论文标题
在伯格曼空间上行动的普通希尔伯特操作员
Generalized Hilbert operator acting on Bergman spaces
论文作者
论文摘要
令$μ$为$ [0,1）$的正borel量度。如果$ f \ in H（\ Mathbb {d}）$和$α> -1 $，则定义如下： $$ \ MATHCAL {i} _ {μ_{μ_{α+1}}（f）（z）= \ int^1_ {0} \ frac {f（t）} {（1-TZ）^{α+1}}} $ \ mathcal {i} _ {μ_{1}} $最近已被广泛研究。在本文中，我们表征了$ \ Mathcal {i} _ {μ_{μ_{α+1}} $的度量$μ$，是一个有限的（sesp。，紧凑）的操作员在bloch space $ \ nathcal $ \ nathcal {b} $和bergman space and bergman space $ a^p} $ a^$ a^$ a^$ a^p}之间。 a^{q}（q \ geq 1）$。我们还研究了伯格曼空间中的类似问题$ a^{p}（1 \ leq p \ leq 2）$。最后，我们确定所有$α> -1 $的$ a^{2} $上的Hilbert-Schmidt类。
Let $μ$ be a positive Borel measure on $[0,1)$. If $f \in H(\mathbb{D})$ and $α>-1$, the generalized integral type Hilbert operator defined as follows: $$\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 $A^{p}(0<p<\infty)$ into $ A^{q}(q\geq 1)$. We also study the analogous problem in Bergman spaces $A^{p}(1 \leq p\leq 2)$. Finally, we determine the Hilbert-Schmidt class on $A^{2}$ for all $α>-1$.
