2

2

z

+ ...

zm

.

Since

F

conclude that

F(z) = azp

p

.

,

in the power series expansion

By normality, these remain bounded as

ranges over the interval

m = p

112

.

Ft's

If

then the coefficient of of

Consequently the

(0,o)

.

Hence

a

m

= 0

unless

cannot be identically zero, by (7.9), we is an integer, and

The estimate (7.9) on the imaginary axis becomes

Re(aip)yp < -IYIp

-
,

For this to hold, the integer Now suppose that

c > 0

on a compact space

Ac

Xc

for a homomorphism c c MA

ac

must be even.

p

is not an even integer.

p

shown that for each constant algebra

.

We have

,

there exists a uniform

,

a representing measure

and a function

,

u

E Re(A) c

c

such that

J

Let

I*u

c

Ipdo

> c

c

c

X = II{Xc : c > 0} X

onto

and let

,

c

on

$ E MA *U

= *u o7r c

c

J

I*U

c

.

c

be the uniform algebra on goTrc

,

g E Ac

where

gener-

X

c > 0

and

.

is multiplicative

ac

Uc = uco7c

belongs to

Re(A)

,

and

The estimate (7.11) shows that

Ipdo > c

IUcIpdo

.

J

Hence the conjugation operator is not bounded on the norm of

X

is a representing measure for some

a

Furthermore

.

be the projection of

c

of the measures

a

A , so that

(7.11)

.

c

let 7r

,

A

ated by the functions Then the product

Ipdo

Iu

J

Re(A)

in

,

Lp(a).

As a generalization of the preceding theorem, we prove the following.

7.5 Theorem.

Suppose

0 < p <-

and

0 < r < -

.

The con-

jugation operator u - *u is continuous, from the Lp-norm r_norm, for all uniform algebras A , all to the L E MA and all representing measures there is an even integer

2m

a

for

such that

¢

,

if and only if

r

<_

2m <_ p

.

113

Proof.

If

<_ 2m <_ p

r

operator from

LP

the continuity of the conjugation

,

follows from Theorem 7.1 and

Lr

to

Holder's inequality.

Assume conversely that the conjugation operator is continuous.

We consider only the case

that there exists

((

IJ

I*ulyda)

We may assume A ,

and o

,

1/p <

b

/r

.

such that for all

c > 0

11/r

I*ulydol

<_ p

r

(c

J

Iulpdo\)

,

u E Re (A)

r

< c

Iulpdo

u E Re(A)

,

J

is convex, so that the tangent line to the

The function

tb

graph of

lies beneath the graph:

tb

-co < t <

b t + 1 -b <_ tb

co

This leads to the estimate

I*ulydo+l -b < c

b

J

Iulpdo

,

u c Re(A)

1

By Corollary 7.3, there exists an entire function that

F(O) ? 0

and

Re(F(z)) <_ cIxIp+b - l -blylr 0 < t < -

For

the entire function

fies

Re(Ft(z)) < clxlp + b/tp

and

114

such

F

z = x+iy E C Ft(z) = F(tz)/tp

.

(7.12) satis-

Ft(0) ? 0

.

It follows that the

is the power series expansion of

F(z) = Eakzk

If

the coefficient of aktk p

is

form a normal family as

Ft's

zk

ak = 0

k > p

for

Replacing

by

F

coefficients of

F

[F(z) + F(-z)]/2

asymptotic to

7.6 Corollary. r

.

1

If

r

as

yJ

<_ 2m

0 < p < 1

,

a

.

fails for some

7.5.

J

by

F

is an even poly-

F

is an even integer

F

-- + .

F(iy)

is

Comparing this

then the Kolmogoroff estimate

1/p c

J luldo

u E Re(A)

,

and some representing

Furthermore, the Zygmund estimate

l*uldo <_ a + SJ Jul log+lulda

Proof.

we can assume that the

.

fails for some uniform algebra A

J

,

F

Along the imaginary axis

(-1)ma2my2m

I*ulpd ol

measure

and

In particular, the degree of 2m <_ p

is a polynomial

,

we can assume that

,

t -> +-.

Furthermore, replacing

are real.

Ft

.

[F(z) + F(z)]/2

with (7.12), we obtain

CJ

p

.

then

,

in the power series expansion of

whose degree does not exceed

2m , and

F

By normality, these remain bounded as

.

It follows that

nomial.

t - +-

A

and

a

,

u E Re(A)

.

The first statement follows immediately from Theorem Since

Jul log+lulda s I Jul3/2do

the Zygmund estimate is valid only when the conjugation operator is continuous from the

L3/2-norm

to the

L1-norm.

115

This also fails in general, by the preceding theorem. 11

If we restrict our attention to positive functions in Re(A)

,

then the M.Riesz Theorem remains valid, at least in

the case that

is not an odd integer.

p

Let

7.7 Theorem.

A

be a uniform algebra, and let

a representing measure for some 1

< p < -

and that

,

is a constant

E MA

.

Suppose that

is not an odd integer.

p

be

a

Then there

such that

c

p

I*ulpdo < c

I

(

p

u c Re(A)

for all

constants

Iulpdo satisfying +-

tend to

c

as

u > 0 p

.

The best possible

tends to an odd integer.

p

Choose

Proof.

y > 0

and

a

real such that

0<6<_7T/2

a cos(p@) -< ylcos Blp -Isin Blp

.

(7.13)

To see that this choice is possible, observe that cos(p ¶72) a 0 a

,

,

since

is not an odd integer.

p

Hence we can choose

possibly negative, so that (7.13) is valid near

and then we can choose

y > 0

B =fr/2,

so large that the estimate is

valid on the remainder of the interval.

principal branch of the function

azp

,

In terms of the the estimate (7.13)

becomes

Re(azp) <- iIxIP -lyIP

where

z = x +iy

,

belongs to the right half-plane

(7.14)

{x > 0)

Now there are two cases to consider. If

116

a > 0

,

then

azp

is positive on the positive real

Applying Theorem 7.2, with

axis.

z0 = Jude

and

0 < Y J

we obtain

,

lulpdo

- J

u c Re(A)

for all

the right half-plane

D

I*ulpdo

u > 0

such that

.

A slightly better

estimate P

a

( udo/ l

<- Y

J

*P0

lulpdo J

J

is obtained by substituting and integrating.

x = u

and

y = *u

in (7.14)

In any event, we obtain the required esti-

mate in the case at hand.

Suppose on the other hand that

a < 0

.

We will use the

estimate

px - p + l

(7.15)

<_ xp ,

which reflects the fact that the tangent line to the graph of

xp

at

x = 1

lies beneath the graph.

From (7.14) and

(7.15) we see that the function

F(z) = azp - a(pz -p + l)

satisfies Re F(z) s (y-a)lxlp -lylP

in the right half-plane.

Note also that

ing Theorem 7.2, with

the right half-plane and

D

F(l) = 0

.

Applyz0 = 1

we obtain

0 s (Y-a)

J

lulpdo - J

l*ulPda

117

for all

u > 0

satisfying

u E Re(A)

fuda = 1

and

.

Since

the inequality is homogeneous, we can drop the condition Judo = 1

,

and we obtain the desired estimate.

It remains to prove the final assertion of the theorem.

We do this by means of an example, in which resenting measure for

A(A)

with respect to the disc algebra

0

.

Fix an odd integer tion

is a rep-

a

h

on

to

h

h(rei6) = C

where

,

n ? 1

Define a func-

.

by

DA

Me i6) = l ei6 We extend

2n+l

n

-11

2n =

C

n

C

d

k=-n

k

eik6

to be a harmonic polynomial, by setting

dkrIkj eik6

G-n

Fix

0 < a < 1

,

and let

measure on the circle resents

0

.

Define

g(aei6) = h(1 ei6)

a

We claim that polynomials.

n

9A g

be the normalized arc-length = {jzj =a}

a

on

aA a

,

so that

n

rep-

by

.

is orthogonal to all harmonic

gn - hd6/2Tr

Indeed, if

f = Eckrkleik6

is a harmonic poly-

nomial, then

f(ei0)h(ei6) 26 =

2

I

J

c.d-j and this coincides with

J

118

f(aei6)g(ae16)dn =

k

a-Ikleik6)d6

Now choose

Ma

1 -eg

so that

c > 0

is strictly positive on

and define

,

o = (1 - eg) n + ch 2e = n + E (h Ze - gn) h ? 0

Since

on

aA

representing measure for F6 E A(A)

,

0

u6 > 0

for

6 > 0

so that

,

2n+ 1

then

c

.

