Published Paper
Inserted: 1 mar 2024
Last Updated: 26 apr 2026
Journal: Potential Anal.
Volume: 63
Pages: 1441-1469
Year: 2025
Doi: https://doi.org/10.1007/s11118-025-10207-2
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.
Keywords: Wasserstein distance, Heat semigroup, Infinitesimal Hilbertianity, smoothing property, indeterminacy, nodal set, Buser inequality
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