Boundeness of a family of Hilbert-type operators and of its Bergman-type analogue

TL;DR

This paper investigates the boundedness and exact norms of a family of Hilbert-type and Bergman-type operators, providing precise conditions and results for their boundedness in specific parameter regimes.

Contribution

It introduces a generalized family of Hilbert and Bergman-type operators, establishing their boundedness criteria and calculating exact norms in certain cases.

Findings

Exact norm formulas for diagonal case operators

Necessary and sufficient conditions for boundedness of Bergman-type operators

Boundedness criteria in the upper triangle case

Abstract

In this note, we frst consider boundedness properties of a family of operators generalizing the Hilbert operator in the upper triangle case. In the diagonal case, we give the exact norm of these operators under some restrictions on the parameters. We secondly consider boundedness properties of a family of positive Bergman-type operators of the upper-half plane. We give necessary and sufficient conditions on the parameters under which these operators are bounded in the upper triangle case.