Functions invariant under the Berezin transform
The author shows that a bounded function \(f\) satisfies \(Bf= f\), where \(B\) is the Berezin transform on the unit disc, if and only if \(f\) is harmonic. There is an equivalent formulation of this result [\textit{S. Axler} and \textit{Ž. Čučković}, Integral Equations Oper. Theory 14, No. 1, 1-12 (1991; Zbl 0733.47027); \textit{W. Rudin}, ``Function theory in the unit ball of \(\mathbb{C}^ N\)'' (1980; Zbl 0495.32001)]: If \(f\) is bounded and satisfies the invariant version of the area mean value property, then \(f\) is harmonic. The main tool employed is Fourier analysis on the Lie group of all Möbius transformations.
invariant version of the area mean value property
scientific article; zbMATH DE number 599764