Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Properties of Lipschitz smoothing heat semigroups

Nicolò De Ponti, Giorgio Stefani

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.

Properties of Lipschitz smoothing heat semigroups

Abstract

We prove several functional and geometric inequalities only assuming the linearity and a quantitative -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, spaces, and some sub-Riemannian structures, such as Carnot groups, the Grushin plane and the group.
Paper Structure (41 sections, 29 theorems, 142 equations)

This paper contains 41 sections, 29 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Framework
  3. Main aim and results
  4. Organization of the paper
  5. Preliminaries
  6. Function spaces
  7. Space of measures
  8. Wasserstein 1-distance
  9. BL* distance
  10. AC measures
  11. Slope
  12. Relaxed gradient
  13. Cheeger energy
  14. Laplacian operator
  15. Heat semigroup
  16. ...and 26 more sections

Key Result

Theorem 2.1

The topology induced by the $\mathrm{BL}^\star$ distance coincides with the topology induced by the weak convergence eq:def_weak_conv on $\mathscr{M}_+(X)$.

Theorems & Definitions (70)

  • Theorem 2.1
  • Corollary 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • Definition 3.1: $\mathrm{L}^\infty$-to-$\mathrm{Lip}$
  • ...and 60 more