Discrete Riesz Potentials on Discrete Weighted Morrey Spaces

Abstract

Let $0<\alpha <1$. We obtain the boundedness of the discrete fractional Hardy-Littlewood maximal operators ${\mathcal{M}}_{\alpha }$ on discrete weighted Lebesgue spaces. From this and a discrete version of Whitney decomposition theorem, we deduce the boundedness of the discrete Riesz potentials $I_{\alpha }$ on discrete weighted Lebesgue spaces. The boundedness of $I_{\alpha }$ on discrete weighted Morrey spaces is further obtained. Moreover, the boundedness of ${\mathcal{M}}_{\alpha }$ is also obtained which is new even for unweighted case.