Berezin transform, Mellin transform and Toeplitz operators (Q371639)

Let \(B\) be the Berezin transform associated with the Bergman space. The authors of the article under review improve a theorem of \textit{P. Ahern} [J. Funct. Anal. 215, No. 1, 206--216 (2004; Zbl 1088.47014)]. Namely, they show that, if \(u\in L^1\) and \(Bu\) is a harmonic function, then \(u\) itself is harmonic. The authors also extend another theorem of Ahern.