Iterates of a Berezin-type transform

Let BB be the open unit ball of RnRn and dV   denote the Lebesgue measure on RnRn normalized so that the measure of BB equals 1. Suppose f∈L1(B,dV)f∈L1(B,dV). The Berezin-type transform of f is defined byBf(x)=∫Bf(y)(1−|x|2)2(1−2x⋅y+|x|2|y|2)n/2+1dV(y),x∈B. We prove that if f∈C(B¯) then the iterates BkfBkf converge to the Poisson extension of the boundary values of f  , as k→∞k→∞. This can be viewed as a higher dimensional generalization of a previous result obtained independently by Engliš and Zhu.