Iterates of a Berezin-type transform ✩

Let B be the open unit ball of Rn and dV denote the Lebesgue measure on Rn normalized so that the measure of B equals 1. Suppose f ∈ L1(B, dV). The Berezin-type transform of f is defined by Bf (x) = B f (y) (1 − |x|2)2 (1−2x · y + |x|2|y|2)n/2+1 dV (y), x ∈ B. We prove that if f ∈ C(B) then the iteratesBkf converge to the Poisson extension of the boundary values of f , as k→∞. This can be viewed as a higher dimensional generalization of a previous result obtained independently by Engliš and Zhu. © 2006 Elsevier Inc. All rights reserved.