Iu6IPda

Iv6IPda /

>

converges in

F6

is a

by

,

J ,

a

.

F6 = (l+z)/(l+6-z) = u6+iv6 Then

Since

is orthogonal to analytic polynomials,

gn - hde/2Tr

Define

is a positive measure.

a

,

.

.

If

p <

J

LP(a)

as

6

decreases to

to

0 ,

F = (l+z)/(1-z) = u + iv

.

Here we have used the dominated convergence theorem, and the fact that ? I

c

has a zero of order

h

v1Pda / I Iulpda

,

2n

at

z = 1

.

Hence

and one computes, noting that

P

u = 0

on

2A

,

that this latter quantity is on the order of

2n p I

do + J Iv(aeie) IpEl - eg(aeie) )]do

11 - eie l

1

f

This tends to

Iu(aeie)IP[1-Eg(aeie)]de +-

as

p

increases to

2n+ 1

.

11

Estimates of Zygmund While the Zygmund estimate fails in general, it is valid for functions with positive real part.

7.8 Theorem.

For each

a > 2/7

,

there exists

S > 0

such

119

that

J

S+ Y

I*ulda

u log+u do J

for all uniform algebras and all

u e Re(A)

Let

Proof.

6

< 8

A , all representing measures

such that

satisfy

6

u > 0

.

> 2/7

.

o

The estimate

Isin 81 < 6 cos 8 log(cos 8) + 60 sin8

is valid near

6 = tir/2

.

Consequently we can choose

a > 0

so large that

I sin 0 I < 6 cos 8 log(cos 0) + 68 sin 0 + a cos8 ,

<-

18 I

9r/2

.

This leads to the estimate

Re (6z log z -a z) = dr cos 8 log r - 6r0 sin 8 - ar cos 8 < 6r cos 0 log r + 6r cose log(cos 8) - r I sin 8

= 6xlogx - IyI z = x +iy

for

y = *u

,

and

x > 0

J

x = u

and

and integrating, we obtain

t log t ? -1

for

J

t > 0

u log u do -

,

e > 0

,

there exists

J

I*ulda

we obtain

I*uldo < 6 + 6 I u log udo + au(k)

For any

120

Substituting

.

6u(b) log uO) - au(k) < 6

Since

,

c > 0

.

such that

(7.16)

uW = J

e+cfu

udo <_

log+u do

.

Combining this estimate with (7.16), and choosing S + aE = S

,

so that

E

we obtain the Zygmund inequality. 0

There is a converse estimate to Theorem 7.8, namely, that

u log u do <

I

2

I

J

log

I*ulda + 2

(7.17)

J

whenever

o

represents

u E Re(A)

and

q

is positive.

To

prove this, observe that

r cos 0 log r + r0 sin 0

Re(-z log z)

- x log x + 0y + r cos 0 log(cos 0)

<_-xlogx+2 IYl Substituting

z = u +i*u

and integrating, we obtain

-UM log uW s- J u log u do + 2 J which is the same as (7.17).

i*ulda

,

t log t ? -1/e

Since

(7.17)

,

yields the estimate

u log+u da < 2

l*ulda + 1

satisfying

u E Re(A)

for all

1

u > 0

and

1

.

There is another estimate, that is also due to Zygmund.

7.9 Theorem.

Let

A

be a uniform algebra, and let

representing measure for lu l

<_

1

,

E MA

and if 0 <_ a < 7/2

,

.

If

u E Re(A)

a

be a

satisfies

then 121

ea* u do <_ 2/cosa

.

1

Proof.

In this case, we take

{-l < Re(z) < +1}

F(z) = 2 -

Note that

,

D

to be the vertical strip

and we consider

eiaz - e-iaz =

F(x) > 0

2(l - cos(az))

-1 < x < 1

for

.

.

The estimates

e±ay cos a s e±ay cos(ax) = Re(e+iaz)

combine to yield

ealyl cos a s (e ay + e-ay)cos a <_

Re(eiaz + e-iaz)

Hence

Re(F(z))

<_

2 - ealyl cos a

-1 < Re(z) < 1

,

This leads to the estimate

0 5 2 - cos a

whenever

uldo

ea

u E Re(A)

satisfies

-1 < u < 1

.

n

The Kolmogoroff Estimate The Kolmogoroff estimate is easiest of all.

7.10 Theorem.

l*ulPda < J

0 < p < 1

If

1

cos( 2 )

(Uda)

for all uniform algebras

122

,

I

then

( 7.18)

A , all representing measures

a

u > 0

satisfying

u E Re(A)

and all

The estimates

Proof.

cos(t) lylp

cos(P2)rp <_ cos(pO)rp = Re(zp)

yield n Re(1-z ) s 1-cos(2) lyl

Applying Theorem 7.2, with

0 <_

1 -cos(t) J u E Re(A)

whenever

F(z) = 1 -zp , we obtain

l*ulpdo u > 0

satisfies

yields (7.18) in the case

Judo = 1

.

(7.18) is hcmogeneous, the restriction

and

Judo = 1

This

.

Since the inequality is un-

Judo = 1

necessary.

Weak-type Estimates Next we treat weak-type estimates, showing first that they fail in general, then that they obtain providing Re(A)

u E

is positive.

7.11 Theorem.

Let

0 < a < 1

> 0

,

and

A > 0

.

Then

the estimate

a({l*ul > A))

<_ a + S

J

luldo

fails for some uniform algebra measure

Proof.

a

A

(7.19)

and some representing

.

Suppose (7.19) is valid for all

A

Corollary 7.3, there is an entire function

and F

a

.

By

such that

123

F(0) > 0

,

and

(7.20)

Re(F(z)) 5 a + SIxI - X

where

is the characteristic function of the set

X

IIm(z)I > X}

{z

:

(Note that Corollary 7.3 is valid also for

.

upper semi-continuous functions.

This can be seen by ap-

proximating such a function from above by continuous functions.)

For fixed

t > 0

,

the entire function

Ft(z) = F(tz)/t

satisfies

Re(Ft(z) <_ t +

Again the family

{Ft

:

of the Taylor coefficients of that

is linear, say

F

stituting

z = iy

a -1 < 0

,

Ft

as

tends to

t

F(z) = a + bz

The boundedness

where

,

a ? 0

shows

Sub-

.

into (7.20), we obtain

a+yRe(ib) _
is normal.

t ? 1}

,

IyH

>x

.

this is impossible. D

7.12 Theorem.

If

A

is a uniform algebra, and

representing measure for

for all

Proof.

u E Re(A)

,

satisfying

A > 0

u > 0

,

.

Consider the harmonic function

U(z) = 1 + I arg(z - iA) - 1 arg(z + iA) 7T

124

is a

then

A ,

of{I*uI ? a}) < y J udo

a

Tr

(7.21)

on the right half-plane. [0,1]

Its range lies in the interval

it is zero on the interval

,

on the intervals

(-ice,-iA)

and

,

(ia,i°)

.

and it is

1

Furthermore,

on the right half-plane, and

U >_ X/2

U(1) = 1 - 2 arc tan A <- 2/(71) 7T

Hence the function

F(z) = a - 1 + - log(z + ia) satisfies

Re F < 4x and

X

F(l) ? 0

0 <_

a

whenever

I

.

We conclude that

(7.22)

udo - of{I*ul > a})

u E Re(A)

satisfies

u > 0

and

Judo = 1

.

The

general case (7.21) is obtained from (7.22) by replacing u+ i*u

by

(u +i*u)/t

and

A

by

Alt , where

t = Judo

0

Notes on Sources We close with some comments on the origin of the material in this section.

The approach that we have followed is that laid out by B.Cole in a colloquium talk at Tulane in December, 1970.

In

particular, Cole obtained Theorem 7.2 and used it to show that many of the classical estimates fail for general representing measures.

The story will be continued in the next

lecture, which deals with Jensen measures. The M.Riesz inequality was announced in 1924 [9]. According

125

to Riesz, he prepared the details for publication in that

year, but then he delayed submission of the manuscript for two years, so that the proofs appeared only in 1927 [11]. The 1927 paper of Riesz is a classic.

