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Andrea Mondino - One of the best experts on this subject based on the ideXlab platform.

  • sharp and rigid isoperiMetric inequalities in Metric Measure spaces with lower ricci curvature bounds
    Inventiones Mathematicae, 2017
    Co-Authors: Fabio Cavalletti, Andrea Mondino
    Abstract:

    We prove that if \((X,\mathsf {d},\mathfrak {m})\) is a Metric Measure space with \(\mathfrak {m}(X)=1\) having (in a synthetic sense) Ricci curvature bounded from below by \(K>0\) and dimension bounded above by \(N\in [1,\infty )\), then the classic Levy-Gromov isoperiMetric inequality (together with the recent sharpening counterparts proved in the smooth setting by Milman for any \(K\in \mathbb {R}\), \(N\ge 1\) and upper diameter bounds) holds, i.e. the isoperiMetric profile function of \((X,\mathsf {d},\mathfrak {m})\) is bounded from below by the isoperiMetric profile of the model space. Moreover, if equality is attained for some volume \(v \in (0,1)\) and K is strictly positive, then the space must be a spherical suspension and in this case we completely classify the isoperiMetric regions. Finally we also establish the almost rigidity: if the equality is almost attained for some volume \(v \in (0,1)\) and K is strictly positive, then the space must be mGH close to a spherical suspension. To our knowledge this is the first result about isoperiMetric comparison for non smooth Metric Measure spaces satisfying Ricci curvature lower bounds. Examples of spaces fitting our assumptions include Measured Gromov–Hausdorff limits of Riemannian manifolds satisfying Ricci curvature lower bounds, Alexandrov spaces with curvature bounded from below, Finsler manifolds endowed with a strongly convex norm and satisfying Ricci curvature lower bounds; the result seems new even in these celebrated classes of spaces.

  • sharp geoMetric and functional inequalities in Metric Measure spaces with lower ricci curvature bounds
    arXiv: Metric Geometry, 2015
    Co-Authors: Fabio Cavalletti, Andrea Mondino
    Abstract:

    For Metric Measure spaces verifying the reduced curvature-dimension condition $CD^*(K,N)$ we prove a series of sharp functional inequalities under the additional assumption of essentially non-branching. Examples of spaces entering this framework are (weighted) Riemannian manifolds satisfying lower Ricci curvature bounds and their Measured Gromov Hausdorff limits, Alexandrov spaces satisfying lower curvature bounds and more generally $RCD^*(K,N)$-spaces, Finsler manifolds endowed with a strongly convex norm and satisfying lower Ricci curvature bounds, etc. In particular we prove Brunn-Minkowski inequality, $p$-spectral gap (or equivalently $p$-Poincare inequality) for any $p\in [1,\infty)$, log-Sobolev inequality, Talagrand inequality and finally Sobolev inequality. All the results are proved in a sharp form involving an upper bound on the diameter of the space; if this extra sharpening is suppressed, all the previous inequalities for essentially non-branching $CD^*(K,N)$ spaces take the same form of the corresponding ones holding for a weighted Riemannian manifold verifying curvature-dimension condition $CD(K,N)$ in the sense of Bakry-Emery. In this sense inequalities are sharp. We also discuss the rigidity and almost rigidity statements associated to the $p$-spectral gap. Finally let us mention that for essentially non-branching Metric Measure spaces, the local curvature-dimension condition $CD_{loc}(K,N)$ is equivalent to the reduced curvature-dimension condition $CD^*(K,N)$. Therefore we also have shown that sharp Brunn-Minkowski inequality in the \emph{global} form can be deduced from the \emph{local} curvature-dimension condition, providing a step towards (the long-standing problem of) globalization for the curvature-dimension condition $CD(K,N)$.

