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

Abstract

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

10 pages