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

Abstract

We study 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 $\mathscr H^p$, $0<p<\infty$. 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 boundedness and compactness of $\mathbb{A}_{\mu,l}$ 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, the boundedness and compactness are equivalent for these operators. We also give direct proofs of compactness under vanishing Carleson measure and compact $H_{\mathrm i}^p$-Carleson embedding hypotheses. As an application, we establish an estimate for the essential norms of bounded Volterra operators. This leads to a complete characterization of compactness for Volterra operators on $\mathscr{H}^p$ and provides a different proof of compactness for those with Dirichlet series symbols in $\operatorname{VMOA}(\mathbb C_0)$.