We mention several of

the highlights.

First Riesz obtains the estimate (7.1) for even integers,

with a proof along roughly the same lines as the proof of Theorem 7.1.

(Our proof of Theorem 7.1 is Cole's simplifi-

cation of Riesz's proof.) (7.1) in case

p

Riesz goes on to give a proof of

is not an odd integer, that is based on

contour integration.

He then handles the exceptional cases

by duality.

Riesz returns to the proof covering even integers in order to estimate the constants as

m -> +°

.

cp

,

and he finds that

c2m = 0(m)

He remarks that it would be interesting to de-

termine how the best possible constant depends on

p

,

and

in a footnote "added in proof", he cites his paper [10] of 1926, in which he obtains his celebrated convexity theorem.

Riesz did not use the convexity theorem to obtain his estimates on conjugate harmonic functions, but rather he was apparently led to the convexity theorem in seeking to understand his estimates.

The idea of basing the proof of the M.Riesz Theorem on the estimate (7.13) is due to A.P.Calder6n[3]. It was Bochner'2] who observed that Riesz's proof for the case of even integers extends to a uniform algebra setting.

Except for the context, Bochner's proof is identical to that of Riesz.

Bochner was apparently unaware of this, and he

omits reference to the Riesz paper.

An example in which the M.Riesz estimate of Theorem 7.7 fails, in the case

p = 3

,

was given by K.Yabuta[12].

The

example we have given, covering all odd integers, is due to H.Kbnig[6].

The example sheds light on the failure at pre-

cisely the odd integers of the complex-variable technique of

126

M.Riesz, in proving his classical estimate. The estimate of Theorem 7.8 is due to Zygmund[13].

A

proof based on contour integration was obtained by J.E.

Littlewood[8], and the proof given here is due to Calder6n It was apparently M.Riesz who observed (cf. [15, vol.

[3].

I, p.381]) that the converse of the Zygmund estimate is

valid, so that in particular if LI(de)

,

and if

u > 0

,

then

u

*u

and

u log+u

E

belong to

L1(de)

.

The estimate of Theorem 7.9 is also due to Zygmund[14]. The weak-type estimate of Theorem 7.12 and the estimate of Theorem 7.10 are due to A.N.Kolmogoroff[5].

In [5], Kol-

mogoroff first obtained the weak-type estimate, and he deduced from this the boundedness of the conjugation operator

from L1(dO)

to

Lp(dO)

,

0 < p < 1

.

Littlewood[7] gave

a proof of the Kolmogoroff estimate using complex variable techniques, and this proof was simplified by G.H.Hardy[4] to the now standard proof.

Strictly speaking, Kolmogoroff's weak-type estimate precedes the other estimates we have considered.

His results

were submitted for publication early in 1923.

It should be

noted though that A.Besicovitch[l] had already obtained a weak-type estimate for the Hilbert transform.

References 1.

Besicovitch, A.

Sur la nature des fonctions a carr6

sommable mesurables, Fund. Math. 4 (1923), 172-195. 2.

Bochner, S.

Generalized conjugate and analytic func-

tions without expansions, Proc. Nat. Acad. Sci. 44 (1959), 855-857. 3.

Calderon, A.P.

On theorems of M.Riesz and Zygmund,

Proc. A.M.S. 1 (1950), 533-535. 4.

Hardy, G.H.

Remarks on three recent notes in the Jour-

nal, J. London Math. Soc. 3 (1928), 166-169. 5.

Kolmogoroff, A.N.

Sur les fonctions harmoniques

127

conjugees et les series de Fourier, Fund. Math. 7 (1925), 23-28. 6.

Kbnig, H.

On the Marcel Riesz estimation for conjugate

functions in the abstract Hardy theory, Commentations Math. (1978). 7.

Littlewood, J.E.

On a theorem of Kolmogoroff, J. London

Math. Soc. 1 (1926), 229-231. 8.

Littlewood, J.E.

On a theorem of Zygmund, J. London

Math. Soc. 4 (1929), 305-307. 9.

Riesz, M.

Les fonctions conjugees et les series de

Fourier, C. R. Acad. Sci. Paris 178 (1924), 1464-1467. 10.

Riesz, M.

Sur les maxima des formes bilineaires et sur

les fonctionnelles lineaires, Acta Math. 49 (1926), 456-497. 11.

Riesz, M.

Sur les fonctions conjugees, Math.Zeitschrift

27 (1927), 218-244. 12.

Yabuta, K.

M.Riesz's theorem in the abstract Hardy

space theory, Arch. Math. 29 (1977), 308-312. 13.

Zygmund, A.

Sur les fonctions conjugees, Fund. Math. 13

(1929), 284-303. 14.

Zygmund, A.

Trigonometric Series, 2nd ed., Cambridge

University Press, 1968.

128

8 The conjugation operation for Jensen measures

While the M.Riesz and Zygmund estimates fail in general, they turn out to be valid for Jensen measures, and the constants are the same as those that arise in the case of the This is a consequence of the implication

disc algebra.

"(iii) implies (i)" of Theorem 8.3, which is due to B.Cole.

Before proving Cole's theorem, we present yet another proof of the M.Riesz Theorem, which will serve to illustrate the underlying idea.

The M.Riesz Estimate for Jensen Measures Let us first consider the classical case.

Fix

1 < p < 2

on the right half-plane by

h

r sin 0) = rp cos(p6)

h(r cos A ,

and extend

Define

.

h

to

so that

C

181

,

<_

Tr/2

,

is symmetric about the

h

imaginary axis:

h(x,y) = h(-x,y)

Note that

h

,

x +iy e C

.

is continuous, and that

h

is harmonic except

We claim that

h

is subharmonic.

on the imaginary axis.

Consider first the behaviour of imaginary axis

ae

{6 = 7r/2}

.

h

near the positive

Since

rp cos(p6) = -prp sin(p6)

has negative sign for

0 = Tr/2

,

the values of

rp cos(pe)

are less than those of the reflected function rp cos[p(7-0)]

129

+E , while

< 8 <

for

2 rp cos(p8)

rp cos[p(r-e)]

2-E < 8 < 2

for

It follows that

.

maximum of the harmonic functions [p(7-0)]

is dominated by

2 rp cos(p8)

near the positive imaginary axis.

is subharmonic there.

h

Similarly,

the negative imaginary axis.

is the

h

and

rp cos

Consequently

h

is subharmonic near

To check that

satisfies

h

the mean value estimate at the origin, we simply compute (r/2

r f

-r

h(r cos 8, r sin 8)d8 = 2rp

-r/2

cos(p8)d8

> 0 = h(0,0)

Hence

h

.

is subharmonic.

As in the preceding chapter, we choose

a > 0

and

y > 0

such that

a cos(p8) <- ylcos SIp - Isin 8Ip

0

In terms of the subharmonic function

h ,

<-

.

2

(8.1)

the inequality be-

comes

ah(x,y) <_ ylxlp - lylp

is a trigonometric polynomial, then

u

If

(8.2)

h(u,*u)

is sub-

harmonic, so that

Iu(0)Ip = h(u(0),0) s J h(u,*u) ?e

Substituting

x = u

and

y = *u

(8.3)

in (8.2), and integrating,

we obtain a

r

h(u,*u)

8

y

r

(uIp dl

-

I

I*uIp Ze

Now (8.3) and (8.4) lead to the estimate

130

(8.4)

*uIp LA + aIu(0)Ip <_

J

Iulp d9

(8.5)

Y 1

which proves in particular the M.Riesz Theorem, for 1< p < 2

Handling the range

by duality, we obtain what

2

is currently the "best" proof of the classical M.Riesz Theorem. 1

For the range

the above proof remains valid when arbitrary Jensen measure.

is replaced by an

dO/2ir

In fact, the only step of the

proof that does not extend immediately to arbitrary representfh(u,*u)da

ing measures is the estimate

.

In

the case of Jensen measures, this estimate is a simple consequence of the results of Chapter 3.

We state the result sep-

arately, for emphasis.

8.1 Lemma.

If

J

for all

is a Jensen measure for

a

hof do

`f E A

and all functions

a neighbourhood of the range

Proof.

