Dirichlet型空间之间的Rhaly算子：一个完整的分类
Rhaly operators between Dirichlet-type spaces: a complete classification
- School of Mathematics and Statistics, Lingnan Normal University（岭南师范学院数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文完整刻画了Dirichlet型空间之间Rhaly算子的有界性与紧致性，以二进块H^q范数给出准则，并应用于加权Bergman空间及Cesàro型算子，还构造了使Rhaly算子无界的符号。
AI中文摘要：
设\\(F_\eta(z)=\sum_{n\ge0}\eta_nz^n\in\Hol(\D)\\)，并令\\(\mathcal R_{(\eta)}\\)为相应的Rhaly算子。我们给出了\\(\mathcal R_{(\eta)}:\mathcal D^p_\alpha\to\mathcal D^q_\beta\\)在\\(1<p,q<\infty\\)且\\(\alpha,\beta>-1\\)时有界性和紧致性的完整刻画。这些准则用\\(F_\eta\\)的二进块（dyadic blocks）的\\(H^q\\)-范数表示，并且我们获得了算子和本质范数的估计。作为应用，我们得到了加权Bergman空间之间Rhaly算子的有界性和紧致性准则，以及由正测度诱导的Cesàro型算子的相应准则，包括\\(C_\mu:B^p\to B^p\\)的对数尾部条件，其中\\(B^p=\mathcal D^p_{p-2}\\)。我们还对每个\\(p>2\\)构造了一个符号\\(F_\eta\in H^\infty\cap\lambda^p_{1/p}\\)，使得\\(\mathcal R_{(\eta)}\\)在\\(H^p\\)上无界。
英文摘要：
Let \(η=(η_n)_{n\ge0}\) be a complex sequence such that \(F_η(z)=\sum_{n\ge0}η_nz^n\in\Hol(\D)\), and let \(\mathcal R_{(η)}\) be the associated Rhaly operator. We give a complete characterization of boundedness and compactness of \(\mathcal R_{(η)}:\mathcal D^p_α\to\mathcal D^q_β\) for \(1<p,q<\infty\) and \(α,β>-1\). The characterization is formulated in terms of the \(H^q\)-norms of the dyadic blocks of \(F_η\). For \(-1<α<p-2\), boundedness and compactness coincide and reduce to \(F_η\in\mathcal D^q_β\). For \(α=p-2\) and for \(α>p-2\), boundedness is characterized by membership in analytic truncated Besov spaces and analytic Besov spaces, respectively; compactness has the same characterization for \(q<p\) and is characterized by the corresponding little spaces for \(p\le q\). We also obtain operator norm and essential norm estimates. As applications, we obtain boundedness and compactness characterizations for Rhaly operators between weighted Bergman spaces and for Cesàro-type operators induced by positive measures. In particular, we characterize \(C_μ:B^p\to B^p\), answering a question left open by Sun, Ye and Zhou and later noted by Tang. For each \(p>2\), we also construct a symbol \(F_η\in H^\infty\capλ^p_{1/p}\) for which \(\mathcal R_{(η)}\) is not bounded on \(H^p\).
