The extreme rays of the positive pluriharmonic functions

2.13.

THE EXTREME RAYS OF THE POSITIVE PLURIHARMONIC FUNCTIONS*

I. that R e g

Let n ~ 2 and consider the class > 0 and g(0) = I,

(and compact

where

B

N(~)

of all holomorphic

is the open unit ball in

in the compact open topology).

functions g on

C~

Thus

N[~)

We think that the structure of

terest and importance. Thus we ask: What are the extreme points of known. Of course if n = I, and if

N~)?

B

such

is convex

N~)

is of in-

Very little is

q=Ci+~)/(~-[), then g is extreme

~j all

and i f

if and only if f(z) = cz where

g is

the Cayley transform

extreme points

of

Let k = (kl,...,k If(z) l ~ 1 if

~

(1) of f ,

C~

(i) . It is proved in [I] that if f(z) =

then

~ ~),

that

We h a v e

~c~S~)

n = I, then

N) be a m u l t i i n d e x .

Thus

and c o n s i d e r

monomiais f ( z )

I~l~(~lkl) - ~ where by Ikl we mean k l

is the c l a s s

~ ) ;

however i t

is a corollary

that

=r

= c z k in

+ ... + kn. ~E~(~)

Thus

~

of

such t h a t

Let

Ck=(k/Ikl) - ~

i f and o n l y i f

of t h e j u s t - m e n t i o n e d

where the closure is in the compact open topology. ~)

E(~)

N~)

and l e t g ( z ) = (1 + c k z k ) / ( 1 -- c k z k ) . I t i s p r o v e d i n [2] components of k are r e l a t i v e l y p r i m e and p o s i t i v e . 2.

where

t h e o r e m of

the

[2]

~(~)~$~)

.

[If

.]

It is also known that if g is an extreme point of N(~) and if (I) holds [i.e., if f = (g -- 1)/(g + I)], then f is irreducible. This is a special case of Theorem 1.2 of [3]. The term "irreducible" is defined in [4]. If n = I, then g is extreme if and only if f irreducible. However for n ~ 2, the fact that f is irreducible does not imply that g is extreme. 3.

The extreme points g in Sec.

by letting

~ )

act on

(g + I) is holomorphic

on

I and the extreme points that can be obtained from them

N(B) have the property that the Cayley transform f = (g -- I)/ ~U~.

Is this the case for every g in

E~)

?

if the answer

is yes, then it would follow (since n ~ 2) that the F. and M. Riesz theorem holds for those Radon measures on ~ whose Poisson integrals are pluriharmonic. In particular there would be no singular Radon measures #0 with this property, which in turn would imply that there are no nonconstant inner functions on LITERATURE CITED i I 9

2. 3. 4.

F. Forelli, "Measures whose Poisson integrals are pluriharmonic. II," Illinois J. Math., 19, 584-592 (1975). F. Forelli, "Some extreme rays of the positive pluriharmonic functions" (unpublished). F. Forelli, "A necessary condition on the extreme points of a class of holomorphic functions," Pacific J. Math., 73, 81-86 (1977). P. Ahern and W. Rudin, "Factorizations of bounded holomorphic functions," Duke Math. J., 3-9, 767-777 (1972).

*FRANK FORELLI. 53106.

University Of Wisconsin, Department of Mathematics,

Madison, Wisconsin

2297