Singular Berezin Transforms

We give examples of pseudoconvex Reinhardt domains where the Berezin transform has integral kernel with singularities and, hence, fails to be a smoothing map. On the other hand, we show that this can never happen for a plane domain – in fact, then the Bergman kernel is always either identically zero or strictly positive everywhere on the diagonal – and also prove that, in contrast to the example by Wiegerinck from 1984, on any pseudoconvex Reinhardt domain the Bergman space can be finite-dimensional only if it reduces to the constant zero. Received: February 02, 2007. Accepted: May 28, 2007.