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

TL;DR

This paper introduces a new class of fractional series operators on integer lattices and proves their boundedness between specific Hardy and Lebesgue spaces, extending previous results in the field.

Contribution

It defines fractional series operators generated by invertible matrices on b^n and establishes their boundedness properties between Hardy and Lebesgue spaces.

Findings

Operators are bounded from $H^p(\u00bb^n)$ to $\u03bb^q(\u00bb^n)$ under certain conditions.

Generalizes previous boundedness results for fractional operators on integer lattices.

Extends the theory of fractional operators with matrix-generated transformations.

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].