Cluster Set, Essential Range,
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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