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

In [J. Class. Anal. 26 (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 $\frac{n-1}{n} \lt p \leq 1$, $\frac{1}{q} = \frac{1}{p} - \frac{\alpha }{n}$ and $0 \lt \alpha \lt n$. In this note, we extend this result to the full range $0 \lt p \leq 1$.