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

For $0\le \alpha <n$ and $m\in \mathbb{N}\cap (1-\frac{\alpha }{n},\mathrm{\infty })$, 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 $H^p({\mathbb{Z}}^n)\to {\mathrm{\ell }}^q({\mathbb{Z}}^n)$ 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].