Two Conjectures by Albert Baernstein II

3.9.

TWO CONJECTURES BY ALBERT BAERNSTEIN

II*

In [I] I proved a factorization theorem for zero-free univalent disk ~ F(O) = I .

Let So denote the set of all functions F analytic and I--I in ~

THEOREM lytic in

functions

~

I.

If ~ o

, then, for each ~

,~(0,O

in the unit

with O ~ F ( ~ ) ,

, there exist functions B and Q ana-

such that

where 5 r

, and

laa,~O.14~.

The "Koebe function" for the class So is k(z) = [(I + z)/(1 -- z)] 2 which maps the slit plane Theorem I . Conjecture

[W~'l~Wl4~] I.

9

~

onto

This suggests that it might be possible to let I § I in

If FG~o , then there exist functions B and Q analytic in ~

such that

F(~).= B(;~)Q(~), zeD, where

5 ~ "4,,I/S~H",

and I ~ I < U .

We do not insist that B or Q be univalent, nor that Q(0) = i. However, when the functions are adjusted so that IQ(0) 1 = I, then ilBll~ and llB-Zlloo should be bounded independently of F. Using the fact that QI/2 has positive real coefficients {a n } of Q satisfy Janl ~< 4n, n >i wood's conjecture asserts that this inequality A proof of Conjecture I could possibly tell us wood's conjecture, and this in turn might lead Bieberbach's conjecture.

part, it is easy to show that the power series I, with equality when Q(z) = k(z). Littleis true for coefficients of functions in So. something new about how to attempt Littleto fresh ideas about h o w t o prove (the stronger)

Theorem I is easily deduced from a decomposition theorem obtained by combining results of Helson and Szeg~ [2] and Hunt, Muckenhoupt,

and Wheeden

[3].

Suppose

J-~Li(l') ' and f real

valued Consider the zero-free analytic function F defined by ~ ( ~ ) = e ~ ( ~ ) + ~ ( % ) ) , ~ c 9 where f(z) denotes the harmonic extension of f(e ie) and f the conjugate of f. Also, let S(F) denote the set of all functions obtained by "hyperbolically translating" F and then normalizing,

s(F): { and let HP denote the usual Hardy space. following way. THEOREM 2.

For ~ L i ~ [ )

(I) f = uz + ue where (2) S(F) U

S(I/F)

Theorem I follows,

Part of Theorem I of [3] can be phrased in the

the following are equivalent. I&~,~2EL~[)

is a bounded

and llu211~ < ~/2.

subset of H I.

since F I/e satisfies

(2) when

~

and 0 < I < I.

Theorem 2 may be regarded as a sharpened form of the theorem of Fefferman and Stein [4], which asserts that f = uz + u2 for some pair of bounded functions if and only if f is of bounded mean oscillation. To obtain Conjecture I in the same fashion as Theorem |, we need a result like Theorem 2 in which the <~/2 of (I) is replaced by ~ / 2 . Consideration of F(z) = (I + z)/(1 -- z) leads to the following guess. 9 ALBERT BAERNSTEIN II.

Washington University,

St. Louis, MO 63130.

2255.

Conjecture 2.

For

~ m

(I') f = ul + u2 where (2') S(F) U Statement

the following are equivalent. ~i,~r

and llu2]]~~ ~/2.

S(]/F) is a bounded subset of weak H I.

(2') means the following:

There is a constant C such that for every ~, ~ +

, and

every 0 , ~ U ~ ( 4 / F )

It is not hard (2') § (1') is

to prove, using subordination, that t r u e , t h e n so i s C o n j e c t u r e 1.

(1')

implies

(2').

If

the

implication

Condition (2') can be restated in a number of e q u i v a l e n t ways. We m e n t i o n one w h i c h i s c l o s e l y r e l a t e d t o t h e s u b h a r m o n i e m a x i m a l t y p e f u n c t i o n u s e d by t h e a u t h o r i n [5] and e l s e where. (2") There is a constant C such that

9

~E

~+e~ ]~e +ClEI I

for every measurable set For

~e~

~,E=T

, and every

~,~ED

9

, Theorem 6 of [5] asserts that (2") holds with C = 0.

In both the Fefferman--Stein and Helson--Szeg8 theorems the splitting f complished via duality and pure existence proofs from functional analysis. considerable interest if, given f, f r BMO, one could show how to actually bounded functions ul and u2. We remark that if f r BMO then some constant satisfies (2").

= ul § u2 is acIt would be of construct the multiple of f

I can prove that (2") § (I') provided we assume also that f is monotone on there exist 0z < 02 < 01 + 2~ such that

T

, i.e.,

By composing with a suitable MSbius transformation, we may assume 01 = 0, 0 2 = ~. Then, when C = 0, u2 can be constructed as follows. Let 0 ~ (0, ~) and x ~ (--I, I) be related by (I + x)(1 -- x) -I = 11 + ei81"11 -- eiSl -I Let V be the harmonic function in ~ with boundary values V(e i0) = f(x), 0 < 8 < ~, and V(e -i0) = V(eiS). Then it turns out that IVI ~ ~/2 and f -- V = 0(I), so that u2 =--V gives us (I'). It follows that Conjecture I is true for functions F, F complement of a "monotone slit."

~

So, which map

~

onto the

LITERATURE CITED I. 2. 3. 4. 5.

2256

A. Baernstein II, "Univalence and bounded mean oscillation," Mich. Math. J., 23, 217223 (1976). H. Helson and G. SzegS, "A problem of prediction theory," Ann. Mat. Pure Appl., 51, (4), 107-138 (1960). R. Hunt, B. Muckenhoupt, and R. Wheeden, "Weighted norm inequalities for the conjugate function and Hilbert transform," Trans. Am. Math. Soc., 176, 227-251 (1973). C. Fefferman and E. M. Stein, "HP spaces of several variables," Acta Math., 129, 137-193 (1972). A. Baernstein II, "Integral means, univalent functions and circular symmetrization," Acta Math., 133, 139-169 (1974).