Rhaly operators between Dirichlet-type spaces: a complete classification

$F_{\eta }(z)={\sum }_{n\ge 0}{\eta }_n{z}^n\in \text{Hol}(\text{D})$, and let ${\mathcal{R}}_{(\eta )}$ be the associated Rhaly operator. We give a complete characterization of boundedness and compactness of ${\mathcal{R}}_{(\eta )}:{\mathcal{D}}_{\alpha }^p\to {\mathcal{D}}_{\beta }^q$ for $1<p,q<\mathrm{\infty }$ 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 $B^p={\mathcal{D}}_{p-2}^p$. We also construct, for each $p>2$, a symbol $F_{\eta }\in H^{\mathrm{\infty }}\cap {\lambda }_{1\mathrm{/}p}^p$ for which ${\mathcal{R}}_{(\eta )}$ is not bounded on $H^p$.