Generalized Hilbert operator acting on Bergman spaces

Pengcheng Tang; Xuejun Zhang

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arXiv · 2208.10747

Generalized Hilbert operator acting on Bergman spaces

Pengcheng Tang·Xuejun Zhang

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Abstract

Let $μ$ be a positive Borel measure on $[0,1)$. If $f \in H(\mathbb{D})$ and $α>-1$, the generalized integral type Hilbert operator defined as follows: $$\mathcal{I}_{μ_{α+1}}(f)(z)=\int^1_{0} \frac{f(t)}{(1-tz)^{α+1}}dμ(t), \ \ \ z\in \mathbb{D} .$$ The operator $\mathcal{I}_{μ_{1}}$ has been extensively studied recently. In this paper, we characterize the measures $μ$ for which $\mathcal{I}_{μ_{α+1}}$ is a bounded (resp., compact) operator acting between the Bloch space $\mathcal {B}$ and Bergman space $ A^{p}$, or from $A^{p}(0 -1$.

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Pengcheng Tang, Xuejun Zhang. 2024-12-24. Generalized Hilbert operator acting on Bergman spaces. https://arxiv.org/abs/2208.10747

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