Local Poincaré inequalities from stable curvature conditions on metric spaces

By T. Rajala. 2011

196 citations recorded by the source corpus.

We prove local Poincaré inequalities under various curvature-dimension conditions which are stable under the measured Gromov–Hausdorff convergence. The first class of spaces we consider is that of weak CD(K, N) spaces as defined by Lott and Villani. The second class of spaces we study consists of spaces where we have a flow satisfying an evolution variational inequality for either the Rényi entropy functional $${\fancyscript{E}_{N}(\rho m) = -\int_{X} {\rho}^{1-1/N} dm}$$ or the Shannon entropy functional $${\fancyscript{E}_{\infty}({\rho} m) = \int_{X}{\rho}\log {\rho} dm.}$$ We also prove that if the Rényi entropy functional is strongly displacement convex in the Wasserstein space, then at every point of the space we have unique geodesics to almost all points of the space.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/30a9d7dfe0ede71073be9fdb769a5f964ef6953f

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In Applied Mathematics

Citations in the corpus

Monge problem in metric measure spaces with Riemannian curvature-dimension condition (2013, unknown): “The case of a general norm as cost function on Rn has been solved first in the particular case of crystalline norms in [2], and then in fully generality independently by L. Caravenna in [7] and by T. Champion and L. De Pascale in [12].”
Characterization of Low Dimensional RCD*(K, N) Spaces (2015, result): “MR3205729 [3] L. Ambrosio, N. Gigli, A. Mondino, and T. Rajala, Riemannian Ricci curvature lower bounds in metric measure spaces with σ-finite measure, Trans.”
Geometry and Analysis of Dirichlet forms (2012, methodology): “4], m satisfies the doubling property, and by [37], (X, d, m) supports a weak (1, 1)-Poincaré inequality.”
Second order differentiation formula on RCD*$(K,N)$ spaces (2018, background): “1 Introduction In the last ten years there has been a great interest in the study of metric measure spaces with Ricci curvature bounded from below, see for instance [44], [55], [56], [29], [5], [6], [27], [7], [48], [49], [30], [24], [26], [38], [9], [47], [13], [12].”
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.”
Perimeter as relaxed Minkowski content in metric measure spaces (2016, methodology): “t}) for L1-a.e. t > 0. Concerning assumption (4.2), we point that it holds on doubling RCD(K,∞) spaces (see [AGS11b]) for the deﬁnition of the latter. Indeed, on one side we know by Rajala’s paper [Ra] that on these spaces a 1-1 weak Poincar´e inequality holds, so that thanks to Cheeger’s results”
Heat Kernel Bounds on Metric Measure Spaces and Some Applications (2014, background): “t Sturm’s results in [33, 34, 35] are valid for a metric measure space (X,d,µ) satisfying the RCD∗(0,N) condition with N ∈ [1,∞), since where doubling property and Poincare´ inequality hold; see e.g. [15, 29]. However, for example, the constant in the exponential term in the Gaussian lower bound in ”
Structure theory of metric measure spaces with lower Ricci curvature bounds (2014, background): “While a great deal of progress has been made in this latter general framework, see for instance [3, 5, 6, 7, 8, 9, 10, 14, 15, 27, 28, 29, 30, 32, 33, 34, 40, 41, 42, 46], the structure theory on such metric-measure spaces is still much less developed than in the case of smooth limits.”
Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds (2015, background): “While a great deal of progress has been made in this latter general framework from both the analytic, geometric and structural points of view, see for instance [2, 3, 4, 5, 6, 7, 9, 14, 15, 18, 27, 33, 35, 37, 36, 43, 50, 59, 63, 66], the isoperimetric problem has remained elusive.”
The abstract Lewy-Stampacchia inequality and applications (2014, background): “Key facts are that CD(K,N), N < ∞, spaces are doubling ([32]) and support a weak local 1-2 Poincaré inequality ([22], [28]) and a number of consequences about the behavior of harmonic functions can be deduced from these informations, see [8] for an overview on the subject.”
Rectifiability of the reduced boundary for sets of finite perimeter over $\RCD(K,N)$ spaces. (2019, background): “Furthermore, it was proved in [45] that it satisfies a local Poincaré inequality.”
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.”