On the discrete 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 $1\le p\le q\le \infty$. 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 $1\le p\le q\le \infty$. 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.

Another author derives more general theorems than ours by employing a simpler method