Fractional series operators on $\mathbb{Z}^n$

Abstract

For $0 \leq \alpha < n$ and $m \in \mathbb{N} \cap (1 - \frac{\alpha}{n}, \, \infty)$, we introduce a class of fractional series operators $T_{\alpha, m}$ defined on $\mathbb{Z}^n$ which are generated by certain $m$-invertible matrices with integer coefficients. In this note, we prove that $T_{\alpha, m}$ is a bounded operator $H^p(\mathbb{Z}^n) \to \ell^q(\mathbb{Z}^n)$ for $0 < p < \frac{n}{\alpha}$ and $\frac{1}{q} = \frac{1}{p} - \frac{\alpha}{n}$. This generalizes the results obtained by the author in [Acta Math. Hungar., 168 (1) (2022), 202-216].