Localization operators on discrete Orlicz modulation spaces

TL;DR

This paper develops the theory of Orlicz modulation spaces on discrete groups, studies their properties, and analyzes the boundedness and compactness of associated localization operators in this setting.

Contribution

It introduces Orlicz modulation spaces on $\\mathbb{Z}^n$, explores their fundamental properties, and investigates the boundedness and compactness of time-frequency localization operators on these spaces.

Findings

Orlicz modulation spaces are closely related to classical modulation spaces for certain Young functions.

Localization operators are bounded on Orlicz modulation spaces under specific symbol classes.

These operators are also shown to be compact and belong to Schatten--von Neumann classes.

Abstract

In this paper, we introduce Orlicz spaces on $ \mathbb Z^n \times \mathbb T^n $ and Orlicz modulation spaces on $\mathbb Z^n$, and present some basic properties such as inclusion relations, convolution relations, and duality of these spaces. We show that the Orlicz modulation space $M^{\Phi}(\mathbb Z^n)$ is close to the modulation space $M^{2}(\mathbb Z^n)$ for some particular Young function $\Phi$. Then, we study a class of pseudo-differential operators known as time-frequency localization operators on $\mathbb Z^n$, which depend on a symbol $\varsigma$ and two windows functions $g_1$ and $g_2$. Using appropriate classes for symbols, we study the boundedness of the localization operators on Orlicz modulation spaces on $\mathbb Z^n$. Also, we show that these operators are compact and in the Schatten--von Neumann classes.