Division by inner functions
by
Daniel Girela
Universidad de Málaga
Coauthors: Cristóbal González (Universidad de Málaga)

deIm Im We denote by \Delta the unit disc {z in C : |z| < 1} and by H\sp p (0 < p <= \infty) the classical Hardy spaces of analytic functions in \Delta. A function I, analytic in \Delta, is said to be an inner funtion if I in H\sp\infty and has radial limits of absolute value 1 almost everywhere.

Definition A subspace X of H¹ is said to have the f-property if h/I in X whenever h in X and I is an inner function with h/I in H¹. 

This notion was introduced by Havin. The spaces H\sp p (1 <= p <= \infty), the Dirichlet space \Cal D, BMOA, VMOA, the Q_p-spaces (0 < p < 1) and many other spaces, such as various Lipschitz spaces, have the f-property.

One of the first examples of a space not possesing the f-property was given by Anderson (1979). The space in question is \Cal B₀ \cap H\sp \infty where \Cal B₀ is the little Bloch space. Anderson proved this using results of Shapiro and Kahane about the existence of certain singular measures. Is is worth noticing that Anderson's result can also be deduced using the fact that \Cal B₀ contains infinite Blaschke products and that any such Blaschke product contains a subproduct which does not belong to \Cal B₀. This suggests a way of finding spaces without the f-property. If X is a subspace of H\sp 1 which contains an infinite Blaschke product B in such a way that a certain subproduct B₁ od B does not lie in X then X does not have the f-property.

If \phi is a positive and increasing function defined in [0, 1) such that \phi(r) --> \infty, as r --> 1, we let \Cal L(1, \phi) denote the space of all functions f, analytic in \Delta, for which M₁(r, f')\overset --> = \int_-\pi\sp \pi|f'(re\spi\theta)|d\theta = \og (\phi(r)), as r --> 1. For certain functions \phi the space \Cal L(1, \phi) has the f-property. However, the author proved that for any \phi as above \Cal L(1, \phi) contains an interpolating Blaschke product with positive zeros. The following question arises:

Question. Is there some \phi so that if B is the interpolating Blaschke in \Cal L(1, \phi) just mentioned then B contains a certain subproduct which does not lie in \Cal L(1, \phi)?

Girela and C. González have recently proved that if B is an interpolating Blaschke product with positive zeros then M₁(r, f')\asymp n(r, B) (here, n(r, B) is the number of zeros of B in |z| < r). Using this result we:

(1) Give a new and simpler proof of the existence of infinite Blaschke products belonging to \Cal L(1, \phi).

(2) Show that the answer to the Question is negative.

In spite of this fact we can prove that indeed there exists a function \phi so that the space \Cal L(1, \phi) fails to have the f-property.

Date received: June 6, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahs-85.