Interpolating sequences for weighted Bergman spaces of the ball.

Abstract

Let $B_{\alpha}^{p}$ be the space of $f$ holomorphic in the unit ball of $\Bbb C^n$ such that $(1-|z|^2)^\alpha f(z) \in L^p$, where $0<p\leq\infty$, $\alpha\geq -1/p$ (weighted Bergman space). In this paper we study the interpolating sequences for various $B_{\alpha}^{p}$. The limiting cases $\alpha=-1/p$ and $p=\infty$ are respectively the Hardy spaces $H^p$ and $A^{-\alpha}$, the holomorphic functions with polynomial growth of order $\alpha$, which have generated particular interest. In §1 we first collect some definitions and well-known facts about weighted Bergman spaces and then introduce the natural interpolation problem, along with some basic properties. In §2 we describe in terms of $\alpha$ and $p$ the inclusions between $B_{\alpha}^{p}$ spaces, and in §3 we show that most of these inclusions also hold for the corresponding spaces of interpolating sequences. §4 is devoted to sufficient conditions for a sequence to be $B_{\alpha}^{p}$-interpolating, expressed in the same terms as the conditions given in previous works of Thomas for the Hardy spaces and Massaneda for $A^{-\alpha}$. In particular we show, under some restrictions on $\alpha$ and $p$, that finite unions of $B_{\alpha}^{p}$-interpolating sequences coincide with finite unions of separated sequences. In his article in Inventiones, Seip implicitly gives a characterization of interpolating sequences for all weighted Bergman spaces in the disk. We spell out the details for the reader's convenience in an appendix (§5).