Cluster set, essential range and distance estimates in BMO
Let F be a function - usually harmonic or holomorphic and satisfying a growth condition such as ``bounded characteristic'' - on the unit disc. Assume that F has finite radial limit at almost every boundary point of the disc, and denote the boundary function by \(F^*\). This paper studies the relationship between the cluster set of F and the essential range of \(F^*\). In particular, when do these two sets coincide? One main result is that when F is the Poisson integral of a function of vanishing mean oscillation (VMO in the sense of Sarason) the two sets do indeed coincide. A consequence is that any function of vanishing mean oscillation has connected essential range. In addition, one obtains a new proof of the fact that the only inner functions u with boundary function \(u^*\) of bounded mean oscillation are the finite Blaschke products. Other function algebraic consequences are derived as well. Central to the proof of these results is the following estimate: the distance from \(F^*\) to the space VMO is bounded below by the Hausdorff distance between the cluster set and the essential range. Additional results along the above lines are obtained on the ball in \({\mathbb{C}}^ n\). This paper is a pleasure to read.
scientific article; zbMATH DE number 4032049