Necessary and Sufficient Conditions for the Boundedness of Some Discrete Operators on Morrey Sequence Spaces

Abstract

The aim of this paper is to study the boundedness properties of the discrete Riesz potential, the fractional maximal operator, and their commutators on Morrey sequence spaces. We establish both necessary and sufficient conditions for the boundedness of the operators in the discrete Morrey framework. We also introduce the absolute higher-order commutators of the discrete Riesz potential and the fractional maximal operator. Moreover, we derive discrete analogues of the John–Nirenberg and Fefferman–Stein inequalities and employ them to prove the boundedness of the absolute higher-order commutators on Morrey sequence spaces. In addition, a necessary condition for the boundedness of the absolute higher-order commutators is obtained. As a consequence, we deduce the boundedness of the higher-order commutators of the discrete Riesz potential and the fractional maximal operator on Morrey sequence spaces, thereby extending several classical results from the continuous to the discrete setting.

Keywords

• Riesz potential
• fractional maximal operator
• commutator
• Lebesgue space
• Morrey space
• sequence space

References

1. [1] I. Gabisonija and A. Meskhi, Two-weighted inequalities for a discrete Hilbert transform, Proc. A. Razmadze Math. Inst. 116, 107-122, 1998