Pseudo-differential operators associated with the gyrator transform on modulation spaces with Shubin-type symbols

TL;DR

This paper develops a theory of pseudo-differential operators linked to the gyrator transform on modulation spaces, extending previous results to a more flexible functional framework.

Contribution

It introduces and analyzes pseudo-differential operators associated with the gyrator transform on modulation spaces using Shubin symbols, extending prior work to a broader setting.

Findings

Proves boundedness of gyrator-based pseudo-differential operators on modulation spaces.

Establishes continuity and invertibility of the gyrator transform on modulation spaces.

Extends earlier results from Schwartz and Sobolev spaces to modulation spaces.

Abstract

We develop a theory of pseudo-differential operators associated with the gyrator transform on modulation spaces. The gyrator transform is a two-dimensional linear canonical transform which can be viewed as a rotation in the time-frequency plane and is closely related to the fractional Fourier transform. Motivated by the global structure of the gyrator kernel, we work with Shubin global symbol classes on $\mathbb{R}^4$. We first recall basic properties of modulation spaces and establish continuity and invertibility of the gyrator transform on these spaces, using its representation as a metaplectic operator. Then we introduce pseudo-differential operators defined via the gyrator transform and a Shubin symbol, and we prove boundedness results on modulation spaces and on gyrator-based modulation-Sobolev spaces. Our work extends and generalises earlier results of Mahato, Arya and Prasad on Schwartz and Sobolev spaces \cite{MahatoGyrator} to the more flexible framework of modulation spaces.