Hilbert-type operator induced by radial weight

TL;DR

The paper studies the Hilbert-type operator $H_\omega$ induced by radial Bergman kernels in the unit disk, establishing precise weight-structure criteria for boundedness across Hardy and Bergman spaces. Central to the results is the doubling class $\widehat{\mathcal{D}}$ of weights, which exactly characterizes $H_\omega: H^{\infty} \to \mathcal{B}$ and governs Hardy-space boundedness via a Muckenhoupt-type inequality involving $\omega$ and its tail $\widehat{\omega}$. The authors derive sharp kernel estimates for $K_t^{\omega}$ and $G_t^{\omega}$, enabling equivalences between boundedness on $H^p$, $HL(p)$, and $D^p_{p-1}$ and the two-weight conditions, with analogous Bergman-space results. They also connect $H_\omega$ to the Bergman projection and its maximal version, showing a unified picture where operator boundedness across these spaces reduces to a common set of weight criteria, extending classical Hilbert-operator theory to weighted Bergman contexts.

Abstract

We consider the Hilbert-type operator defined by $$ H_ω(f)(z)=\int_0^1 f(t)\left(\frac{1}{z}\int_0^z B^ω_t(u)\,du\right)\,ω(t)dt,$$ where $\{B^ω_ζ\}_{ζ\in\mathbb{D}}$ are the reproducing kernels of the Bergman space $A^2_ω$ induced by a radial weight $ω$ in the unit disc $\mathbb{D}$. We prove that $H_ω$ is bounded from $H^\infty$ to the Bloch space if and only if $ω$ belongs to the class $\widehat{\mathcal{D}}$, which consists of radial weights $ω$ satisfying the doubling condition $\sup_{0\le r<1} \frac{\int_r^1 ω(s)\,ds}{\int_{\frac{1+r}{2}}^1ω(s)\,ds}<\infty$. Further, we describe the weights $ω\in \widehat{\mathcal{D}}$ such that $H_ω$ is bounded on the Hardy space $H^1$, and we show that for any $ω\in \widehat{\mathcal{D}}$ and $p\in (1,\infty)$, $H_ω:\,L^p_{[0,1)} \to H^p$ is bounded if and only if the Muckenhoupt type condition \begin{equation*} \sup\limits_{0<r<1}\left(1+\int_0^r \frac{1}{\widehatω(t)^p} dt\right)^{\frac{1}{p}} \left(\int_r^1 ω(t)^{p'}\,dt\right)^{\frac{1}{p'}} <\infty, \end{equation*} holds. Moreover, we address the analogous question about the action of $H_ω$ on weighted Bergman spaces $A^p_ν$.

Theorems & Definitions (20)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma A
• Lemma B
• Lemma 5
• proof
• Lemma 6
• proof