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

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 next investigate compactness of these operators. For every $0<p<\infty$, we characterize boundedness and compactness of $\mathbb{A}_{μ,p}$ on both $\mathscr H^p$ and the Hardy space $\mathscr H^p_0$ 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 $\mathscr H^p$ with Dirichlet series symbols in $\operatorname{VMOA}(\mathbb C_0)$.