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Table of Contents
Intro
Contents
Introduction to the Notes of the School on Analysis and Geometry in Metric Spaces
Geometric Inequalities on Riemannian and Sub-Riemannian Manifolds by Heat Semigroups Techniques
1 Introduction
2 Subelliptic Diffusion Operators
2.1 Diffusion Operators
2.2 Subelliptic Diffusion Operators
2.3 The Distance Associated to Subelliptic Diffusion Operators
2.4 Essentially Self-Adjoint Subelliptic Operators
2.5 The Heat Semigroup Associated to a Subelliptic Diffusion Operator
3 The Heat Semigroup on a Complete Riemannian Manifold and Its Geometric Applications
3.1 The Laplace-Beltrami Operator
3.2 The Heat Semigroup on a Compact Riemannian Manifold
3.3 Bochner's Identity
3.4 The Curvature Dimension Inequality
3.5 Stochastic Completeness
3.6 Convergence to Equilibrium, Poincaré and Log-Sobolev Inequalities
3.7 The Li-Yau Inequality
3.8 The Parabolic Harnack Inequality
3.9 The Gaussian Upper Bound
3.10 Volume Doubling Property
3.11 Upper and Lower Gaussian Bounds for the Heat Kernel
3.12 The Poincaré Inequality on Domains
6 Weak Besicovitch Covering Property
References
Sobolev Spaces in Extended Metric-Measure Spaces
1 Introduction
1.1 Main Notation
2 Topological and Metric-Measure Structures
2.1 Metric-Measure Structures
2.1.1 Topological and Measure Theoretic Notions
2.1.2 Extended Metric-Topological (Measure) Spaces
2.1.3 Examples
2.1.4 The Kantorovich-Rubinstein Distance
2.1.5 The Asymptotic Lipschitz Constant
2.1.6 Compatible Algebra of Functions
2.1.7 Embedding and Compactification of Extended Metric-Measure Spaces
2.1.8 Notes
2.2 Continuous Curves and Nonparametric Arcs
2.2.1 Continuous Curves
2.2.2 Arcs
2.2.3 Rectifiable Arcs
2.2.4 Notes
2.3 Length and Conformal Distances
2.3.1 The Length Property
2.3.2 Conformal Distances
2.3.3 Duality for Kantorovich-Rubinstein Cost Functionals Induced by Conformal Distances
2.3.4 Notes
3 The Cheeger Energy
3.1 The Strongest Form of the Cheeger Energy
3.1.1 Relaxed Gradients and Local Representation of the Cheeger Energy
3.1.2 Invariance w.r.t. Restriction and Completion
3.1.3 Notes
Contents
Introduction to the Notes of the School on Analysis and Geometry in Metric Spaces
Geometric Inequalities on Riemannian and Sub-Riemannian Manifolds by Heat Semigroups Techniques
1 Introduction
2 Subelliptic Diffusion Operators
2.1 Diffusion Operators
2.2 Subelliptic Diffusion Operators
2.3 The Distance Associated to Subelliptic Diffusion Operators
2.4 Essentially Self-Adjoint Subelliptic Operators
2.5 The Heat Semigroup Associated to a Subelliptic Diffusion Operator
3 The Heat Semigroup on a Complete Riemannian Manifold and Its Geometric Applications
3.1 The Laplace-Beltrami Operator
3.2 The Heat Semigroup on a Compact Riemannian Manifold
3.3 Bochner's Identity
3.4 The Curvature Dimension Inequality
3.5 Stochastic Completeness
3.6 Convergence to Equilibrium, Poincaré and Log-Sobolev Inequalities
3.7 The Li-Yau Inequality
3.8 The Parabolic Harnack Inequality
3.9 The Gaussian Upper Bound
3.10 Volume Doubling Property
3.11 Upper and Lower Gaussian Bounds for the Heat Kernel
3.12 The Poincaré Inequality on Domains
6 Weak Besicovitch Covering Property
References
Sobolev Spaces in Extended Metric-Measure Spaces
1 Introduction
1.1 Main Notation
2 Topological and Metric-Measure Structures
2.1 Metric-Measure Structures
2.1.1 Topological and Measure Theoretic Notions
2.1.2 Extended Metric-Topological (Measure) Spaces
2.1.3 Examples
2.1.4 The Kantorovich-Rubinstein Distance
2.1.5 The Asymptotic Lipschitz Constant
2.1.6 Compatible Algebra of Functions
2.1.7 Embedding and Compactification of Extended Metric-Measure Spaces
2.1.8 Notes
2.2 Continuous Curves and Nonparametric Arcs
2.2.1 Continuous Curves
2.2.2 Arcs
2.2.3 Rectifiable Arcs
2.2.4 Notes
2.3 Length and Conformal Distances
2.3.1 The Length Property
2.3.2 Conformal Distances
2.3.3 Duality for Kantorovich-Rubinstein Cost Functionals Induced by Conformal Distances
2.3.4 Notes
3 The Cheeger Energy
3.1 The Strongest Form of the Cheeger Energy
3.1.1 Relaxed Gradients and Local Representation of the Cheeger Energy
3.1.2 Invariance w.r.t. Restriction and Completion
3.1.3 Notes