# Volterra type operators on weighted Dirichlet spaces

## TL;DR

This paper fully characterizes when Volterra type operators are bounded or compact between weighted Dirichlet spaces, advancing understanding of their operator theory and related measures.

## Contribution

It provides complete criteria for boundedness and compactness of Volterra operators on weighted Dirichlet spaces, extending prior partial results.

## Key findings

- Complete characterization of boundedness of Volterra operators
- Complete characterization of compactness of Volterra operators
- Analysis of order boundedness of Volterra operators

## Abstract

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 $0<p<q<\infty$), which essentially complete their works. Furthermore, we investigate the order boundedness of Volterra type operators between weighted Dirichlet spaces.