Rhaly operators between Dirichlet-type spaces: a complete classification

RESUMO

\(F_\eta(z)=\sum_{n\ge0}\eta_nz^n\in\Hol(\D)\), and let \(\mathcal R_{(\eta)}\) be the associated Rhaly operator. We give a complete characterization of boundedness and compactness of \(\mathcal R_{(\eta)}:\mathcal D^p_\alpha\to\mathcal D^q_\beta\) for \(1<p,q<\infty\) and \(\alpha,\beta>-1\). The criteria are expressed in terms of the \(H^q\)-norms of the dyadic blocks of \(F_\eta\), and we obtain estimates for the operator and essential norms. As applications, we obtain boundedness and compactness criteria for Rhaly operators between weighted Bergman spaces and for Cesàro-type operators induced by positive measures, including logarithmic tail conditions for \(C_\mu:B^p\to B^p\), where \(B^p=\mathcal D^p_{p-2}\). We also construct, for each \(p>2\), a symbol \(F_\eta\in H^\infty\cap\lambda^p_{1/p}\) for which \(\mathcal R_{(\eta)}\) is not bounded on \(H^p\).