An extension to non-nilpotent groups of Rothschild-Stein lifting method
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910620595519488 |
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| author | Galeotti, Mattia |
| author_facet | Galeotti, Mattia |
| contents | In their celebrated paper of 1976, Rothschild and Stein prove a lifting procedure that locally reduces to a free nilpotent Lie algebra any family of smooth vector fields $X_1,\dots,X_q$, over a manifold $M$. Then, a large class of differential operators can be lifted, and fundamental solutions on the lifted space can be re-projected to fundamental solutions of the given operators on $M$. In case that the Lie algebra $\mathfrak g=\mbox{Lie}(X_1,\dots,X_q)$ is finite dimensional but not nilpotent, this procedure could introduce a strong tilting of the space. In this paper we represent a global construction of a Lie group $G$ associated to $\mathfrak g$ that avoid this tilting problem. In particular $\mbox{Lie}(G)\cong\mathfrak g$ and a right $G$-action exists over $M$, faithful and transitive, inducing a natural projection $E\colon G\to M$. We represent the group $G$ as a direct product $M\times G^z$ where the model fiber $G^z$ has a group structure. We prove that for any simply connected manifold $M$ -- and a vast class of non-simply connected manifolds -- a fundamental solution for a differential operator $L=\sum_{α\in\mathbb N^q} r_α\cdot X^α$ of finite degree over $M$ can be obtained, via a saturation method, from a fundamental solution for the associated lifted operator over the group $G$. This is a generalization of Biagi and Bonfiglioli analogous result for homogeneous vector fields over $M=\mathbb R^n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_19619 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An extension to non-nilpotent groups of Rothschild-Stein lifting method Galeotti, Mattia Analysis of PDEs Differential Geometry 35A08, 17B66 (Primary) 22E05, 57R25, 57R27 (Secondary) In their celebrated paper of 1976, Rothschild and Stein prove a lifting procedure that locally reduces to a free nilpotent Lie algebra any family of smooth vector fields $X_1,\dots,X_q$, over a manifold $M$. Then, a large class of differential operators can be lifted, and fundamental solutions on the lifted space can be re-projected to fundamental solutions of the given operators on $M$. In case that the Lie algebra $\mathfrak g=\mbox{Lie}(X_1,\dots,X_q)$ is finite dimensional but not nilpotent, this procedure could introduce a strong tilting of the space. In this paper we represent a global construction of a Lie group $G$ associated to $\mathfrak g$ that avoid this tilting problem. In particular $\mbox{Lie}(G)\cong\mathfrak g$ and a right $G$-action exists over $M$, faithful and transitive, inducing a natural projection $E\colon G\to M$. We represent the group $G$ as a direct product $M\times G^z$ where the model fiber $G^z$ has a group structure. We prove that for any simply connected manifold $M$ -- and a vast class of non-simply connected manifolds -- a fundamental solution for a differential operator $L=\sum_{α\in\mathbb N^q} r_α\cdot X^α$ of finite degree over $M$ can be obtained, via a saturation method, from a fundamental solution for the associated lifted operator over the group $G$. This is a generalization of Biagi and Bonfiglioli analogous result for homogeneous vector fields over $M=\mathbb R^n$. |
| title | An extension to non-nilpotent groups of Rothschild-Stein lifting method |
| topic | Analysis of PDEs Differential Geometry 35A08, 17B66 (Primary) 22E05, 57R25, 57R27 (Secondary) |
| url | https://arxiv.org/abs/2403.19619 |