Let

then

e MA ,

f*a

h

f(MA)

that are subharmonic in of

MA

on

f

be the probability measure on

.

f(MA)

de-

fined so that

J wd(f*a) = J wof do

If

g

gof

.

is a rational function with poles off belongs to

f*a

the algebra

f(MA)

,

then

A , so that

J

Hence

w e C(f(MA))

,

loglgofldo =

I

is a Jensen measure for R(f(MA))

.

loglgld(f*a)

f(¢)

with respect to

The lemma now follows from

131

Theorem 3.4, applied to

v = f*a

p =

and

11

Thus we see that subharmonic functions give rise to estimates for integrals, and the particular subharmonic function used above gives rise to the estimate

J

y

I*ulpda +

where

Iulpda

u E Re(A)

,

is any Jensen measure for

a

E MA

,

Cole's theorem

.

will tell us in addition that all integral estimates arise in the above manner from subharmonic functions.

Cole's Theorem The proof of Cole's theorem depends on the following lemma from potential theory.

Let

8.2 Lemma.

K

be a compact subset of H

is connected, and let

C\K

function on

Let

K .

h

such that

Q

be a continuous, real-valued

be the upper envelope of the family

of functions that are subharmonic in a neighbourhood of and are dominated by h

H

on

K .

Then

is subharmonic on the interior of

Furthermore,

h

is continuous,

h

h = H on 8K

K , and

is harmonic on the open set

Sketch of proof.

The fact that

the fact that every point of

3K

h = H

K

on

{h< H1

8K

.

follows from

is a "stable" boundary

point, that is, a regular boundary point for the outer Dirichlet problem.

Another way of looking at it, from the

point of view of Chapter 5, is to observe that solution data

H

H

of the

on all of

R(K)-Dirichlet K .

h

is the

problem with boundary

In this case, each point of

a Jensen boundary point, so that

h = H

on

8K

.

The crux of the proof involves showing that the set

132

8K

is

{h< H}

This depends on a standard estimate for

is open.

harmonic measure, and the details are given in [4].

Once we know that harmonic on

{h
is open, we deduce that

{h< H}

h

is

since it is the upper envelope of a

,

Perron family of subharmonic functions.

Since

is lower

h

semi-continuous, it is continuous at each point of the set {h= H}

,

hence everywhere on

is subharmonic on

h

of subharmonic functions,

Being the upper envelope

K .

K° 11

8.3 Theorem (Cole's Theorem).

valued function on

C

H

be a continuous, real-

Then the following are equivalent.

.

There exists a subharmonic function

(i)

h ? 0

h <_ H , while

that

Let

(ii)

If

A

Jensen measure for

J H(u,*u)da ? 0

e MA , and

a

is a

then

u E Re(A)

,

such

0

on the real axis.

is a uniform algebra, ,

on

h

.

(iii) For all trigonometric polynomials

u

r2Tr

H(u(eie),*u(eie))d6 ? 0

.

0

Proof.

Suppose first that (i) is valid.

measure for

e MA

,

u e Re(A)

let

,

Let

and set

Using Lemma 8.1, we obtain 0:5 h(f

5 fhof da

The latter integral is the same as

!H(u,*u)da

be a Jensen

a

f = u+i*ue A.

fHof da ,

.

so that (ii)

is proved.

Since (iii) is a special case of (ii), it remains to show that (iii) implies (i).

For this, we suppose that (i) fails.

We must show that (iii) also fails. and let hR debe the closed disc {IzI <_ R} AR note the upper envelope of the functions that are subharmonic

Let

in a neighbourhood of

,

AR

and are dominated by

H

on

AR

.

133

is continuous on

hR

By Lemma 8.2,

harmonic on the interior of R

As

- ,

tends to

harmonic function H

minorant of

on

C

the assertion that

h

R > 0

exist

H(x0,OY < 0

If

u = x0

tion

H(x0,0)

The hypothesis (i) is equivalent to

.

on R .

0

Note that

hR(xo) < H(x0,0)

containing

x0

n A

.

By Lemma 8.2,

AR , and

U

is an open

U

is harmonic on

hR

hR = H

and

,

{z E AR

on

U

3U

be the universal covering map of the open unit onto

U

the functions

,

normalized so that

hRon

we see that

hRon

,

Ti

(d6)

A

n(0) = x0 .

2U

Consider

.

Since the nontangen, whenever they exist,

have the same nontangential on

8A

.

Since

hRon

is har-

we have

0 > hR(x0) = (hRon)(0) =

= J

Approximating

on

lie in

Hon

and

boundary values a.e. A

Hon

and

tial boundary values of

monic on

.

is continuous on

hR

.

then (iii) fails for the constant func-

subset of the interior of

Let

hR(x0) < 0

such that

be the connected component of the set

U

hR(z) < H(z)}

disc

(i) fails, there

Since

.

In particular,

Let

decrease to a sub-

Hence we assume that

.

>_ 0

,

hR

that is the greatest subharmonic

0

x0 E R n AR

and

is sub-

AR .

the functions

on

h

hR

AR , and

n

I

(h on)(ei6)

d0

(Hon)(eie) d6 2n

in the weak-star topology of

H_(d6)

analytic polynomials, we find an analytic polynomial

by g

such

that

(Hog)(ei6)d6

J 134

< 0

.

(8.6)

Since

is real, we can assume that

fr(O)

Then

takes the form

g

u + i*u

is real.

g(O)

and the inequality (8.6)

,

becomes

J H(u,*u)dO < 0 Hence (iii) fails.

El

Several variants of Theorem 8.3 are possible.

For in-

stance, one can consider an estimate of the form

J

Hof da ? 0 f e A

for

domain

such that and

D

is included in some specified

f(MA)

S c D

in some specified subset

f(4)

The

.

validity of this estimate for all Jensen measures is equivalent to the existence of a subharmonic function such that

h <_ H

on

h ? 0

D , while

Theorem 8.3 remains valid if

S

on

D

.

is only upper semi-

H

One merely approximates

continuous.

on

h

H

from above by con-

tinuous functions.

With a slight modification, Theorem 8.3 is also valid in H

the case that 0

to

(-co,+oo]

.

is a lower semi-continuous function from In this case, the function

h

of Theorem

8.3(i) is not necessarily subharmonic, but rather lower semicontinuous and quasi-subharmonic. Chapter 5,

h

In the terminology of

is a log-envelope function.

Following Cole, we now deduce the various classical estimates for the conjugation operator in the case of a Jensen measure.

8.4 Theorem.

For each

S >

2/'ir

that for all uniform algebras measures

a

for

4

,

,

A ,

there exists

4 e MA

y > 0

such

and Jensen

we have

135

I

I*ulda <_ gJ Jul log+luldo + y

0 < p < 1

Moreover, if

,

u E Re(A)

,

there are constants

(8.7)

.

such that

c

p <

(J I*uIPda)l/P

cp J

u E Re(A)

,

luldo

1 < p < - , there are constants

If

c > 0

There is a constant

a({I*ul > a})

u e Re(A)

,

such that if

lulda

f

such that

c

(J I*uIPda)l/P < cp(J lulpda)1/P

,

(8.8)

.

A > 0

u E Re(A)

,

(8.9)

.

then

(8.10)

.

Furthermore, any constants which serve in these estimates for the disc algebra and the measure

dO/27

also serve for

any uniform algebra and any Jensen measure.

Proof.

Consider first the Zygmund estimate (8.7).

case of the disc algebra, one writes u+ = max(u,0)

and

u- _ -min(u,0)

.

u = u+ - u- ,

In the

where

The estimate of

Theorem 7.8 for close approximants of

u+

and

u-

leads

immediately to the estimate (8.7), for the disc algebra.

The validity of the Zygmund estimate for a Jensen measure then follows from Theorem 8.3, since the estimate can be

a

cast in the form

JH(u,*u)da ? 0

.

The Kolmogoroff estimate (8.8) is established for the disc algebra as above.

However, this estimate cannot be

expressed in the form Substituting

t = fl*uIPd6/2v

1 t + 1 - 1 < tl /p p

we obtain

136

p

JH(u,*u)da ? 0

,

-
.

