Comparing three possible hypoelliptic Laplacians on the 5-dimensional Cartan group via div-curl type estimates
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| Format: | Preprint |
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2024
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| _version_ | 1866915318878699520 |
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| author | Baldi, Annalisa Tripaldi, Francesca |
| author_facet | Baldi, Annalisa Tripaldi, Francesca |
| contents | On general Carnot groups, the definition of a possible hypoelliptic Hodge-Laplacian on forms using the Rumin complex has been considered by Rumin, who introduced a 0-order pseudodifferential operator on forms. However, for questions regarding regularity for example, where one needs sharp estimates, this 0-order operator is not suitable. Up to now, there have only been very few attempts to define hypoelliptic Hodge-Laplacians on forms that would allow for such sharp estimates. Indeed, this question is rather difficult to address in full generality, the main issue being that the Rumin exterior differential $d_c$ is not homogeneous on arbitrary Carnot groups.
In this note, we consider the specific example of the free Carnot group of step 3 with 2 generators, and we introduce three possible definitions of hypoelliptic Hodge-Laplacians. We compare how these three possible Laplacians can be used to obtain sharp div-curl type inequalities akin to those considered by Bourgain & Brezis and Lanzani & Stein for the de Rham complex, or their subelliptic counterparts obtained by Baldi, Franchi & Pansu for the Rumin complex on Heisenberg groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_14316 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Comparing three possible hypoelliptic Laplacians on the 5-dimensional Cartan group via div-curl type estimates Baldi, Annalisa Tripaldi, Francesca Analysis of PDEs Differential Geometry 58A10, 35R03, 26D15, 43A80, 53C17 On general Carnot groups, the definition of a possible hypoelliptic Hodge-Laplacian on forms using the Rumin complex has been considered by Rumin, who introduced a 0-order pseudodifferential operator on forms. However, for questions regarding regularity for example, where one needs sharp estimates, this 0-order operator is not suitable. Up to now, there have only been very few attempts to define hypoelliptic Hodge-Laplacians on forms that would allow for such sharp estimates. Indeed, this question is rather difficult to address in full generality, the main issue being that the Rumin exterior differential $d_c$ is not homogeneous on arbitrary Carnot groups. In this note, we consider the specific example of the free Carnot group of step 3 with 2 generators, and we introduce three possible definitions of hypoelliptic Hodge-Laplacians. We compare how these three possible Laplacians can be used to obtain sharp div-curl type inequalities akin to those considered by Bourgain & Brezis and Lanzani & Stein for the de Rham complex, or their subelliptic counterparts obtained by Baldi, Franchi & Pansu for the Rumin complex on Heisenberg groups. |
| title | Comparing three possible hypoelliptic Laplacians on the 5-dimensional Cartan group via div-curl type estimates |
| topic | Analysis of PDEs Differential Geometry 58A10, 35R03, 26D15, 43A80, 53C17 |
| url | https://arxiv.org/abs/2407.14316 |