A note on a theorem of Zhu on weighted bergman projections on the polydisc

Abstract

We prove the following result: Suppose f is holomorphic in Dn and α = (α1-,..., αn), with α > -1 for j = 1,..., n. Then (Formula Presented) for some φ ∈ L∞ (Dn), if and only if the functions (Formula Presented) are bounded for every S ⊆ {1, 2,... ,n}, where Xs(•) denotes the characteristic function of the set S, and |S| is the cardinal number of S. This result improves Theorem 4 in the paper K. Zhu, Weighted Bergman projections on the polydisc, Houston J. Math. 20 (2) (1994), 275-292.