On a particular integral operator

TL;DR

This paper studies a specific integral operator, transforming it into weighted composition operators, and establishes boundedness criteria on weighted Dirichlet spaces, extending and unifying previous results in the field.

Contribution

It introduces a new boundedness criterion for a class of integral operators on weighted Dirichlet spaces, using transformations to weighted composition operators.

Findings

Established boundedness criteria for the integral operator on weighted Dirichlet spaces

Unified previous results by extending the class of bounded operators

Applied the criteria to specific integral operators to demonstrate boundedness

Abstract

We consider an integral operator $\mathcal{I}$, special instances of which was studied in various contexts. Using an appropriate transformation we write this operator in terms of weighted composition operators. Then, we provide a boundedness criterion on weighted Dirichlet spaces, and we apply this result in order to prove that certain integral operators are bounded on that spaces, unifying this way and extending previous results.