Diagonal systems of differential operators on compact Lie groups
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866913809540579328 |
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| author | da Silva, Paulo L. Dattori Kirilov, Alexandre da Silva, Ricardo Paleari |
| author_facet | da Silva, Paulo L. Dattori Kirilov, Alexandre da Silva, Ricardo Paleari |
| contents | We investigate the global hypoellipticity and global solvability of systems of left-invariant differential operators on compact Lie groups. Focusing on diagonal systems, we establish necessary and sufficient conditions for these global properties. Specifically, we show that global solvability is characterized by a Diophantine condition on the symbol of the system, while global hypoellipticity further requires that a set depending on the symbol to be finite. As an application, we provide a complete characterization of these properties for systems of vector fields defined on products of tori and spheres. Additionally, we present illustrative examples, including systems involving higher-order differential operators. Finally, we extend our analysis to triangular systems on compact Lie groups, introducing an additional condition related to the boundedness of the dimensions of the representations in these Lie groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_16623 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Diagonal systems of differential operators on compact Lie groups da Silva, Paulo L. Dattori Kirilov, Alexandre da Silva, Ricardo Paleari Analysis of PDEs 35R03, 35H10, 58D25, 43A80 We investigate the global hypoellipticity and global solvability of systems of left-invariant differential operators on compact Lie groups. Focusing on diagonal systems, we establish necessary and sufficient conditions for these global properties. Specifically, we show that global solvability is characterized by a Diophantine condition on the symbol of the system, while global hypoellipticity further requires that a set depending on the symbol to be finite. As an application, we provide a complete characterization of these properties for systems of vector fields defined on products of tori and spheres. Additionally, we present illustrative examples, including systems involving higher-order differential operators. Finally, we extend our analysis to triangular systems on compact Lie groups, introducing an additional condition related to the boundedness of the dimensions of the representations in these Lie groups. |
| title | Diagonal systems of differential operators on compact Lie groups |
| topic | Analysis of PDEs 35R03, 35H10, 58D25, 43A80 |
| url | https://arxiv.org/abs/2412.16623 |