Interpolated measures with bounded density in metric spaces satisfying the curvature-dimension conditions of Sturm

By T. Rajala. 2011

104 citations recorded by the source corpus.

We construct geodesics in the Wasserstein space of probability measure along which all the measures have an upper bound on their density that is determined by the densities of the endpoints of the geodesic. Using these geodesics we show that a local Poincar\'e inequality and the measure contraction property follow from the Ricci curvature bounds defined by Sturm. We also show for a large class of convex functionals that a local Poincar\'e inequality is implied by the weak displacement convexity of the functional.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/dd88bef48700d709b65f93f39815a9edb68b6ce6

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below (2011, result): “See also [35] for more recent results in this direction, independent of the non-branching assumption.”
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.”
Non-branching geodesics and optimal maps in strong $$CD(K,\infty )$$CD(K,∞)-spaces (2012, result): “Although in some of the results this assumption has recently been removed (see for example [15, 14]), in many it still remains.”
Constancy of the Dimension for RCD(K,N) Spaces via Regularity of Lagrangian Flows (2018, unknown): “The results are mainly taken from [36, 37, 58].”
Geometry and Analysis of Dirichlet forms (2012, methodology): “utely continuous curve in P2(X). Moreover, (5.13) also implies that |T˙ tµ| 2 ≤ Z X 1 Ttf dΓ(Ttf, Ttf), which is as desired. For the proof of Theorem 5.1, we still need a very recent result of Rajala [37, 38]: λ- displacement convexity of U∞ implies that (X, d, m) supports a weak (1, 1)- and hence a”
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].”
On the differential structure of metric measure spaces and applications (2012, result): “Recent results by Rajala (see [40]) suggest that infinitesimal Hilbertianity plus CD(K,∞) are actually sufficient to recover all the properties obtained in [6], without any strengthening of the curvature condition, but the situation, as of today, is not yet completely clear2.”
Cones over metric measure spaces and the maximal diameter theorem (2013, background): “pu(x) = limsup y→x |u(y)−u(x)| d X(x,y) . Lipuis an upper gradient for u[18, proposition 1.11]. T. Rajala and M.-K. von Renesse proved the following results that we state only for K≥ 0. Theorem 4.11 ([40]). Suppose that (X,d X,m X) satisﬁes CD(K,N) with K ≥ 0 and N ≥ 1. Then (X,d X,m X) supports a w”
Nonsmooth differential geometry - An approach tailored for spaces with Ricci curvature bounded from below (2014, background): “It is a remarkable result of Rajala [48] the fact that on CD(K,∞) spaces, K ∈ R, for any couple of probability measures with bounded support and bounded density, a W2-geodesic of bounded compression connecting them always exists.”
On the Bakry–Émery Condition, the Gradient Estimates and the Local-to-Global Property of $$\mathsf{RCD}^*(K,N)$$RCD∗(K,N) Metric Measure Spaces (2016, background): “he non-branching assumption. Actually, some results of the CD(K,N) theory initially proved under the non-branching assumption have been recently proved by Rajala without making use of this assumption [27], [28]. But, a recent remarkable paper by the same author [29] provides for all K∈ Rand N≥ 1 a (”
Independence on p of weak upper gradients on RCD spaces (2014, background): “We conclude recalling another useful regularity property of RCD spaces, this one concerning displacement interpolation of measures, see [8] for a proof: Proposition 2.”
The Li-Yau Inequality and Heat Kernels on Metric Measure Spaces (2014, background): “Rajala [36, 37] showed that a local weak L2-Poincaré inequality holds on RCD∗(K,N) spaces, and a uniformL2-Poincaré inequality holds if K ≥ 0.”
Cheeger-harmonic functions in metric measure spaces revisited (2013, background): “Moreover, it is proved by Rajala [25, 26] that a local weakL2-Poincaré inequality holds on them, and a uniformL2-Poincaré inequality holds ifK ≥ 0.”