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

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