Beurling type density theorems for weighted Lp spaces of entire functions

Abstract

Set $\Delta=\partial^{2} / \partial_{z} \partial_{\bar{z}}$ and let $\varphi$ be a subharmonic function satisfying $0<m \leq \Delta \varphi(z) \leq M$ for all $z \in \mathbb{C},$ where $m$ and $M$ are positive constants. $\|f\|_{\varphi, p}^{p}=\int_{\mathrm{C}}|f|^{p} e^{-p \theta} d \sigma$ for $p<\infty$ where $d \sigma$ is a Lebesgue area measure on $\mathrm{C}$, and $\|f\|_{\varphi, \infty}=\sup _{z}|f(z)| e^{-\varphi(z)} . \mathcal{F}_{\varphi, p}^{p}$ denotes a set of all analytic functions $f$ with $\|f\|_{\varphi, p}<\infty .$ For $\Gamma=\left\{\gamma_{n}\right\}$ a sequence of distinct points from $\mathbb{C}$ $\left\|\left.f\left|\Gamma \|_{\varphi, p}^{p}=\sum_{n}\right| f\left(\gamma_{n}\right)\right|^{p} e^{-p \varphi\left(\gamma_{n}\right)} \text { for } p<\infty, \text { and }\right\| f\left|\Gamma \|_{\varphi, \infty}=\sup _{n}\left[f\left(\gamma_{n}\right) | e^{-\varphi\left(\gamma_{n}\right)}, \text { where we permit } f\text { to be a function defined on }\right.\right.$ some set containing $\Gamma$. We say that the sequence $\Gamma$ is sampling for $\mathcal{F}_{\varphi}^{p}$ if w have $\|f | \Gamma\|_{\varphi, p} \sim\|f\|_{\varphi, p}$ for $f \in \mathcal{F}_{\varphi}^{p}$. For a fixed $\Gamma$, we denote by $n(z, r)$ the number of points of the sequence $\Gamma$ in $D(z, r),$ which is the disc of center $z$ and radius $r .$ Set $D_{\varphi^{\prime}}^{-}(\Gamma)=\liminf _{r \rightarrow \infty} \inf _{z \in \mathbb{C}} n(z, r) / \int_{D(z, r)} \Delta \varphi$ and $D_{\varphi^{\prime}}^{+}(\Gamma)=\lim \sup _{r \rightarrow \infty} \sup _{z \in \mathbb{C}} n(z, r) / \int_{D(z, r)} \Delta \varphi .$ The authors show the following: (1) If a sequence $\Gamma$ is sampling for $\mathcal{F}_{\varphi}^{p}$ then it contains a uniformly separated subseqence $\Gamma^{\prime}$ satisfying $D_{\varphi}^{-}\left(\Gamma^{\prime}\right)>2 / \pi .$ (2) If a sequence $\Gamma$ is interpolating for $\mathcal{F}_{\varphi}^{p}$ then it is uniformly separated and satisfies $D_{\varphi}^{+}(\Gamma)<2 / \pi$. In ( 1 ) and ( 2 ), it is known that the converses are also true.