Pseudo-differential operators associated with the gyrator transform on modulation spaces with Shubin-type symbols
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911404960776192 |
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| author | Pasawan, Durgesh |
| author_facet | Pasawan, Durgesh |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_00799 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pseudo-differential operators associated with the gyrator transform on modulation spaces with Shubin-type symbols Pasawan, Durgesh Functional Analysis 2020: 42B35, 47G30, 35S05, 46F12, 42C15 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. |
| title | Pseudo-differential operators associated with the gyrator transform on modulation spaces with Shubin-type symbols |
| topic | Functional Analysis 2020: 42B35, 47G30, 35S05, 46F12, 42C15 |
| url | https://arxiv.org/abs/2601.00799 |