Local Poincaré inequalities from stable curvature conditions on metric spaces

Rajala, Tapio

doi:10.1007/s00526-011-0442-7

Cited by 150 publications

(126 citation statements)

References 20 publications

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“…By the lower semicontinuity of E N we then have (6) for all t ∈ [0, 1]. Next we prove (2) for N = N and all t ∈ [0, 1].…”

Section: Convexity-inequalities For Intermediate Timesmentioning

confidence: 94%

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Improved geodesics for the reduced curvature-dimension condition in branching metric spaces

Rajala¹

2013

Discrete &Amp; Continuous Dynamical Systems - A

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In this note we show that in metric measure spaces satisfying the reduced curvature-dimension condition CD * (K, N ) we always have geodesics in the Wasserstein space of probability measures that satisfy the critical convexity inequality of CD * (K, N ) also for intermediate times and in addition the measures along these geodesics have an upper-bound on their densities. This upper-bound depends on the bounds for the densities of the end-point measures, the lower-bound K for the Ricci-curvature, the upper-bound N for the dimension, and on the diameter of the union of the supports of the end-point measures.

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“…By the lower semicontinuity of E N we then have (6) for all t ∈ [0, 1]. Next we prove (2) for N = N and all t ∈ [0, 1].…”

Section: Convexity-inequalities For Intermediate Timesmentioning

confidence: 94%

“…Since µ 1 2 has been fixed, by Proposition 1 and Lemma 2.5 we can consider the parts of the transports with d(x 0 , x 1 ) ∈ [l , (l + 1) [, l ∈ N, separately. Therefore, by estimating in these parts similarly as for (6) and then letting 0 the inequality (2) follows for t = k2 −n−1 .…”

Section: Convexity-inequalities For Intermediate Timesmentioning

confidence: 95%

Improved geodesics for the reduced curvature-dimension condition in branching metric spaces

Rajala¹

2013

Discrete &Amp; Continuous Dynamical Systems - A

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“…Also, in [27], [40] a synthetic notion CD(K, N ) of having Ricci curvature bounded from below by K and dimension bounded above by N was given (in [27] only the case CD(0, N ) was considered, for N < ∞), and a number of geometric consequences of these notions, like Brunn-Minkowski and Bishop-Gromov inequalities, have been derived. In [26] it was also proved that, at least under the nonbranching assumption, the CD(K, N ) condition implies also the Poincaré inequaliy, see also [33] for some recent progress in this direction.…”

Section: Introductionmentioning

confidence: 98%

Metric measure spaces with Riemannian Ricci curvature bounded from below

Ambrosio¹,

Gigli²,

Savaré³

2014

Duke Math. J.

454

688

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In this paper we introduce a synthetic notion of Riemannian Ricci bounds from below for metric measure spaces (X, d, m) which is stable under measured Gromov-Hausdorff convergence and rules out Finsler geometries. It can be given in terms of an enforcement of the Lott, Sturm and Villani geodesic convexity condition for the entropy coupled with the linearity of the heat flow. Besides stability, it enjoys the same tensorization, globalto-local and local-to-global properties. In these spaces, that we call RCD(K, ∞) spaces, we prove that the heat flow (which can be equivalently characterized either as the flow associated to the Dirichlet form, or as the Wasserstein gradient flow of the entropy) satisfies Wasserstein contraction estimates and several regularity properties, in particular Bakry-Emery estimates and the L ∞ − Lip Feller regularization. We also prove that the distance induced by the Dirichlet form coincides with d, that the local energy measure has density given by the square of Cheeger's relaxed slope and, as a consequence, that the underlying Brownian motion has continuous paths. All these results are obtained independently of Poincaré and doubling assumptions on the metric measure structure and therefore apply also to spaces which are not locally compact, as the infinite-dimensional ones.

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“…Besides many useful geometric and functional applications of this notion [26,40,48], one of its strongest features is its stability under measured Gromov-…”

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confidence: 99%

Bakry–Émery curvature-dimension condition and Riemannian Ricci curvature bounds

Ambrosio¹,

Gigli

Savaré³

2015

Ann. Probab.

228

377

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The aim of the present paper is to bridge the gap between the Bakry-Émery and the Lott-Sturm-Villani approaches to provide synthetic and abstract notions of lower Ricci curvature bounds.We start from a strongly local Dirichlet form E admitting a Carré du champ Γ in a Polish measure space (X, m) and a canonical distance dE that induces the original topology of X. We first characterize the distinguished class of Riemannian Energy measure spaces, where E coincides with the Cheeger energy induced by dE and where every function f with Γ(f ) ≤ 1 admits a continuous representative.In such a class, we show that if E satisfies a suitable weak form of the Bakry-Émery curvature dimension condition BE(K, ∞) then the metric measure space (X, d, m) satisfies the Riemannian Ricci curvature bound RCD(K, ∞) according to [Duke Math. J. 163 (2014) 1405-1490], thus showing the equivalence of the two notions.Two applications are then proved: the tensorization property for Riemannian Energy spaces satisfying the Bakry-Émery BE(K, N ) condition (and thus the corresponding one for RCD(K, ∞) spaces without assuming nonbranching) and the stability of BE(K, N ) with respect to Sturm-Gromov-Hausdorff convergence.

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Local Poincaré inequalities from stable curvature conditions on metric spaces

Cited by 150 publications

References 20 publications

Improved geodesics for the reduced curvature-dimension condition in branching metric spaces

Improved geodesics for the reduced curvature-dimension condition in branching metric spaces

Metric measure spaces with Riemannian Ricci curvature bounded from below

Bakry–Émery curvature-dimension condition and Riemannian Ricci curvature bounds

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