A note about the discrete Riesz potential on ${\mathbb{Z}}^n$

Abstract

In this note we prove that the discrete Riesz potential $I_{\hat{I}}\pm$ defined on ${\mathbb{Z}}^n$ is a bounded operator $H^p({\mathbb{Z}}^n)\to {\ell }^q({\mathbb{Z}}^n)$ for $0<p\le 1$ and $\frac{1}{q}=\frac{1}{p}-\frac{\hat{I}}{\pm }n$, where $0<\hat{I}\pm <n$.

10 pages