Area operators on Hardy spaces of Dirichlet series II: counterexamples and compactness criteria

Abstract

We study the area operators ${\mathbb{A}}_{\mu ,l}$, $0<l<\infty$, induced by positive Borel measures on the right half-plane and acting on the Hardy spaces of Dirichlet series ${\mathcal{H}}^p$, $0<p<\infty$. 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 next investigate compactness of these operators. For every $0<p<\infty$, we characterize boundedness and compactness of ${\mathbb{A}}_{\mu ,p}$ on both ${\mathcal{H}}^p$ and the Hardy space ${\mathcal{H}}_0^p$ of Dirichlet series vanishing at $+\infty$; in particular, boundedness and compactness coincide for these operators. For general $0<p,l<\infty$, we further establish sufficient conditions for compactness in terms of vanishing Carleson measures and compact $H_{\mathrm{i}}^p$-Carleson embeddings. As an application, we also give a different proof of a known compactness result for Volterra operators on ${\mathcal{H}}^p$ with Dirichlet series symbols in $\mathrm{VMOA}({\mathbb{C}}_0)$.