  • riemannian ricci curvature lower bounds in Metric Measure spaces with finite Measure
    Transactions of the American Mathematical Society, 2015
    Co-Authors: Luigi Ambrosio, Andrea Mondino, Nicola Gigli, Tapio Rajala
    Abstract:

    In prior work (4) of the first two authors with Savare, a new Riemannian notion of lower bound for Ricci curvature in the class of Metric Measure spaces (X,d,m) was introduced, and the corresponding class of spaces denoted by RCD(K,∞). This notion relates the CD(K,N) theory of Sturm and Lott-Villani, in the case N = ∞, to the Bakry-Emery approach. In (4) the RCD(K,∞) property is defined in three equivalent ways and several properties of RCD(K,∞) spaces, including the regularization properties of the heat flow, the connections with the theory of Dirichlet forms and the stability under tensor products, are provided. In (4) only finite reference Measures m have been considered. The goal of this paper is twofold: on one side we extend these results to general σ-finite spaces, on the other we remove a technical assumption appeared in (4) concerning a strengthening of the CD(K,∞) condition. This more general class of spaces includes Euclidean spaces endowed with Lebesgue Measure, complete noncompact Riemannian manifolds with bounded geometry and the pointed Metric Measure limits of manifolds with lower Ricci curvature bounds.

  • sharp and rigid isoperiMetric inequalities in Metric Measure spaces with lower ricci curvature bounds
    arXiv: Metric Geometry, 2015
    Co-Authors: Fabio Cavalletti, Andrea Mondino
    Abstract:

    We prove that if $(X,\mathsf{d},\mathfrak{m})$ is a Metric Measure space with $\mathfrak{m}(X)=1$ having (in a synthetic sense) Ricci curvature bounded from below by $K>0$ and dimension bounded above by $N\in [1,\infty)$, then the classic L\'evy-Gromov isoperiMetric inequality (together with the recent sharpening counterparts proved in the smooth setting by E. Milman for any $K\in \mathbb{R}$, $N\geq 1$ and upper diameter bounds) hold, i.e. the isoperiMetric profile function of $(X,\mathsf{d},\mathfrak{m})$ is bounded from below by the isoperiMetric profile of the model space. Moreover, if equality is attained for some volume $v \in (0,1)$ and $K$ is strictly positive, then the space must be a spherical suspension and in this case we completely classify the isoperiMetric regions. Finally we also establish the almost rigidity: if the equality is almost attained for some volume $v \in (0,1)$ and $K$ is strictly positive, then the space must be mGH close to a spherical suspension. To our knowledge this is the first result about isoperiMetric comparison for non smooth Metric Measure spaces satisfying Ricci curvature lower bounds. Examples of spaces fitting our assumptions include Measured Gromov-Hausdorff limits of Riemannian manifolds satisfying Ricci curvature lower bounds and Alexandrov spaces with curvature bounded from below; the result seems new even in these celebrated classes of spaces.

Karl-theodor Sturm - One of the best experts on this subject based on the ideXlab platform.

Fabio Cavalletti - One of the best experts on this subject based on the ideXlab platform.

  • sharp and rigid isoperiMetric inequalities in Metric Measure spaces with lower ricci curvature bounds
    Inventiones Mathematicae, 2017
    Co-Authors: Fabio Cavalletti, Andrea Mondino
    Abstract:

    We prove that if \((X,\mathsf {d},\mathfrak {m})\) is a Metric Measure space with \(\mathfrak {m}(X)=1\) having (in a synthetic sense) Ricci curvature bounded from below by \(K>0\) and dimension bounded above by \(N\in [1,\infty )\), then the classic Levy-Gromov isoperiMetric inequality (together with the recent sharpening counterparts proved in the smooth setting by Milman for any \(K\in \mathbb {R}\), \(N\ge 1\) and upper diameter bounds) holds, i.e. the isoperiMetric profile function of \((X,\mathsf {d},\mathfrak {m})\) is bounded from below by the isoperiMetric profile of the model space. Moreover, if equality is attained for some volume \(v \in (0,1)\) and K is strictly positive, then the space must be a spherical suspension and in this case we completely classify the isoperiMetric regions. Finally we also establish the almost rigidity: if the equality is almost attained for some volume \(v \in (0,1)\) and K is strictly positive, then the space must be mGH close to a spherical suspension. To our knowledge this is the first result about isoperiMetric comparison for non smooth Metric Measure spaces satisfying Ricci curvature lower bounds. Examples of spaces fitting our assumptions include Measured Gromov–Hausdorff limits of Riemannian manifolds satisfying Ricci curvature lower bounds, Alexandrov spaces with curvature bounded from below, Finsler manifolds endowed with a strongly convex norm and satisfying Ricci curvature lower bounds; the result seems new even in these celebrated classes of spaces.

