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].
Categories: math.CA