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

TL;DR

This paper establishes the boundedness of the discrete Riesz potential operator on integer lattice spaces, extending classical harmonic analysis results to discrete settings for certain p and q ranges.

Contribution

It proves the boundedness of the discrete Riesz potential from Hardy spaces to cute spaces on bZ^n, a result previously known in continuous contexts.

Findings

Boundedness of I_lpha from H^p(bZ^n) to ll^q(bZ^n)

Valid for 0 < p  1 and 1/q = 1/p - lpha/n

Extends harmonic analysis to discrete lattice settings.

Abstract

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