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 $\mathscr H^p$, $0<p<\infty$. 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 $\mathbb{A}_{\mu,p}$ 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 $\operatorname{VMOA}(\mathbb C_0)$.
Let $g$ be analytic in the unit disc and consider the generalized Hilbert operator $$ \mathcal{H}_g(f)(z)=\int_0^1 f(t)g'(tz)\, dt. $$ The boundedness of $\mathcal H_g$ on $H^p$ is characterized by the mean Lipschitz condition $g\in\Lambda\left(p,\frac{1}{p}\right)$ when $1<p\leq2$, while the problem remains open for $...
D. Norrbo, J. A. Pel'aez, Fang-Lei Wu· 2 citations· ⚡1
We quantify the failure of interpolation between Hardy spaces $H^p(\mathbb T^\infty)$ on the infinite-dimensional torus. For any set $A\subseteq[1,\infty]$ such that $A\setminus\{\infty\}$ is closed in $[1,\infty)$, we show that there exists an operator that is densely defined on $H^p(\mathbb T^\infty)$ for all $1\leq...
In this paper, we investigate the analytic properties of the Hardy-type operator $-\Delta + \mu H_0$ on the $d$-dimensional integer lattice graph $\mathbb{Z}^d$, where $\mu \geq -1$ and $H_0$ is a critical Hardy potential at infinity--arising naturally from discrete Hardy inequalities. We derive sharp fixed-pole Green...
We solve the local embedding problem for Hardy spaces of Dirichlet series, which is a dimension-free trace problem asking whether the global $\mathscr{H}^p$-norm controls local $L^p$-mass on the critical line $\operatorname{Re}s=1/2$. More precisely, for every $2<p<\infty$, there exists a constant $C_p<\infty$ such tha...
Bo-Nan Chen, Xiang Fang, Feng Guo et al.· 0 citations
Let $I=(a,b)$ be an open interval with finite or infinite endpoints. For $1<p_1,p_2<\infty$ and $0<q<\infty$, we study the weighted bilinear Hardy operator $ \mathcal H_2(f,g)(x)=(\int_a^x f)(\int_a^x g) $ from $L^{p_1}(v_1;I)\times L^{p_2}(v_2;I)$ to $L^q(u;I)$, including the quasi-Banach target range $0<q<1$. With $\...
We study closed range and essential norms of bounded composition operators $C_\varphi$ on weighted Dirichlet spaces $\mathcal{D}_{\alpha}$. For $-1<\alpha<0$, we first consider maps whose images are obtained by removing a compact subset from a simply connected subdomain of $\mathbb{D}$. In this setting, closed range is...
Cai-Xing Gu, Li He, Xiao-Feng Wang et al.· 0 citations
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