Differential Complexes in Time-Periodic Gelfand-Shilov Spaces
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| Format: | Preprint |
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2026
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| author | Silva, Fernando de Ávila Cappiello, Marco Kirilov, Alexandre Tokoro, Pedro Meyer |
| author_facet | Silva, Fernando de Ávila Cappiello, Marco Kirilov, Alexandre Tokoro, Pedro Meyer |
| contents | We study the global solvability of a class of differential complexes on the product manifold $\mathbb{T}^m \times \mathbb{R}^n$ associated with systems of evolution operators of the form $L_r = \partial_{t_r} + ia_r(t)P(x,D_x), r=1,\ldots,m,$ where the coefficients $a_r$ are real-valued Gevrey functions on the torus and $P(x,D_x)$ is a globally elliptic normal differential operator on $\mathbb{R}^n$. Within the framework of time-periodic Gelfand--Shilov spaces, we introduce a natural differential complex generated by these operators and investigate its solvability in both functional and ultradistributional settings. We provide a complete characterization of global solvability in terms of a Diophantine condition involving the constant part of the associated $1$-form and the spectrum of $P$. We also analyze global hypoellipticity of the complex. These results extend previous works on scalar operators and constant coefficient systems to the setting of differential complexes with time-dependent real coefficients. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_09646 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Differential Complexes in Time-Periodic Gelfand-Shilov Spaces Silva, Fernando de Ávila Cappiello, Marco Kirilov, Alexandre Tokoro, Pedro Meyer Analysis of PDEs Primary 35B10, 58J10 Secondary 58A10, 46F05 We study the global solvability of a class of differential complexes on the product manifold $\mathbb{T}^m \times \mathbb{R}^n$ associated with systems of evolution operators of the form $L_r = \partial_{t_r} + ia_r(t)P(x,D_x), r=1,\ldots,m,$ where the coefficients $a_r$ are real-valued Gevrey functions on the torus and $P(x,D_x)$ is a globally elliptic normal differential operator on $\mathbb{R}^n$. Within the framework of time-periodic Gelfand--Shilov spaces, we introduce a natural differential complex generated by these operators and investigate its solvability in both functional and ultradistributional settings. We provide a complete characterization of global solvability in terms of a Diophantine condition involving the constant part of the associated $1$-form and the spectrum of $P$. We also analyze global hypoellipticity of the complex. These results extend previous works on scalar operators and constant coefficient systems to the setting of differential complexes with time-dependent real coefficients. |
| title | Differential Complexes in Time-Periodic Gelfand-Shilov Spaces |
| topic | Analysis of PDEs Primary 35B10, 58J10 Secondary 58A10, 46F05 |
| url | https://arxiv.org/abs/2602.09646 |