Interpolating sequences for weighted Bergman spaces of the ball

Abstract

Let $B_α^{p}$ be the space of $f$ holomorphic in the unit ball of $\Bbb C^n$ such that $(1-|z|^2)^αf(z) \in L^p$, where $0<p\leq\infty$, $α\geq -1/p$ (weighted Bergman space). In this paper we study the interpolating sequences for various $B_α^{p}$. The limiting cases $α=-1/p$ and $p=\infty$ are respectively the Hardy spaces $H^p$ and $A^{-α}$, the holomorphic functions with polynomial growth of order $α$, 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 $α$ and $p$ the inclusions between $B_α^{p}$ 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 $B_α^{p}$-interpolating, expressed in the same terms as the conditions given in previous works of Thomas for the Hardy spaces and Massaneda for $A^{-α}$. In particular we show, under some restrictions on $α$ and $p$, that finite unions of $B_α^{p}$-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&#39;s convenience in an appendix (§5).