The inner function problem in balls

CHAPTER 13 INNER FUNCTIONS IN THE SPHERE The problems discussed in the two sections of this chapter are related in some way or other to the conjecture that in a multidimensional sphere there are no nonconstant inner functions. It seems useful to complete the references of Secs. 1.13 and 2.13 by the following works: I) A. E. Tumanov, Usp. Mat. Nauk, 29, No. 4, 158-159 (1974); 2) A. Sadullaev, Mat. Zametki, I-9, No. I, 63-66 (1976); 3) G. M. Khenkin and E. M. Chirka, in: Contemporary Problems of Mathematics, Vol. 4, VINITI, Moscow (1975); 4) N. Sibony, Lecture Notes in Mathematics, No. 578, Springer, Berlin (1977), pp. 14-27"; 5) E. Bedford and B. Taylor, "Variation of properties of the complex Monge--Ampere equation" (preprint), Michigan Univ. (1977). In connection with the problem of the description of the isometries of the LP-spaces of analytic functions, touched upon at the end of the first section, we mention the paper of A. I. Plotkin, cited after Sec. 5.1. Finally, the subject of this chapter is discussed on pp. 367-368 of B. V. Shabat's book Introduction to Complex Analysis. Part II [in Russian], Nauka, Moscow (1976).

*G. M. Khenkin has informed us that the arguments of this paper prove in fact the inexistence of inner functions satisfying (in the multidimensional sphere) the H~Ider condition of order I/2 (and not only the Lipschitz one, as indicated by the author).

2294

1.13.

THE INNER FUNCTION PROBLEM IN BALLS~

The open (Euclidean) unit ball in (with n at least 2) is denoted by B A nonconstant bounded holomorphic function f with domain B is called ~ # ~ if its radial limits ~(I~)-~- ~ i

~C ~ )

satisfy

I~(IAf)I=~ a.e. on S = ~B, where "a.e." refers to the rotation-

invariant probability measure d on S. Conjecture

I.

There are no inner functions in B.

Here is some evidence in support of the conjecture: (i) If f is inner in B, and if V is an open subset of is dense in the unit disc ~).

~

that intersects S, then ~(~n~)

Proof. If not, then V contains one-dimensional discs D with ~Dc~, such that f ID is a one-variable inner function whose range is not dense in ~) , an impossibility. In other words, at every boundary point of B, the cluster set of f is the whole closed unit disc. No inner function behaves well at any boundary point. (ii) If f is inner in B and if E is the set of all w, w E has no interior (relative to S). Proof.

e~

at which

~C~)I=~ ~

then

If not, an application of Baire's theorem leads to a contradiction with (1).

Conjecture I could be proved by proving it under some additional hypotheses, were an inner function f in B, then there would exist (a) a zero-free

for if there

inner function, namely exp [(f + 1)/(f -- I)];

(b) an inner function g with one-dimensional

l~vl~ = 0 , via Frostman's theorem; for almost all S discs D through the origin, gID would be a Blaschke product;

(c) an inner function h that satisfies (b) and is not a product of two inner functions i.e., h is irreducible, in the terminology of [I]; (d) a nonconstant bounded pluriharmonic

function u with u* = I or 0 a.e. on S, namely

~=~e(@o~) , where ~ is a conformal map of ~ onto the strip 0 < x < I [i.e., there would be a set E, E c S, o(E) = I/2, whose characteristic function has a pluriharmonic Poisson integral]; (e) a function F = (I + f)/(1 --f) with R e F

> 0 in B but R e F *

This Re f would be the Poisson integral of a singular measure. equivalent to

= 0 a.e. on S. Hence Conjecture

I is

Conjecture I' If D is a positive measure on S whose Poisson integral is pluriharmonic then ~ cannot be singular with respect to o. Forelli [3, 4] has partial results that support the following conjecture viously implies 1'): Conjecture 2. ~ << o.

(which ob-

If ~ is a real measure on S, with pluriharmonic Poisson integral,

then

Conjecture 2 leads to some related HI-problems: Conjecture 3. Conjecture 3'. that

~I~I~O~OIIR4}~ S

If f is holomorphic

in B and Re f > 0, then ~H~C~).

There is a constant c, c < ~ (depending only on the dimension n) such for all

~, ~ ( ~ )

(the ball algebra).

5

~WALTER RUDIN. Department of Mathematics, California, 92037.

University of California,

San Diego, La Jolla,

2295

Conjecture Clearly,

3".

If f is holomorphic

in B, f = u + iv, and

3' implies 3, and 3" is a reformulation

Let N(B) be the Nevanlinna class in B ( ~ mist of all

~ ,~N(~),

Conjecture 4.

for which

HI(B).

of 3 that might be easier to attack.

is bounded,

[ ~ i ~ i I is uniformly

integrable.

I.

I leads to the problem of finding the extreme points of the unit ball of

(When n = I, these are exactly the outer functions

Conjecture

as r § I), and let N,(B) con-

N(B) = N,(B).

This would imply I', hence Conjecture

I~.~O

IvJ ~ u, then ~ e H~8)-

5.

Every

~ , ~H~(5),

of norm I.)

Let

~=[~(~):

with HfHI = I is an extreme point of ~.

It is very easy to see that 5 implies I. If ~eA(5) it is known (and easy to prove) It is tempting to try to extend this to H~(B): Is it true for every f,

that f(S) = f(B).

~Hm(~), that the essential range of f* on S is equal to the closure of f(B)? An affirmative answer would of prove it, one would presumably need Does there exist c, c > 0 (depending for every f, f ~ A(B), with f(0) = inner if

~ ( ~ 5

Conjecture 6.

course be a much stronger result than Conjecture I. To a more quantitative version of f(s) = f(B). For example: only on the dimension n) such that ~({Ifl < I/2}) > c 0, Ifl < I? Finally, call a holomorphic mapping ~:B + B

for almost all w, w e~ If ~ is inner, then

~ is one-to-one and onto.

This implies I, as well as the conjecture that every isometry of HP(B) into HP(B) is actually onto, when p ~ 2 (see [5]). If "inner" is replaced by "proper," then Conjecture 6 is true, as was proved by Alexander [2]. LITERATURE I 9

2. 3. 4. 5.

2296

CITED

P. R. Ahern and W. Rudin, "Factorizations of bounded holomorphic functions," Duke Math. J., 39, 767-777 (1972). ~n H. Alexander, "Proper holomorphic mappings in ," Indiana Univ. Math. J., 26, 137-146 (1977). F. Forelli, "Measures Whose Poisson integrals are pluriharmonic," Iii. J. Math., 18, 37, 373-388 (1974). F. Forelli, "Measures whose Poisson integrals are pluriharmonic. II," Iii. J. Math., 19, 584-592 (1975). W-? Rudin, "LP-isometries and equimeasurability," Indiana Univ. Math. J., 25, 215-228 (1976).