We argue as follows.

in the inequality

(8.11)

P-

P l l*ulpda + 1-

when

a = d6/27r

and

cp

J

lulda

(8.12)

is a trigonometric polynomial.

u

the estimate (8.12) can be written in the form 0

.

a

and all

u e Re(A)

(8.11) becomes equality for

Jensen measures Il*ulpda = 1

.

a

Since the inequality

.

t = 1

and for all

,

we obtain (8.8) for all

,

u E Re(A)

such that

Since (8.8) is appropriately homogeneous in

the condition

,

fH(u,*u)da

By Theorem 8.3, (8.12) is valid for all Jensen

measures

u

Now

Il*ulpda = 1

can be dropped, and the

Kolmogoroff estimate is proved. We have already given a proof of the M.Riesz estimate (8.9) in the case of the disc algebra, where duality allows us to deduce the estimate in the case in the case

1 < p

2

<_

2

<-

p < -

from that

Theorem 8.3 then shows that the

.

estimate extends to Jensen measures. The weak-type estimate (8.10) is proved in the case of the disc algebra by applying Theorem 7.12 to the positive and negative parts of (8.10) in the form

IXI

u

For the general case, we express

.

IHc(u,*u)da ? 0 ,

lYl

<

IYI

-

,

where

(8.13)

Hc(x,Y) = ixI - c/X

,

x

.

We have already remarked that the statements (ii) and (iii)

of Theorem 8.3 are equivalent when ous.

H

is only semi-continu-

Consequently (8.10) is valid also for all Jensen

measures.

[1

Sharp Constants A striking feature of the proof of Theorem 8.3 is that it points towards the extremal functions giving the sharp constants in the classical estimates.

The proof shows that if

an integral estimate fails, then it already fails for very

137

u +i*u , namely, functions that are cover-

special functions ing maps of

onto certain domains in the complex plane.

A

Symmetry considerations may limit substantially the possible domains.

For instance, the radial behaviour of the function associated with the M.Riesz inequality convinces one that the domains to be checked are wedges symmetric about the x-axis.

We are led then to consider the functions l+z

z

defined for unit disc ga(0) = 1

0 < a < 1

,

The function

.

_

onto the wedge

A

On

.

{arg w = ±a7/2}

3A

,

g

a

maps the open

{ 2- < arg w < 2} ,

the values of

ga

and

lie on the rays

In terms of the decomposition

.

ua+ iv a

ga g

ZE0

(I-Z)

ga ( )

of

a

into real and imaginary parts, this becomes

a

ua = cos (2) I ga

va = ± sin(g) Igal 84

If

a < 1/p Let

va I

fua Ze = ua(0) = 1

Note also that

on

.

,

then

1 < p < 2

ga E Lp(dO) ,

and let

a < 1/p

.

Since

= tan( 2 ) ua

we obtain (1

138

Iv al p Ze /I11/p

= tan(

2

J

I u al

p

Ze

1/p

,

and

ua >_ 0

Letting

increase to

a

1/p

,

we see that the best constant

for the estimate (8.9) satisfies

c

p

cp ? tan(2p)


The argument we have hinted at shows that in fact equality It is easier to obtain the best constant,

holds here.

though, by choosing the constants

a

and

y

of (8.1) ju-

diciously, and in the process a slightly better estimate will emerge.

We wish to choose there exists

a > 0

y > 0

as small as possible, so that

For this, fix a> 0,

satisfying (8.1).

and consider the function

F(6) = [sinP0 +acos(p0)1/cosp0

,

0 5 0 5 71/2

.

One computes that

F'(8)

p Sin cosp

p+1 0 [1 -ag(e)] 0

where

g(0) =

sin((P-l)0) sing-16

Now

g'(0) =

P-1 sin((2-p)0) > 0 sinp0

Consequently one zero.

g

is strictly increasing, and

Furthermore, the zero

1-cg(O0) = 0

,

60

of

F'

F'

has at most

satisfies

or

sin((p-1)60) = a

(8.14)

sine-180

139

Since for

F' F

changes signs at

60

We conclude that if some

.

(8.14), then

words, if

attains its maximum at

F

60

is fixed, and if

60 E [O,7/2]

sinp6 + acos(p0)

In other

.

is defined by

a 00

satisfies

i.e.

,

sinp0O+a cos(p6O)

<

(8.15)

0<6

cos6 Now set

60 E [O,7/2]

attains its maximum at

F

(8.11), then

is a maximum value

60

,

cos p 6

00 = rr/(2p)

0

and define

,

2

a

by (8.14).

The

estimate (8.15) becomes sing-1(2

sinp6 +

cos(p6) <_ tanp(2)cosp8,

)

cos (2) 2p Ti

0 <_ 6 <_ 7r/2

p

This leads to the following form of (8.9): sinp-1(T) f

I'eulP

dO +

2p

Iu(O)Ip <_ tanp(2p)

p cos(-)

J

Iulp d6

1
2

,

2Tf

J

p

(8.16) increases

.

In any event, the estimate (8.16) shows that the best con-

stants for the M.Riesz estimate (8.9) are

cp = tan(Z )

1: 5p52

,

By duality, we obtain as best constants

cp = tan[2(1 - P)]

The functions

g

a

,

2 <_ p <

can also be used to show that the Kol-

mogoroff estimate of Theorem 7.10 is sharp, and that the

140

Zygmund estimate (8.7) is sharp, in that it fails for

2/7

.

Now we turn to the weak-type estimate (8.10).

we study first an auxiliary function Denote the strip

z = x+ iy ' S

if

IxI

< A}

{IIm(z)I ,

F

Poisson integral of the function continuous and subharmonic, and

.

by

and extend

S

F

Define

.

to

on

xl

For this,

F(x) _

to be the

S

8S

Then

.

F

is

is symmetric with respect

F

to both coordinate axes.

From the symmetry of the Poisson kernel for

S

,

one sees

easily that

F(x+iy) ? F(iy)

8

It follows that

x +iy E S

,

(8.17)

.

2

(iy) ? 0

2F

for

-A < y < A

(iy)

<_ 0

.

Since

-A < y < A

AF = 0

on

and

is a concave function on the vertical interval

S

, we obtain

for

-

F

-A < y < = 0

,

F

at y = 0

A

Since

.

is symmetric, while

F

F(iA) = F(-iA)

must attain its maximum value over the interval .

Now consider the function

F - Ixi

on

S

It is super-

.

harmonic, it is harmonic except on the imaginary axis, it vanishes on

8S ,

and it vanishes as

attains its maximum over it coincides there with

Hence it

.

on the imaginary axis.

S

F

x -} ±-

,

Since

its maximum is attained at 0

Hence

F(z) - IxI <_ F(0)

z = x + iy c S

,

(8.18)

.

Now consider the function lyl

IxI

H(z) =

IxI -F(0) Then

F -F(0)

,

< A

,

lyl > A

coincides with

H

for

lyl

? A

,

and the

141

estimate (8.18) shows that F -F(O)

(8.17),

F- F(0)

we obtain

,

and all

a

on

>_ 0

F -F(O)

lyl

< A

By

.

Applying Theorem 8.3 to

]R.

fH(u,*u)da ? 0

u e Re(A)

when

<- H

h =

for all Jensen measures

This yields the weak-type estimate

.

(8.10) in the form

a({I*ul > a}) < F1

where

a

lulda

f

l/F(0)

appearing here is

Indeed, suppose that the weak-type estimate (8.10)

sharp.

is valid with constant

c/1

Define

.

c

Hc

that are dominated by

C

He

by (8.13), and

be the upper envelope of the subharmonic functions

h

on

(8.19)

,

is a Jensen measure.

We claim that the constant

let

u e Re(A)

,

By Theorem 8.3,

h

>_ 0

c

on ]R. Since

Hc,

h

for S

Ixl - A/c

c lyl

? A

is a subharmonic function dominated by

- A/c

Ixl

,

l/F(0)

,

must coincide with

h c

By the maximum principle,

.

In particular,

.

and

h

0 <_ hc(0) <- F(0) - A/c

c .

the conformal map

l/F(0)

on

c/a

S

For this, we consider

.

.

Making the corresponding

change of variable in the integral, we obtain

F(0) = 1

Ixldp 0 = J_

11

r

as

Tr2

dt

Ilogiti j

Tr(l+t2)

log t dt = 8XK

= 8A

11+

t2 Tr

where

142

Hence

of the right half-plane

(2a/7) log C

onto the strip

{Re(d) > 0}

l

5 F - A/c

and the estimate (8.19) is sharp.

