Estimates of discrete Riesz potentials on discrete weighted Lebesgue spaces

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