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arXiv · 2403.00620

Properties of Lipschitz smoothing heat semigroups

Nicolò De Ponti·Giorgio Stefani
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Abstract

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.

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Nicolò De Ponti, Giorgio Stefani. 2024-03-01. Properties of Lipschitz smoothing heat semigroups. https://arxiv.org/abs/2403.00620

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