论文标题
由径向重量引起的希尔伯特型操作员
Hilbert-type operator induced by radial weight
论文作者
论文摘要
我们考虑由Hilbert型操作员定义的 $$ h_Ω（f）（z）= \ int_0^1 f（t）\ left（\ frac {1} {z} {z} \ int_0^z b^z b^z_______t（u）\，du \ right）\，ω（t）在单位光盘$ \ mathbb {d} $中，由径向重量$ω$引起的伯格曼空间$ a^2_Ω$复制核。我们证明$h_Ω$从$ h^\ infty $到bloch空间，并且仅当$ω$属于$ \ widehat {\ mathcal {d}} $ class $ \ mathcal {d}} $，它由径向重量$ω$组成，使双重状态$ \ \ sup_ {0 \ sup_ {0 \ sup_ {0 \ le r r <1} ω（s）\，ds} {\ int _ {\ frac {1+r} {2}}}^1Ω（s）\，ds} <\ infty $。此外，我们描述了\ wideHat {\ Mathcal {d}} $的权重$ω\，以便在hardy space $ h^1 $上限制了$h_Ω$，我们向任何$ω\ in \ in \ wideHat {\ mathcal {\ mathcal {d}}} $和$ p \ in（1，\ inforty）in（1，\ inffty in（1，\ inf），我们证明了这一点。 $h_Ω：\，l^p _ {[0,1）} \ to H^p $在且仅当Muckenhoupt类型条件时才有限 \ begin {equation*} \ sup \ limits_ {0 <r <1} \ left（1+ \ int_0^r \ frac {1} {\wideHatΩ（t）^p} dt \ right）^{\ frac {1} {1} \ left（\ int_r^1Ω（t）^{p'} \，dt \ right）^{\ frac {1} {p'}} <\ infty， \ end {equation*}保持。此外，我们解决了关于$h_Ω$在加权伯格曼空间$ a^p_ν$上的动作的类似问题。
We consider the Hilbert-type operator defined by $$ 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 $A^p_ν$.
