and Distance Estimates in BMO |

Joel H. Shapiro |

Michigan Math.
J. 34 (1987) 323--336 |

Abstract: This paper focuses on functions F holomorphic
or harmonic on the open unit disc U and having finite radial
limits at a.e. boundary point. the main result is that the essential
range of this boundary function coincides with the cluster set
of F if and only if F is the Poisson integral of a function of
vanishing mean oscillation. This implies that every function
of vanishing mean oscillation has connected essential range,
it recovers the well-known fact that among inner functions only
the Blaschke products can have vanishing mean oscillation, and
it has consequences for the algebra QC of quasi-continuous functions
on the unit circle. |

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