On a theorem of Privalov and normal functions

Author:
Daniel Girela

Journal:
Proc. Amer. Math. Soc. **125** (1997), 433-442

MSC (1991):
Primary 30D45, 30D55

DOI:
https://doi.org/10.1090/S0002-9939-97-03544-2

MathSciNet review:
1363422

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Abstract: A well known result of Privalov asserts that if is a function which is analytic in the unit disc , then has a continuous extension to the closed unit disc and its boundary function is absolutely continuous if and only if belongs to the Hardy space . In this paper we prove that this result is sharp in a very strong sense. Indeed, if, as usual, we prove that for any positive continuous function defined in with , as , there exists a function analytic in which is not a normal function and with the property that , for all sufficiently close to .

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Additional Information

**Daniel Girela**

Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain

Email:
Girela@ccuma.sci.uma.es

DOI:
https://doi.org/10.1090/S0002-9939-97-03544-2

Keywords:
Normal functions,
Hardy spaces,
integral means,
theorem of Privalov

Received by editor(s):
November 1, 1994

Received by editor(s) in revised form:
June 25, 1995

Additional Notes:
This research has been supported in part by a D.G.I.C.Y.T. grant (PB91-0413) and by a grant from “La Junta de Andalucía”

Communicated by:
Albert Baernstein II

Article copyright:
© Copyright 1997
American Mathematical Society