Estimates of discrete Riesz potentials on discrete weighted Lebesgue spaces

Abstract

Let \(0<\alpha <1\). We obtain necessary and sufficient conditions for the boundedness of the discrete fractional Hardy–Littlewood maximal operators \(\mathcal {M}_\alpha \) on discrete weighted Lebesgue spaces. From this and a discrete variant of the Whitney decomposition theorem, necessary and sufficient conditions for the boundedness of the discrete Riesz potentials \(I_\alpha \) on discrete weighted Lebesgue spaces are discussed. As an application, the boundedness of \(I_\alpha \) on discrete weighted Morrey spaces is further obtained.