Dirichlet级数的Hardy空间上的面积算子II：反例与紧性准则
Area operators on Hardy spaces of Dirichlet series II: counterexamples and compactness criteria
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中文总结 AI 辅助
本文否证了面积算子的Carleson型猜想，并完整刻画了其在Dirichlet级数Hardy空间上的有界性与紧性等价性，同时给出一般情形下紧性的充分条件及应用。
中文摘要 AI 辅助
我们研究由右半平面上的正Borel测度诱导、作用于Dirichlet级数的Hardy空间$\mathscr H^p$（$0<p<\infty$）上的面积算子$\mathbb{A}_{\mu,l}$（$0<l<\infty$）。首先，我们通过构造一个对所有$0<p,l<\infty$均有效的概率测度，否证了本文作者在先前工作中提出的一个猜想：尽管该测度不满足所提出的Carleson条件，其关联的面积算子却是有界的。接下来，我们研究这些算子的紧性。对于每个$0<p<\infty$，我们刻画了$\mathbb{A}_{\mu,p}$在$\mathscr H^p$及在$+\infty$处消失的Dirichlet级数Hardy空间$\mathscr H^p_0$上的有界性与紧性；特别地，对这些算子而言，有界性与紧性是等价的。对于一般的$0<p,l<\infty$，我们进一步利用消失的Carleson测度和紧的$H_{\mathrm i}^p$-Carleson嵌入建立了紧性的充分条件。作为应用，我们还给出了$\mathscr H^p$上以$\operatorname{VMOA}(\mathbb C_0)$中Dirichlet级数为符号的Volterra算子一个已知紧性结果的不同证明。
英文摘要
We study the area operators $\mathbb{A}_{μ,l}$, $0<l<\infty$, induced by positive Borel measures on the right half-plane and acting on the Hardy spaces of Dirichlet series $\mathscr H^p$, $0<p<\infty$. We first disprove a conjecture proposed by the present authors in an earlier work by constructing a probability measure, valid for all $0<p,l<\infty$, for which the associated area operator is bounded although the measure fails the proposed Carleson conditions. We then characterize the compactness of $\mathbb{A}_{μ,l}$ on both $\mathscr H^p$ and the Hardy space $\mathscr H^p_0$ of Dirichlet series vanishing at $+\infty$, for every $0<p,l<\infty$, via integrability conditions for the associated cone integrals. In particular, boundedness and compactness are equivalent for these operators. We also give direct proofs of compactness under vanishing Carleson measure and compact $H_{\mathbb i}^p$-Carleson embedding hypotheses. The direct proofs yield estimates that can be used for measures depending on the vertical limit character $χ$. As an application, we give a different proof of a known compactness result for Volterra operators on $\mathscr H^p$ with Dirichlet series symbols in $\operatorname{VMOA}(\mathbb C_0)$.
发表机构
- Shaanxi Normal University（陕西师范大学）
- Wuhan University（武汉大学）
- Hebei University of Technology（河北工业大学）
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