The $H^p({\mathbb{Z}}^n)-H^q({\mathbb{Z}}^n)$ boundedness of the discrete Riesz potential

In [J. Class. Anal., vol. 26 (1) (2025), 63-76], we proved that the discrete Riesz potential $I_{\alpha }$ is a bounded operator $H^p({\mathbb{Z}}^n)\to H^q({\mathbb{Z}}^n)$ for $\frac{n-1}{n}<p\le 1$, $\frac{1}{q}=\frac{1}{p}-\frac{\alpha }{n}$ and $0<\alpha <n$. In this note, we extend such boundedness on the full range $0<p\le 1$.