Volterra type operators on weighted Dirichlet spaces

The Carleson measures for weighted Dirichlet spaces had been characterized by Girela and Pel'{a}ez, who also characterized the boundedness of Volterra type operators between weighted Dirichlet spaces. However, their characterizations for the boundedness are not complete. In this paper, we completely characterize the boundedness and compactness of Volterra type operators from the weighted Dirichlet spaces $D_{\alpha }^p$ to $D_{\beta }^q$ ($-1<\alpha ,\beta$ and $0<p<q<\mathrm{\infty }$), which essentially complete their works. Furthermore, we investigate the order boundedness of Volterra type operators between weighted Dirichlet spaces.