Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below

Abstract

This paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces (X,d,m). Our main results are: - A general study of the relations between the Hopf-Lax semigroup and Hamilton-Jacobi equation in metric spaces (X,d). - The equivalence of the heat flow in L^2(X,m) generated by a suitable Dirichlet energy and the Wasserstein gradient flow of the relative entropy functional in the space of probability measures P(X). - The proof of density in energy of Lipschitz functions in the Sobolev space W^{1,2}(X,d,m). - A fine and very general analysis of the differentiability properties of a large class of Kantorovich potentials, in connection with the optimal transport problem. Our results apply in particular to spaces satisfying Ricci curvature bounds in the sense of Lott & Villani [30] and Sturm [39,40], and require neither the doubling property nor the validity of the local Poincaré inequality.

Minor typos corrected and many small improvements added. Lemma 2.4, Lemma 2.10, Prop. 5.7, Rem. 5.8, Thm. 6.3 added. Rem. 4.7, Prop. 4.8, Prop. 4.15 and Thm 4.16 augmented/reenforced. Proof of Thm. 4.16 and Lemma 9.6 simplified. Thm. 8.6 corrected. A simpler axiomatization of weak gradients, still equivalent to all other ones, has been proposed

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• Metric measure spaces with Riemannian Ricci curvature bounded from below
• Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds
• Riemannian Ricci curvature lower bounds in metric measure spaces with $\sigma$-finite measure
• Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures
• Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in RCD(K,\infty) metric measure spaces
• Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds
• Well posedness of Lagrangian flows and continuity equations in metric measure spaces
• Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds
• Structure of measures in Lipschitz differentiability spaces
• New formulas for the Laplacian of distance functions and applications
• Asymptotic isoperimetry on non collapsed spaces with lower Ricci bounds
• On quotients of spaces with Ricci curvature bounded below
• New differential operator and non-collapsed $RCD$ spaces
• Isoperimetric sets in spaces with lower bounds on the Ricci curvature
• Polya-Szego inequality and Dirichlet $p$-spectral gap for non-smooth spaces with Ricci curvature bounded below
• On Interpolation and Curvature via Wasserstein Geodesics
• Sharp Cheeger-Buser type inequalities in $\mathsf{RCD}(K,\infty )$ spaces
• On the extremal points of the ball of the Benamou-Brenier energy
• Infinitesimal Hilbertianity of weighted Riemannian manifolds
• Gravity from thermodynamics: optimal transport and negative effective dimensions
• The Cheeger problem in abstract measure spaces
• Rademacher-type Theorems and Sobolev-to-Lipschitz Properties for Strongly Local Dirichlet Spaces
• A superposition principle for the inhomogeneous continuity equation with Hellinger-Kantorovich-regular coefficients
• Singular Weyl's law with Ricci curvature bounded below
• Curvewise characterizations of minimal upper gradients and the construction of a Sobolev differential
• Maximal volume entropy rigidity for ${\mathsf{RCD}}^{\ast }(-(N-1),N)$ spaces
• A Talenti-type comparison theorem for $\mathrm{RCD}(K,N)$ spaces and applications
• The equality case in Cheeger's and Buser's inequalities on $\mathsf{RCD}$ spaces
• Functional inequalities for the heat flow on time-dependent metric measure spaces
• Harmonic functions and gravity localization
• Optimal transport, gradient estimates, and pathwise Brownian coupling on spaces with variable Ricci bounds
• Super-Ricci flows and improved gradient and transport estimates
• Characterizations of monotonicity of vector fields on metric measure spaces
• Sharp log-Sobolev inequalities in $\mathsf{CD}(0,N)$ spaces with applications
• Sharp Heat Kernel Bounds and Entropy in Metric Measure Spaces
• Sobolev-to-Lipschitz Property on QCD-spaces and Applications
• Angles between curves in metric measure spaces
• On the notion of Laplacian bounds on $\mathrm{RCD}$ spaces and applications
• Weighted Littlewood--Paley inequalities for heat flows in $\RCD$ spaces
• Isoperimetric problem and structure at infinity on Alexandrov spaces with nonnegative curvature
• Convergence of Non-Symmetric Diffusion Processes on RCD spaces
• Cheeger bounds on spin-two fields
• Hölder continuity of tangent cones in RCD(K,N) spaces and applications to non-branching
• Generalized Bakry-Émery curvature condition and equivalent entropic inequalities in groups
• Plans on measures and AM-modulus
• Wasserstein geometry and Ricci curvature bounds for Poisson spaces
• Gradient flow for the Boltzmann entropy and Cheeger's energy on time-dependent metric measure spaces
• An approach to metric space valued Sobolev maps via weak* derivatives
• Sobolev spaces via chains in metric measure spaces
• Principal frequency of clamped plates on RCD(0,N) spaces: sharpness, rigidity and stability
• Curvature bound of Dyson Brownian Motion
• The normal contraction property for non-bilinear Dirichlet forms
• Local Conditions for Global Convergence of Gradient Flows and Proximal Point Sequences in Metric Spaces
• Gradient flows of $(K,N)$-convex functions with negative $N$
• Persistence of Rademacher-type and Sobolev-to-Lipschitz properties
• Boundary Regularity of Harmonic maps from $RCD(K,N)$-space to $CAT(0)$-space
• Integral Varadhan formula for non-linear heat flow
• Margulis Lemma on $\text{RCD}(K,N)$ spaces
• Transport inequalities for random point measures
• Construction of $p$-energy measures associated with strongly local $p$-energy forms
• BV Functions and Sets of Finite Perimeter on Configuration Spaces
• Metric Sobolev spaces I: equivalence of definitions