A Carleson-type condition for interpolation in Bergman spaces

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An analogue of the notion of uniformly separated sequences, expressed in terms of canonical divisors, is shown to yield a necessary and sufficient condition for interpolation in the Bergman space Ap, 0 0 such that ∏ j:j 6=k ∣∣∣∣ zj − zk 1− zjzk ∣∣∣∣ ≥ δ for all k. (1) Carleson’s result can be extended to the Hardy space H, which, for 0 < p < ∞ consists of the functions f analytic in D with ‖f‖pHp = sup 0<r<1 1 2π ∫ 2π 0 |f(reiθ)|pdθ < ∞. We say that Γ is interpolating for H if the interpolation problem f(zj) = aj has a solution f ∈ H for every sequence {aj} with {aj(1− |zj |2)1/p} ∈ `. Shapiro and Shields [15] proved that (1) is necessary and sufficient for Γ to be interpolating for H, 1 ≤ p < ∞, and Kabǎila [10] obtained the same result for 0 < p < 1. Consider now the Bergman space A, which is the set of functions f analytic in D with ‖f‖p = 1 π ∫ D |f(z)|pdA(z) < ∞, where dA denotes Lebesgue area measure. Γ is interpolating for A if for every sequence {aj} satisfying {aj(1 − |zj |2)2/p} ∈ `, there is an f ∈ A with f(zj) = aj . Interpolation sequences for the growth space A−n as well as for A have been characterized completely in [13] using a density condition which will be defined shortly. The purpose of this paper is to identify a condition for interpolation in the Bergman spaces, which, though less 1991 Mathematics Subject Classification. 30H05, 46E15.