It remains to evaluate

K=

- A/c

lxl

log

1 1+t 2

dt = 1 - 2 + 3

2 2 1

S

7

+ .

is the mysterious Catalan constant.

The best constant in

(8.10) is thus given by

c

Tr2

F(0)

A

8KX

One can also conclude that (8.19) is sharp by verifying that (8.19) becomes equality when

f = U+ i*u =

2X

log( 1+w)

is the conformal map of

onto

S

.

Arens-Singer Measures Cole's theorems have an analogue for Arens-Singer measures.

8.5 Theorem. z0 E D D

on

,

.

be a domain in the complex plane, let

D

H

and let

be a continuous, real-valued function

Then the following are equivalent. There exists a harmonic function

(i)

that

Let

h <_ H ,

(ii)

If

h(z) >_ 0

and

A

h

on

.

is a uniform algebra, $ E MA , and

Arens-Singer measure for

¢

such

D

,

is an

o

then

(Hof)da ? 0

J

for all

f E A

satisfying

f(MA) s D

and

zo

.

The proof runs along the same lines as the proofs of Theorems 7.2 and 8.3.

Note that if the domain connected, then on

D

.

h

D

of Theorem 8.5 is simply

is the real part of an analytic function

Comparing Theorems 7.3 and 8.5, we see that in this

case, an integral estimate is valid for all representing measures as soon as it is valid for all Arens-Singer measures.

143

Notes on Sources As mentioned earlier, the extension of the classical inequalities to Jensen measures is due to B.Cole, and has been presented by him in various talks dating back to 1970.

The

proof of the M.Riesz Theorem, given at the beginning of this chapter, together with a determination of the best constant, is due to S.K.Pichorides[5].

Cole had independently dis-

covered the best constant for the M.Riesz estimate.

The best constant for the weak type estimate was disDavis' proof, which depends on

covered by B.Davis[21.

Brownian motion, is discussed by D.L.Burkholder in [1].

The

proof we have given is close to the one given in the classical case by Albert Baernstein, II.

It was modified to cover

Jensen measures by K.Yabuta[6], who had independently discovered Lemma 8.1 as a vehicle for extending classical estimates to Jensen measures (cf. reference [12] of Chapter 7).

Davis[3] has also obtained the best constants for the Kolmogoroff estimate (8.8). for each

S > 2/7

,

Pichorides [5] has discovered,

the best constant

Zygmund estimate (8.7).

that serves in the

The best constant for the Zygmund

f exp(al*ul)do

estimate for

y

given in Theorem 7.9 turns out

to be 2

cos a

-

1

4 Tr

J

0

t(2a/Tr) 1

2

dt =dt 1t2 t(2a/Tr)

4

Tr

1

References 1.

Burkholder, D.L.

Harmonic analysis and probability, in

Studies in Harmonic Analysis, J.M.Ash (ed.), MAA Studies in Mathematics, Vol.13, pp.136-149. 2.

Davis, B.

On the weak type (1,1) inequality for conju-

gate functions, Proc. A.M.S. 44 (1974), 307-311. 3.

144

On Kolmogorov's inequalities I f I p 0 < p < 1 , Trans. A.M.S. 222 (1976), 179-192. Davis, B .

I

I

<_

C

f II 1

4.

Gamelin, T.W. The polynomial hulls of certain subsets 2, Pac. J. Math. 61 (1975), 129-142. of (L

5.

Pichorides, S.K.

On the best values of the constants in

the theorems of M.Riesz, Zygmund and Kolmogorov, Studia Math. 44 (1972), 165-179. 6.

Yabuta, K.

Kolmogorov's inequalities in the abstract

Hardy space theory, Arch. Math. 30 (1978), 418-421.

145

9 Moduli of functions in H2(a)

Let

be a compact Hausdorff space, let

X

algebra on

and let

X ,

E MA

unique representing measure kernel of

,

and let

A and A

the closures of

X

on

Let

.

and HP(a)

Hp(a)

Lp(a)

in

has a

We assume that

.

a

be a uniform

A

A

denote the

denote respectively In the case

.

the closures are taken in the weak-star topology of The following facts about the function theory for

p

C(a)

Hp(a)

are

known and will be used in one form or another.

Every

can be approximated pointwise

f c H (a)

almost everywhere by a sequence

satisfying

IIfn1IX

<

IIfII

{fn}

A

in

.

L2(a) = H2(a) ® H2(a)

loglf(OI 5

(9.2)

f E H1(a)

loglflda

J

The conjugation operator

u - *u

to a continuous operator from

LR(a)

,

0
If

u E LR(a)

If

u e L1(a)

,

.

(9.3)

extends

(9.4)

L11(a)

to

.

then

exp(u+i*u)

eu E L2(a)

and

exp(u+i*u) e H2(a)

(9.1)

,

e

H'(a)

.

then

(9.5)

(9.6)

.

The estimate (9.3) is the Jensen-Hartogs inequality, which is valid since

a

is necessarily a Jensen measure.

The

property (9.4) follows from the Kolmogoroff inequality, once

146

it is shown that

is dense in

Re(A)

L1(a)

and (9.6) follow easily from (9.4).

Both (9.5)

.

For detailed proofs,

see [1] or [2].

We wish to address ourselves to the problem of characterizing the moduli of the functions in

H2(a)

The prop-

.

erty (9.6) provides a partial solution to the problem: if w E LR(a)

then a sufficient condition that there exist

,

such that

f E H2(a)

IfI

log w

is that

= w

be summable.

log w

As a start, we might ask when the summability of also a necessary condition in order that f E H2(a)

is

for some

.

The summability of

is a necessary condition, when

log w

is the disc algebra, and

A

w = Ifl

a = de/27

Szego's Theorem asserts that

log Ifi

is not identically zero.

f E H2(do)

.

In this case,

is summable whenever Szego's Theorem is

proved by merely applying the Jensen-Hartogs inequality to f/zm , where

the function

m

is the order of vanishing of

at the origin.

f

Hp((1+t2)-1dt)

Consider next the transplants Hp(do)

spaces

H ((l+t2)-ldt)

to the upper half-plane.

of the

The algebra

can be identified with the algebra of bounded

analytic functions on the upper half-plane, while the Poisson kernel

point

[7r(l+t2)]-1dt i

The space

.

H-((l+t2)-ldt)

in

H2((l+t2)-ldt)

in

plane.

is a representing measure for the H2((l+t2)-1dt)

is the closure of

L2((l+t2)-ldt), and all the functions extend to be analytic on the upper half-

In this case, Szego's Theorem asserts that LI((l+t2)-ldt)

belongs to

whenever

log Ifl

f c H2((l+t2)-1dt)

is

not identically zero.

A more complicated situation is'provided by the algebras of generalized analytic functions associated with compact groups with archimedean-ordered duals. subgroup of Let

G = F

]R, and outfit ,

r

the dual group of

Let

r

be a dense

with the discrete topology. r

,

and let

a

be the Haar

147

measure on G

G

Each

.

A

and we define

,

a E

to be the uniform algebra on

erated by the characters p(f) = ffda A

ters

X

a

We can embed t -} et

,

where

a> =

<e

a ? 0

If

,

,

a

a ? 0

for

a E

,

A

F

Then

.

.

into

7R

as a dense subgroup via the map

G

is the character on

et

eita

a E

P

.

Xa(et) = elta

then

defined by

F

,

t E]R

extends to be

,

bounded and analytic in the upper half-plane. tions in

gen-

G

is the closed linear span of the charac-

A

for a > 0

,

x

on

Xa

defines a multiplicative linear functional on

and in fact

,

determines a character

F

Thus the func-

are analytic almost-periodic functions in the Similarly, for

upper half-plane.

f c A

fx(t) = f(x+e t)

,

is analytic almost-periodic in the upper half-plane above x +]R

the coset

for all

,

in

x

G

are analytic in the sense that

H2(a)

H2((l+t2)-ldt)

longs to Since

a

The functions in

.

for almost all

x E G

o

G .

for

However

H.Helson and D.Lowdenslager[41 have

Szegb's Theorem fails.

shown there are nontrivial functions fj

.

is a unique representing measure on

the facts (9.1) through (9.6) are valid for

log

be-

fx(t) = f(x+e t)

is not integrable.

moduli of functions in

f

in H2(a)

such that

The characterization of the remained an outstanding prob-

H2(a)

lem until 1973, when Helson[31 succeeded in establishing the following.

