On a theorem of Privalov and normal functions |
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| Authors: | Daniel Girela |
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| Affiliation: | Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain |
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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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| Keywords: | Normal functions Hardy spaces integral means theorem of Privalov |
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