  • sharp geoMetric and functional inequalities in Metric Measure spaces with lower ricci curvature bounds
    arXiv: Metric Geometry, 2015
    Co-Authors: Fabio Cavalletti, Andrea Mondino
    Abstract:

    For Metric Measure spaces verifying the reduced curvature-dimension condition $CD^*(K,N)$ we prove a series of sharp functional inequalities under the additional assumption of essentially non-branching. Examples of spaces entering this framework are (weighted) Riemannian manifolds satisfying lower Ricci curvature bounds and their Measured Gromov Hausdorff limits, Alexandrov spaces satisfying lower curvature bounds and more generally $RCD^*(K,N)$-spaces, Finsler manifolds endowed with a strongly convex norm and satisfying lower Ricci curvature bounds, etc. In particular we prove Brunn-Minkowski inequality, $p$-spectral gap (or equivalently $p$-Poincare inequality) for any $p\in [1,\infty)$, log-Sobolev inequality, Talagrand inequality and finally Sobolev inequality. All the results are proved in a sharp form involving an upper bound on the diameter of the space; if this extra sharpening is suppressed, all the previous inequalities for essentially non-branching $CD^*(K,N)$ spaces take the same form of the corresponding ones holding for a weighted Riemannian manifold verifying curvature-dimension condition $CD(K,N)$ in the sense of Bakry-Emery. In this sense inequalities are sharp. We also discuss the rigidity and almost rigidity statements associated to the $p$-spectral gap. Finally let us mention that for essentially non-branching Metric Measure spaces, the local curvature-dimension condition $CD_{loc}(K,N)$ is equivalent to the reduced curvature-dimension condition $CD^*(K,N)$. Therefore we also have shown that sharp Brunn-Minkowski inequality in the \emph{global} form can be deduced from the \emph{local} curvature-dimension condition, providing a step towards (the long-standing problem of) globalization for the curvature-dimension condition $CD(K,N)$.

  • sharp and rigid isoperiMetric inequalities in Metric Measure spaces with lower ricci curvature bounds
    arXiv: Metric Geometry, 2015
    Co-Authors: Fabio Cavalletti, Andrea Mondino
    Abstract:

    We prove that if $(X,\mathsf{d},\mathfrak{m})$ is a Metric Measure space with $\mathfrak{m}(X)=1$ having (in a synthetic sense) Ricci curvature bounded from below by $K>0$ and dimension bounded above by $N\in [1,\infty)$, then the classic L\'evy-Gromov isoperiMetric inequality (together with the recent sharpening counterparts proved in the smooth setting by E. Milman for any $K\in \mathbb{R}$, $N\geq 1$ and upper diameter bounds) hold, i.e. the isoperiMetric profile function of $(X,\mathsf{d},\mathfrak{m})$ is bounded from below by the isoperiMetric profile of the model space. Moreover, if equality is attained for some volume $v \in (0,1)$ and $K$ is strictly positive, then the space must be a spherical suspension and in this case we completely classify the isoperiMetric regions. Finally we also establish the almost rigidity: if the equality is almost attained for some volume $v \in (0,1)$ and $K$ is strictly positive, then the space must be mGH close to a spherical suspension. To our knowledge this is the first result about isoperiMetric comparison for non smooth Metric Measure spaces satisfying Ricci curvature lower bounds. Examples of spaces fitting our assumptions include Measured Gromov-Hausdorff limits of Riemannian manifolds satisfying Ricci curvature lower bounds and Alexandrov spaces with curvature bounded from below; the result seems new even in these celebrated classes of spaces.

