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 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.