Weak curvature conditions and functional inequalities

By J. Lott, Cédric Villani. 2005

170 citations recorded by the source corpus.

We give sufficient conditions for a measured length space (X, d,ν) to admit local and global Poincare inequalities, along with a Sobolev inequality. We first introduce a condition DM on (X, d,ν) ,d ef ined in terms of transport of measures. We show that DM, together with a doubling condition on ν, implies a scaleinvariant local Poincare inequality. We show that if (X, d,ν) has nonnegative N -Ricci curvature and has unique minimizing geodesics between almost all pairs of points then it satisfies DM, with constant 2 N .T he condition DM is preserved by measured Gromov–Hausdorff limits. We then prove a Sobolev inequality for measured length spaces with N -Ricci curvature bounded below by K> 0. Finally we derive a sharp global Poincare inequality.

Subfields: Applied Mathematics, Genetics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ab731700319298e6638b0593112438c2e29cc67d

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Citations in the corpus

Gradient estimates for heat kernels and harmonic functions (2017, background): “Under the RCD∗(K,N) condition, the (local) doubling condition was established in [86, 108], and the (local) Poincaré inequality was established in [92].”
The Splitting Theorem in Non-Smooth Context (2013, unknown): “Yet, if m is doubling and the space supports a weak-local 1-1 Poincaré inequality (which is always the case for CD(K,N) spaces with N <∞ - see [49] and [58]), then the results of Cheeger in [19] ensure that |Df |p = |Df |p′ m-a.”
Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below (2011, background): “L convergence: we state it for all the stretchable classes of test plans satisfying a condition weaker than bounded compression, inspired to the “democratic” condition introduced by [29].”
Heat flow on Finsler manifolds (2008, background): “Theorem 6.5 (N-Ric ≥ K equals CD(K, N), [Oh4]) For each compact smooth Finsler space (M, F, m), the bound ∞-Ric ≥ K in the sense of Definition 5.1 is equivalent to CD(K, ∞). More generally, N-Ric ≥ K in the sense of Definition 5.1 is equivalent to CD(K, N) in the sense of Lott and Villani [ LV2 ] an”
Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds (2015, unknown): “Then for any p ∈ (1,∞) it holds λ1,p(X,d,m) ≥ λ 1,p K,N,D, where λ1,pK,N,D is defined by λ1,pK,N,D := inf µ∈FK,N,D inf { ´ R |u′|p µ ´ R |u|p µ : u ∈ Lip(R) ∩ L p(µ), ˆ R u|u|p−2µ = 0, u 6= 0 } .”
Bochner-Weitzenb\"ock formula and Li-Yau estimates on Finsler manifolds (2011, background): “equivalent to Lott, Villani and the second author’s curvature-dimension condition CD(K,N). This extends the corresponding result on (weighted) Riemannian manifolds (due to [vRS], [St1], [St2], [St3], [LV2], [LV3]), and has many analytic and geometric applications (see [Oh1], [OS1] or a survey [Oh2])”
Some Geometric Calculations on Wasserstein Space (2006, methodology): “It has had applications to PDE theory [16], metric geometry [8,19,20] and functional inequalities [9,17].”
Uniform convexity and smoothness, and their applications in Finsler geometry (2009, background): “In fact, in a recent work of the author [23], Theorem 7.4 acts as one of key ingredients of extending Lott, Sturm and Villani’s remarkable theory [14, 15 ,29,30] to Finsler manifolds.”
Metric measure spaces with Riemannian Ricci curvature bounded from below (2011, methodology): “In [28] it was also proved that, at least under the nonbranching assumption, the CD.”
Heat Flow on Alexandrov Spaces (2010, background): “(ii) The condition CD(K,N) for N ∈ (1,∞) is also introduced in [41, 24].”
An Overview of the Proof of the Splitting Theorem in Spaces with Non-Negative Ricci Curvature (2014, background): “4)) and (X, d,m) supports a 1-2 weak local Poincaré inequality (see [36], [45] and [44]) already grant important properties of this ow.”
Wasserstein geometry of Gaussian measures (2011, background): “In particular, the singular set is stratified according to the dimension of the support of the Gaussian measures, providing an explicit nontrivial example of Alexandrov space with extremal sets.”