Weighted Berezin transform in the polydisc

For c > −1, let νc denote a weighted radial measure on C normalized so that νc(D) = 1. If f is harmonic and integrable with respect to νc over the open unit disc D, then D(f ◦ ψ)dνc = f (ψ(0)) for every ψ ∈ Aut(D). Equivalently f is invariant under the weighted Berezin transform; Bcf = f . Conversely, does the invariance under the weighted Berezin transform imply the harmonicity of a function? In this paper, we prove that for any 1 p <∞ and c1, c2 > −1, a function f ∈ Lp(D2, νc1 × νc2 ) which is invariant under the weighted Berezin transform; Bc1,c2f = f needs not be 2-harmonic. © 2007 Elsevier Inc. All rights reserved.