Algebras contained within H

5.9.

ALGEBRAS CONTAINED WITHIN H ~*

D=DuT

Let A = {f:f analytic in D , f continuous in tained within H ~176 but there are two intermediate algebras we require some notation. Let B denote the Banach space of functions

Then A is an algebra conthat present some interest. First

f, analytic

in ~

for which the norm

II~IB=I~-(o)I su.p(4-1zf) IJ:'(z)l +

iZl.~,l

is finite.

This is called the Bloch

For a survey of these spaces see [I]. a) ~ o ~

; b) ~ o

space.

.

We also define

The following facts are easily established:

; c) H~o'~---~X

is a subalgebra of H ~.

Similarly we define BMOA (analytic functions of bounded mean osoilT.ation) space of those functions

f, analytic

in m

to be the

, for which the norm

I~I<~ is finite.

Here

II. I12 is the ordinary H 2 norm and

Similarly

The space VMOA consists of those analytic functions in ~ whose boundary values on T vanishing mean oscillation (see [2, p. 591]). It is also easy to see that d) H ~ M 0 ~

; e)H~MOA

It is not difficult

; f) ~ V ~ 0 ~ Y

to establish

is a subalgebra

the following relation

~Y~X~H

have

of H ~

(see, e.g.,

[3])

~.

The algebra X has already been studied. It was shown by Behrens, unpublished, that X consists precisely of those f, f C H ~, whose Gelfand transform f is constant on all the nontrivial Gleason parts of the maximal ideal space of H ~. It is also known [3] that X does not possess the f property or K property in the sense of Havin [4]. It would be nice to have a similar study made of Y. The space Y cannot contain any inner functions [3], other than finite Blaschke products, in contrast to X. But Y does, of course, contain functions having an inner factor -- e.g., the function of [5, p. 29] belongs to A. LITERATURE I ~

2. 3.

CITED

J. M. Anderson, J. Clunie, and Ch. Pommerenke, "On Bloch functions and normal functions," J. Reine Angew. Math., 270, 12-37 (1974). Ch. Pommerenke, "Schlichte Funktionen und analytische Funktionen von beschr~nkter mittlerer Oszillation," Comment. Math. Helvo, 52, 591-602 (1977). J. M. Anderson, "On division by inner factors" (to appear).

*J. M. ANDERSON.

University College, London WCI, England.

2259

.

5.

2260

V. P. Khavin,"On the factorization of analytic functions smooth up to the boundary," Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR, 2_22, 202-205 (1971). V. P. Gurarii, "The factorization of absolutely convergent Taylor series and Fourier integrals," Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR, 3__0, 15-32 (1972).