Properties of Lipschitz smoothing heat semigroups
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929745488248832 |
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| author | De Ponti, Nicolò Stefani, Giorgio |
| author_facet | De Ponti, Nicolò Stefani, Giorgio |
| contents | We prove several functional and geometric inequalities only assuming the linearity and a quantitative $\mathrm{L}^\infty$-to-Lipschitz smoothing of the heat semigroup in metric-measure spaces. Our results comprise a Buser inequality, a lower bound on the size of the nodal set of a Laplacian eigenfunction, and different estimates involving the Wasserstein distance. The approach works in large variety settings, including Riemannian manifolds with a variable Kato-type lower bound on the Ricci curvature tensor, $\mathsf{RCD}(K,\infty)$ spaces, and some sub-Riemannian structures, such as Carnot groups, the Grushin plane and the $\mathbb{SU}(2)$ group. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_00620 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Properties of Lipschitz smoothing heat semigroups De Ponti, Nicolò Stefani, Giorgio Functional Analysis Primary 53C23. Secondary 31E05, 58J35 We prove several functional and geometric inequalities only assuming the linearity and a quantitative $\mathrm{L}^\infty$-to-Lipschitz smoothing of the heat semigroup in metric-measure spaces. Our results comprise a Buser inequality, a lower bound on the size of the nodal set of a Laplacian eigenfunction, and different estimates involving the Wasserstein distance. The approach works in large variety settings, including Riemannian manifolds with a variable Kato-type lower bound on the Ricci curvature tensor, $\mathsf{RCD}(K,\infty)$ spaces, and some sub-Riemannian structures, such as Carnot groups, the Grushin plane and the $\mathbb{SU}(2)$ group. |
| title | Properties of Lipschitz smoothing heat semigroups |
| topic | Functional Analysis Primary 53C23. Secondary 31E05, 58J35 |
| url | https://arxiv.org/abs/2403.00620 |