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

## TL;DR

This paper extends the boundedness results of the discrete Riesz potential operator from a limited range of p to the full range 0 < p ≤ 1 on discrete Hardy spaces, broadening its applicability.

## Contribution

It generalizes previous boundedness results of the discrete Riesz potential to the entire range 0 < p ≤ 1, enhancing understanding of its operator behavior.

## Key findings

- Boundedness holds for the full range 0 < p ≤ 1
- Extended the operator's boundedness from partial to complete p-range
- Provides a broader framework for discrete Riesz potential analysis

## Abstract

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 $\frac{n-1}{n} < p \leq 1$, $\frac{1}{q} = \frac{1}{p} - \frac{\alpha}{n}$ and $0 < \alpha < n$. In this note, we extend such boundedness on the full range $0 < p \leq 1$.