Helson's Theorem. let

w E L2(a)

Let

and A

F, G, a

satisfy w > 0

.

be as above, and

Then the following are

equivalent:

w = jfl

148

for some

f

in

H2 ((J)

;

(9.7)

for almost all

wA

L1((l+t2)-ldt)

belongs to

log w(x+e t)

x c G

is not dense in

(9.8)

;

L2(a)

(9.9)

.

Actually, Helson and Lowdenslager had shown earlier in [4] that (9.8) and (9.9) are equivalent, while (9.8) follows from (9.7) by applying Szego's Theorem to the functions f

c

x

H2((l+t2)-1dt)

.

Helson's contribution in [3] is to

show that (9.8) and (9.9) imply (9.7). In a seminar talk in 1974, B.Cole presented a simplified proof of Helson's Theorem.

It turns out that Cole's proof

extends to a more general context.

The theorem we aim to

establish is the following.

9.1 Theorem.

pose that X

Let E MA

A

be a uniform algebra on

,

and sup-

has a unique representing measure

Suppose furthermore that no function in

.

X

H2(a)

a

on

can

vanish on a set of positive measure, unless it is identically zero.

Let

f c H2(a)

dense in

w c L2(a)

such that L2(o)

w > 0

satisfy IfI

= w

.

Then there exists

if and only if

is not

wA

.

In other words, the conclusion of the theorem is that (9.7) and (9.9) are equivalent.

zero sets of functions in

H2(a)

Since the hypothesis on the is met in the case treated

by Helson, Theorem 9.1 combines with the earlier work of Helson and Lowdenslager to yield Helson's Theorem.

The remainder of this lecture is devoted to proving Theorem 9.1.

It will be obtained as a corollary of the more

general Theorem 9.3.

The idea of the proof is contained in

the following Theorem 9.2.

The Jensen-Hartogs inequality is

crucial to the proof.

9.2 Theorem.

Let

A

be a uniform algebra on

X

,

and

149

e M has a unique representing measure a A M be a closed subspace of L2(a) such that

suppose that on

X

Let

.

AM s M , and let

M vanishes off

every member of

M

in

such that

E

.

Then there exists

almost everywhere on

IFI = 1

E

F

.

The proof breaks into four steps.

Proof.

If the complex numbers

Step I. E

be a set of minimal a-measure such that

E

<

Ia.l

al,a2,...

satisfy

then

,

3

max

Here

torus

I

J

15j<-

log

<_

Ia.I

j =l

T

a.J e

ie dT(61)02,...)

.

(9.10)

is the normalized Haar measure on the infinite

T

T

Applying the Jensen-Hartogs inequality to

a

+e°

,

we

obtain

loglal

<_

logla + be161

J

Le

= J loglaeie + bl 26

(In fact, one can establish easily the identity

max(loglal,loglbl) = I loglaeie

dO + bl

J

which already appears in an equivalent form in Jensen's paper of 1898 cited in Chapter 2.) i01

loglalI

loglale

<_

i6 2

+ ate

+ ...1

d 61 2,r

J

for fixed

Hence

62,63,...

By the uniqueness of Haar measure,

d61

T(61,2,...) =

2n

T(02,03,...)

equality with respect to

150

.

Integrating our in-

T(02,03,...) , we obtain

i0

loglaIei0 1

logla1

+ ate

<_

+ ...I dr(01,02,...)

2

.

J

The same estimate applies to the other

a

's

.

J

There exists

Step II.

everywhere on

E

Indeed, for

h

M

in

such that

h z 0

almost

.

f E M , let

E(f)

be the set on which

f

does not vanish, and set

a = sup{o(E(f))

f E M}

:

{f.}

Choose a sequence

in

g E M , and choose

Let

assume the value

.

M

such that

X. x 0

so that

g/fj

a ? a(E(g+ajf.)) = a(E(g)

a(E(g)\E(f.))

.

Letting

a(E(g)\E(fj)) -} 0

included in

j

u E(fj)

u E(f.)) = o(E(f.)) +

tend to

Since

.

Then

E(g) u E(fj)

- , we find that

so that almost all points of

,

.

does not

on a set of positive measure.

X.

coincides almost everywhere with

E(g+A.fj) so that

o(E(f.)) -} a

g c M

E(g)

are

is arbitrary, we ob-

tain

E =

E(f.)

U

(9.11)

j =l Multiplying the E IlfiIda < For

c > 0

BE _ {x

o(B

,

:

.

fJ.'s

Then

E

by constants, we can assume that If. (x)l

< -

almost everywhere.

define

max Ifj(x)I ? e} l<j<-

log e B

6

JBe

max loglf.(x)Ido(x) 1<j<J

i0 . J

T

f . (x) e

log j=1

dT(0)da(x)

.

J

151

In particular, for T-almost all choices of function

ei6Jf.(x)

E

we see that

does not vanish on

Be

E 's

is the union of the

E

61,62,...

From (9.11)

.

h(x) _

Hence

.

the

,

e

does not vanish on

E e16Jf. (x)

E

for almost all choices

,

J

61,62,...

of

.

There exists an

Step III.

integrable on

E

M

in

f

such that

log Ifl

is

.

This step is the core of the proof.

The idea is due to

Cole, and a similar idea had been used by Helson.

Fix

{e.}

Choose a sequence

as in Step II.

h

positive numbers such that

el = 1

and

of

J J=1

E E.

Accord-

<

J

ing to (9.5), there exist 1/(j+l) <

whenever

1/IhI

gj

such that

e H°°(o)

1/j

Ihl

,

while

J

J

From (9.1) and the hypothesis

otherwise.

H (o)M s M .

that

Hence

fj = gjh

=

Igjl

ig.I = e.

AM s M , we see

belongs to

M

.

More-

over,

j+l If

<

Ih(x)I <

.(x)I = J

e

.Ih(x)I

,

otherwise,

j

and

whenever

IfI(x)I = Ih(x)I

max

If.(x)I ? 1

,

Ih(x)I ? 1

almost all

.

x c E

J

1<_j<`°

Furthermore,

Elf .(x)l

converges to a summable function.

J

Using Step I, we obtain

0 <_

max loglfj(x)I do(x)

J

E lsj
logle J

152

Thus

JT

f .(x)IdT(61,62,...)do(x)

summable function

f=

ie. Jfj j=l e

satisfies

loglfl

L1(OIE)

E

for almost all choices of

e1,e2,...

finite, the series defining belongs to

Choose

L2 (o)

F = f exp(u +i*u) F E M

as in Step III, define

f

u = -loglfl

show that

converges in

f

,

and

f

M .

Step IV. so that

E IIfI2do is

Since

.

on

E

,

Since

.

while

u = 0 on

= 1

IFI

loglf(x)I

off

E

E

.

Set

it suffices to

,

un E LR(a)

For this, define

.

u E LR(o)

by

-n < logIf(x)I < n

,

un(x) = 0

and set

verges to measure.

F

otherwise

,

= f exp(u

n

n

+i*u

n

)

By (9.4),

c M .

in measure, so that

*u

Since

dominated, and

IF nI

Fn

F

<_ max(l,IfI)

converges to

,

*u

con-

n

converges to

n

F

in

the convergence is

,

L2(a)

in

F

.

CI

For the next theorem, it will be convenient to denote the closure of a linear subset

9.3 Theorem.

pose that X

.

Let

Let

A

E MA\X w c L2(a)

L2(a)

of

B

by

[B]

be a uniform algebra on

X ,

.

and sup-

has a unique representing measure on satisfy

minimal measure such that

w ? 0

[wA]

.

Let

E

be a set of

includes all functions in

153

vanishing on

L2(a)

Then

.

there exists an

(i)

on

E

off

f = 0

and

E

f

E

if and only if

Let

Proof.

f

Ifl = w

in

H2(a)

such that

Ifl = w

be the orthogonal complement in

[wA]1

if and only if

dently

such that

has full measure.

