Weighted Berezin transform in the polydisc

초록

For c > -1, let upsilon(c) denote a weighted radial measure on C normalized so that upsilon(c)(D) = 1. If f is harmonic and integrable with respect to upsilon(c) over the open unit disc D, then integral(D)(f o psi) d upsilon(c) = f (psi(0)) for every psi epsilon Aut(D). Equivalently f is invariant under the weighted Berezin transform; B(c)f = 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 < infinity and c(1), c(2) > -1, a function f epsilon L-p (D-2, upsilon c(1) x upsilon c(2)) which is invariant under the weighted Berezin transform; B-c1,B-c2 f = f needs not be 2-harmonic. (C) 2007 Elsevier Inc. All rights reserved.

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