Tauberian theorems for pluriharmonic functions which are BMO or Bloch (Q1820899)

The main result is a necessary and sufficient condition in order that a pluriharmonic function f in the unit ball of \({\mathbb{C}}^ n\), \(n\geq 2\), admits a radial limit at a given boundary point. The function f is supposed to be in the class BMO or in a Bloch space, and the NASC is that certain averages of f converge; hence the title of the paper. The argument proceeds by reducing the averages involved from the boundary of the unit ball in \(C^ n\) to an open set of C where most of the hard work is done. An interesting quantitative version of Wiener's Tauberian Theorem is proved.