Sobolev-to-Lipschitz property on -spaces and applications
Lorenzo Dello Schiavo1, Kohei Suzuki2
1IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria.
Mathematische Annalen
|October 24, 2022
Summary
We establish the Sobolev-to-Lipschitz property for metric measure spaces using a novel quasi curvature-dimension condition. This has key applications for heat semigroups, including Varadhan asymptotics and irreducibility on sub-Riemannian manifolds.
Area of Science:
- Analysis
- Geometric Measure Theory
- Differential Geometry
Background:
- Metric measure spaces are fundamental in geometric analysis.
- The quasi curvature-dimension condition is a recent development in geometric analysis.
- Heat semigroups are crucial for studying diffusion and geometry.
Purpose of the Study:
- To prove the Sobolev-to-Lipschitz property for metric measure spaces.
- To explore applications of this property to heat semigroups.
- To extend results to sub-Riemannian manifolds.
Main Methods:
- Utilizing the quasi curvature-dimension condition.
- Analyzing properties of the heat semigroup.
- Applying the concept of infinitesimal Hilbertianity.
Main Results:
- Established the Sobolev-to-Lipschitz property for spaces satisfying the quasi curvature-dimension condition.
- Demonstrated Varadhan short-time asymptotics for the heat semigroup.
- Proved the irreducibility of the heat semigroup under additional assumptions.
- Showcased applicability to sub-Riemannian manifolds.
Conclusions:
- The Sobolev-to-Lipschitz property is a powerful tool in geometric analysis.
- The quasi curvature-dimension condition provides a unified framework for various geometric properties.
- These findings advance the understanding of heat semigroups on metric measure spaces and sub-Riemannian manifolds.
Keywords:
Quasi curvature-dimension conditionSobolev-to-Lipschitz propertySub-Riemannian geometryVaradhan short-time asymptoticsMore Related Videos
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