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arXiv · 2601.00799

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

Durgesh Pasawan
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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.

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Durgesh Pasawan. 2025-12-05. Pseudo-differential operators associated with the gyrator transform on modulation spaces with Shubin-type symbols. https://arxiv.org/abs/2601.00799

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