E

of the subspace spanned by [wA]1

H2(a)

;

there exists an

(ii)

in

h

for

wh ,

fgwhda = 0

A

in

E

and that

,

such that

IFJ

fFwhda = 0 (9.2).

on

for all

Setting

.

vanish

has minimal measure among sets with

E

= 1

L2(a)

[wA]1

By Theorem 9.2, there is an

this property.

g c

Evi-

.

is a closed invariant subspace of

[wA]1

Our hypothesis shows that all functions in off

Then

.

h e A

for all

L2(a)

F = 0

and

E

E

belongs to

Fw

h E A ,

off

[wA]1

in

F

Since

.

H02 (o)

by

,

f = Fw , we obtain (i), and we also obtain

the "if" statement of (ii).

Suppose there exists an Define a bounded function f = 0

,

while

F = w/f

[wA] = F[fA] c FH2(a) E

,

in H2(a)

f

so that

F

elsewhere.

Evidently

.

so that the definition of

cludes all functions in

F

that

u c H2(a)

(If

fu2do = 0

,

be a point mass.

and

w = Ff

H2(a)

in-

that vanish on

E

How-

H2(o)

is real, then

u = 0 .)

and

,

is unimodular off

ever, the only real-valued functions in constants.

wherever

F = 0

Then

If I =w

shows that

E

L2 (a)

such that

u2

Since

We conclude that

E

X

.

are the E ,

HI(a) a

,

so

cannot

has full measure.

Cliffs, 1969.

11

Proof of Theorem 9.1.

mass at

,

and the theorem is obvious.

Pick

that

¢ E MA\X

9.3.

We must show that

[wA] x L2(a)

154

E X , then

If

.

.

a

is the point

We assume then

as in the statement of Theorem

E E

has full measure if and only if

From part (i) of Theorem 9.3 and the

hypothesis on the zero sets of functions in that either

E

has full measure, or

By the definition of while

[WA] = L2(a)

E

,

E

[wA] s L2(o)

H2(a) , we see

has zero measure. in the former case,

in the latter case. 0

References 1.

Introduction to Function Algebras, W.A.

Browder, A.

Benjamin, Inc., New York, 1969. 2.

Gamelin, T.W. Uniform Algebras, Prentice Hall, Englewood Cliffs, 1969.

3.

Helson, H.

Compact groups with ordered duals IV, Bull.

London Math. Soc. 5 (1973), 67-69. 4.

Helson, H. and Lowdenslager, D.

Prediction theory and

Fourier Series in several variables II, Acta Math. 106 (1961), 175-213.

155

List of notation

R

real line

C

complex plane

A

open unit disc in complex plane

Z

integers

Z+

strictly positive integers

E

closure of

DE

topological boundary of

E°

interior of

X

compact Hausdorff space

C(X)

continuous complex-valued functions on

CR(X)

continuous real-valued functions on

A

uniform algebra on

MA

maximal ideal space of

E

E

A (p.24)

invertible functions in

A-1 loglA-11

{loglfI

:

X

A

space of real parts of functions in

Re(A)

X

X

Shilov boundary of

aA

E

A

A

f E A-1}

A

kernel of

A ®P

algebra of Hartogs polynomials (p.58)

AE

closure in C(E) of the restriction algebra AIE (p.65)

E

A-convex hull of

H (D)

bounded analytic functions on

M(D)

maximal ideal space of

A(A)

disc algebra

R(K)

closure in C(K) of rational functions with poles off K (p.34)

0(K)

functions analytic in a neighbourhood of

156

E MA

E

D (p.46)

H_(D) (p.46)

(p.30)

K (p.88)

H(K)

closure of

H(K)

closure in C (K) of functions harmonic in a neighbourhoodRof K (p.37)

log+u

max(0,log u)

u*

upper semi-continuous regularization of

*u

harmonic conjugate of a harmonic function abstract conjugate function of u (p.107)

u

solution of the R-Dirichlet problem (p.9); solution of the A-Dirichlet problem (p.29, p.56)

u

solution of the generalized Dirichlet problem (p.78)

in

0(K)

C(K)

(p.88)

u

(p.65) u.;

(p.102)

Hu(z)

complex Hessian matrix of

Au

Laplacian of

S

point mass at

supp(o)

closed support of

u (p.83)

u

a

Cauchy transform of

(p.34)

v

Vv

negative of logarithmic potential of

Hp (o)

closure of

H°°(a)

weak-star closure of

Hp(o)

closure of

A

[B]

closure of

B

R

(p.1)

0

aR U

A

v (p.35)

in Lp (a) , 1
Lp(o)

in

in

A

L2 (a)

L_(o) (weak-star, if p = °)

(p.153)

Choquet boundary associated with

R (p.10)

(p.2)

S

R-envelope functions

SC

continuous R-envelope functions

_<

(p.16)

(p.5)

(p.8)

157

Index

A-Dirichlet problem Arens,R.

29,56

32

Arens-Singer boundary

27

Arens-Singer measure

26

Baernstein, Albert, II barrier

144

13

plurisubharmonic subharmonic

84

79

Basener,R.

64

Bedford,E.

104,105

Besicovitch,A.

127

11,16,19,32

Bishop,E. Bochner,S.

32,126

Bremermann,H.

83,91

Bremermann function

74

on a domain in Cn

98

Bremermann's Theorem

93

Bremermann's generalized Dirichlet problem Burkholder,D.L.

Calder6n,A.P. Carleson,L.

144

126,127 47

Catalan constant

142

Cauchy transform

34

Choquet,G.

1

Choquet boundary

10

complex Hessian matrix Cole,B.

158

83

46,64,125,129,135,144,149,152

102

Cole's Theorem conjugation operation for Jensen measures

133

conjugation operation for representing measures corona counterexample for Riemann surface conjugate function

Davie,A.M. Davis,B.

108

46

73

144

Debiard,A.

41,43

deLeeuw,K.

11,19

Diederich,K.

97

Edwards,D.A.

1,3

Edwards' Theorem

Fornaess,J.E.

Garnett,J. Gaveau,B.

3

97

73

41,43

generalized Dirichlet problem generalized peak point generalized peak set

Hadamard,J.

78

24

24

32

Hardy,G.H.

127

Hartogs,F.

32

Hartogs series

57

Hartogs-Laurent polynomials Helson,H.

49

108

conjugation operator

corona problem

109

61

148,149,152

Helson's Theorem Hoffman,K. Hormander,L.

148

32

83

159

Jensen,J.L.W.V.

31,32

Jensen boundary

28

Jensen-Hartogs inequality Jensen measures

2,28

Jensen's inequality

Keldysh,M.V.

7

13

Kolmogoroff,A.N.

127

Kolmogoroff estimate Konig,H.

2,28

115,122,136

126

Littlewood,J.E.

127

local maximum modulus principle locally log-envelope function on a domain in Cn

65 57

98

locally quasi-subharmonic function locally subharmonic function log-envelope function Lowdenslager,D.

Martin,W.T.

28,55

148,149

32

maximal measure

16

McKissick's Swiss cheese Meyer,P.A.

peak point

Phelps,R.

105

24

24 1

Pichorides,S.K.

144

plurisubharmonic barrier plurisubharmonic function

quasi-subharmonic function

160

64

1

Monge-Ampere equation

peak set

57

84

83

54

57

R-Dirichlet problem

9

R-envelope function

5

R-face

12

R-measure

1

regular boundary point representing measure

80 22

107,126,127

Riesz,M.

estimate

107,108,111,125,129,136

roadrunner set Rossi,H.

44

86

local maximum modulus principle

local peak point theorem

65

71

theorem on Shilov boundary of H(K)

SS-set

89

88

Scheinberg,S.

52

sharp constants

137

Shilov boundary

13

of A

23

Sibony,N.

52,54

Singer,I.M.

32

smooth boundary point

85

solution to

Bremermann's generalized Dirichlet problem the A-Dirichlet problem

56

the. generalized Dirichlet problem

the R-Dirichlet problem

strictly pseudoconvex boundary point

subsolution Szego,G.

78

9

strictly plurisubharmonic function

subharmonic function

102

84 85

54

78

32

Taylor,B.A.

104,105

161

upper semi-continuous regularization

Vladimirov,V.S.

Walsh,J.B.

32,83

103

weak-type estimate Wermer,J.

64

Yabuta,K.

126,144

Zygmund,A.

107,127

Zygmund estimates

162

123,136

107,115,119,136

65