A molecular decomposition for $H^p(\mathbb{Z}^n)$

TL;DR

This paper establishes a molecular decomposition theorem for discrete Hardy spaces on integer lattices and demonstrates the boundedness of the discrete Riesz potential operator between these spaces.

Contribution

It introduces a molecular reconstruction theorem for $H^p(Z^n)$ and applies it to prove boundedness of the discrete Riesz potential operator.

Findings

Molecular decomposition for $H^p(Z^n)$ in the specified range.

Boundedness of the discrete Riesz potential operator between Hardy spaces.

Extension of atomic decomposition techniques to molecular frameworks.

Abstract

In this work, for the range $\frac{n-1}{n} < p \leq 1$, we give a molecular reconstruction theorem for $H^p(\mathbb{Z}^n)$. As an application of this result and the atomic decomposition developed by S. Boza and M. Carro in [Proc. R. Soc. Edinb., 132 A (1) (2002), 25-43], we prove that the discrete Riesz potential $I_{\alpha}$ defined on $\mathbb{Z}^n$ is a bounded operator $H^p(\mathbb{Z}^n) \to H^q(\mathbb{Z}^n)$ for $\frac{n-1}{n} < p < \frac{n}{\alpha}$ and $\frac{1}{q} = \frac{1}{p} - \frac{\alpha}{n}$, where $0 < \alpha < n$.