Renjin Jiang - One of the best experts on this subject based on the ideXlab platform.

  • cheeger harmonic functions in Metric Measure spaces revisited
    Journal of Functional Analysis, 2014
    Co-Authors: Renjin Jiang
    Abstract:

    Abstract Let ( X , d , μ ) be a complete Metric Measure space, with μ a locally doubling Measure, that supports a local weak L 2 -Poincare inequality. By assuming a heat semigroup type curvature condition, we prove that Cheeger-harmonic functions are Lipschitz continuous on ( X , d , μ ) . Gradient estimates for Cheeger-harmonic functions and solutions to a class of non-linear Poisson type equations are presented.

Zhenqing Chen - One of the best experts on this subject based on the ideXlab platform.

  • stability of heat kernel estimates for symMetric jump processes on Metric Measure spaces
    arXiv: Probability, 2016
    Co-Authors: Zhenqing Chen, Takashi Kumagai, Jian Wang
    Abstract:

    In this paper, we consider symMetric jump processes of mixed-type on Metric Measure spaces under general volume doubling condition, and establish stability of two-sided heat kernel estimates and heat kernel upper bounds. We obtain their stable equivalent characterizations in terms of the jumping kernels, variants of cut-off Sobolev inequalities, and the Faber-Krahn inequalities. In particular, we establish stability of heat kernel estimates for $\alpha$-stable-like processes even with $\alpha\ge 2$ when the underlying spaces have walk dimensions larger than $2$, which has been one of the major open problems in this area.

  • discrete approximation of symMetric jump processes on Metric Measure spaces
    arXiv: Probability, 2010
    Co-Authors: Zhenqing Chen, Takashi Kumagai
    Abstract:

    In this paper we give general criteria on tightness and weak convergence of discrete Markov chains to symMetric jump processes on Metric Measure spaces under mild conditions. As an application, we investigate discrete approximation for a large class of symMetric jump processes. We also discuss some application of our results to the scaling limit of random walk in random conductance.

  • on heat kernel estimates and parabolic harnack inequality for jump processes on Metric Measure spaces
    Acta Mathematica Sinica, 2009
    Co-Authors: Panki Kim, Zhenqing Chen, Takashi Kumagai
    Abstract:

    In this paper, we discuss necessary and sufficient conditions on jumping kernels for a class of jump-type Markov processes on Metric Measure spaces to have scale-invariant finite range parabolic Harnack inequality.

  • heat kernel estimates for jump processes of mixed types on Metric Measure spaces
    Probability Theory and Related Fields, 2007
    Co-Authors: Zhenqing Chen, Takashi Kumagai
    Abstract:

    In this paper, we investigate symMetric jump-type processes on a class of Metric Measure spaces with jumping intensities comparable to radially symMetric functions on the spaces. The class of Metric Measure spaces includes the Alfors d-regular sets, which is a class of fractal sets that contains geoMetrically self-similar sets. A typical example of our jump-type processes is the symMetric jump process with jumping intensity \(e^{-c_0 (x, y)|x-y|}\, \int_{\alpha_1}^{\alpha_2} \frac{c(\alpha, x, y)}{|x-y|^{d+\alpha}} \, \nu (d\alpha)\) where ν is a probability Measure on \([\alpha_1, \alpha_2]\subset (0, 2)\) , c(α, x, y) is a jointly measurable function that is symMetric in (x, y) and is bounded between two positive constants, and c 0(x, y) is a jointly measurable function that is symMetric in (x, y) and is bounded between γ1 and γ2, where either γ2 ≥ γ1 > 0 or γ1 = γ2 = 0. This example contains mixed symMetric stable processes on \({\mathbb{R}}^n\) as well as mixed relativistic symMetric stable processes on \({\mathbb{R}}^n\) . We establish parabolic Harnack principle and derive sharp two-sided heat kernel estimate for such jump-type processes.