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

Abstract

In this work, for the range $\frac{n-1}{n}<p\le 1$, 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 $0<\alpha <n$.

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