On generalized Dirichlet integrals in the smooth and in the o-minimal setting
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915040134692864 |
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| author | Dall'Ara, Gian Maria |
| author_facet | Dall'Ara, Gian Maria |
| contents | Given a compact manifold $M$ equipped with smooth vector fields $X_1,\ldots, X_r$, we consider the generalized Dirichlet energy \[\mathbf{E}(f)= \sum_{j=1}^r\int_M |X_jf|^2\, dm,\] where $dm$ is a volume form, and ask if the set \[ \mathcal{B}=\{f\in L^2(M)\colon\,\mathbf{E}(f)+\lVert f\rVert_{L^2(M)}^2\leq 1 \} \] is precompact in $L^2(M)$. We find a geometric sufficient condition in terms of "iterated characteristic sets" and use it to show that, if the vector fields are tame (in the sense of o-minimality) and satisfy the Hörmander condition of some order $s$ on a dense set of points, then the only obstruction to precompactness is the existence of a characteristic submanifold (i.e. a nonempty submanifold of positive codimension to which each $X_j$ is tangent). Implications for global regularity of sum-of-squares operators not necessarily satisfying Hörmander condition are discussed in an appendix. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_19540 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On generalized Dirichlet integrals in the smooth and in the o-minimal setting Dall'Ara, Gian Maria Analysis of PDEs Classical Analysis and ODEs Given a compact manifold $M$ equipped with smooth vector fields $X_1,\ldots, X_r$, we consider the generalized Dirichlet energy \[\mathbf{E}(f)= \sum_{j=1}^r\int_M |X_jf|^2\, dm,\] where $dm$ is a volume form, and ask if the set \[ \mathcal{B}=\{f\in L^2(M)\colon\,\mathbf{E}(f)+\lVert f\rVert_{L^2(M)}^2\leq 1 \} \] is precompact in $L^2(M)$. We find a geometric sufficient condition in terms of "iterated characteristic sets" and use it to show that, if the vector fields are tame (in the sense of o-minimality) and satisfy the Hörmander condition of some order $s$ on a dense set of points, then the only obstruction to precompactness is the existence of a characteristic submanifold (i.e. a nonempty submanifold of positive codimension to which each $X_j$ is tangent). Implications for global regularity of sum-of-squares operators not necessarily satisfying Hörmander condition are discussed in an appendix. |
| title | On generalized Dirichlet integrals in the smooth and in the o-minimal setting |
| topic | Analysis of PDEs Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2411.19540 |