On the discrete Hilbert-type operators

TL;DR

This paper investigates the boundedness of discrete Hilbert-type operators on weighted sequence spaces, providing necessary and sufficient conditions, and sharp norm estimates, highlighting differences from integral operator cases.

Contribution

It introduces new criteria for the boundedness of discrete Hilbert-type operators and extends previous integral operator results to the discrete setting with sharper estimates.

Findings

Established necessary and sufficient conditions for $l^{p}-l^{q}$ boundedness.

Found sharp norm estimates for specific cases.

Highlighted differences between discrete and integral Hilbert-type operators.

Abstract

Recently, Bansah and Sehba studied in [3] the boundedness of a family of Hilbert-type integral operators, where they characterized the $L^{p}-L^{q}$ boundedness of the operators for $1\leq p\leq q\leq \infty$. In this paper, we deal with the corresponding discrete Hilbert-type operators acting on the weighted sequence spaces. We establish some sufficient and necessary conditions for the $l^{p}-l^{q}$ boundedness of the operators for $1\leq p\leq q\leq \infty$. We find out that the conditions of the boundedness of discrete Hilbert-type operators are different from those of the boundedness of Hilbert-type integral operators. Also, for some special cases, we obtain sharp norm estimates for discrete Hilbert-type operators. Finally, it is pointed out that certain extensions of the theorems given in [3] can be established by using our different arguments.