Generalized Hilbert Operator Acting on Hardy Spaces
Huiling Chen, Shanli Ye
TL;DR
The paper characterizes measures $\mu$ for which the generalized Hilbert operator $\mathcal{H}_{\mu,\alpha}$ acts boundedly or compactly between Hardy spaces, and it determines the operator's essential norm. By representing $\mathcal{H}_{\mu,\alpha}$ via a Hankel-type matrix and leveraging Carleson-type measures, Fefferman duality, and integral representations, the authors derive sharp Carleson-type criteria for various ranges of $p$ and $q$, including $0<p\le1$ and $1\le p\le2$, and identify the Hilbert-Schmidt class on $H^2$. The essential norm is shown to be governed by tail measures $\mu([s,1))$ scaled by exponents that depend on $p$, $q$, and $\alpha$, yielding precise asymptotics as $s\to1^-$. Together, these results advance the operator-theoretic understanding of generalized Hilbert operators on Hardy spaces and connect boundedness, compactness, and tail behavior of the underlying measure.
Abstract
Let $α>0$ and $μ$ be a positive Borel measure on the interval $[0,1)$. The Hankel matrix $\mathcal{H}_{μ,α}=(μ_{n,k,α})_{n,k\ge0}$ with entries $μ_{n,k,α}=\int_{[0,1)}^{}\frac{Γ(n+α)}{Γ(n+1)Γ(α)}t^{n+k}dμ(t)$, induces, formally, the generalized-Hilbert operator as $$ \mathcal{H}_{μ,α}\left ( f \right ) \left ( z \right ) =\sum_{n=0}^{\infty} \left (\sum_{k=0}^{\infty} μ_{n,k,α}a_k \right )z^n,z\in\mathbb{D} $$ where $f(z)={\textstyle \sum_{k=0}^{\infty }} a_kz^k$ is an analytic function in $\mathbb{D}$. This article is devoted study the measures $μ$ for which $\mathcal{H}_{μ,α}$ is a bounded(resp., compact) operator from $H^p(0<p\le1)$ into $H^p(1\le q<\infty)$. Then, we also study the analogous problem in the Hardy spaces $H^p(1\le p\le2)$. Finally, we obtain the essential norm of $\mathcal{H}_{μ,α}$ from $H^p(0<p\le1)$ into $H^p(1\le q<\infty)$.