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Bang-Xian Han

Publications and source records attributed to Bang-Xian Han.

At least 37 records · Page 2Linked to original sources
arXiv2025

On the geometry of Wasserstein barycenter I

We study the Wasserstein barycenter problem in the setting of non-compact, non-smooth extended metric measure spaces. We introduce a couple of new concepts and obtain the existence, uniqueness, absolute continuity of the Wasserstein barycenter, and prove Jensen's inequality in an abstract framework. This generalized several results on Euclidean space, Riemannian manifolds and Alexandrov spaces, to metric measure spaces satisfying Riemannian Curvature-Dimension condition à la Lott--Sturm--Villani, and some extended metric measure spaces including abstract Wiener spaces and configuration spaces over Riemannian manifolds. We also introduce a new curvature-dimesion condition, we call Barycenter-Curvature-Dimension condition. We prove its stability under measured-Gromov--Hausdorff convergence and prove the existence of the Wasserstein barycenter under this new condition. In addition, we get some geometric inequalities including a multi-marginal Brunn--Minkowski inequality and a functional Blaschke--Santaló type inequality.

math.MG↗
arXiv2024

ABP estimate on metric measure spaces via optimal transport

By using optimal transport theory, we establish a sharp Alexandroff--Bakelman--Pucci (ABP) type estimate on metric measure spaces with synthetic Riemannian Ricci curvature lower bounds, and prove some geometric and functional inequalities including a functional ABP estimate. Our result not only extends the border of ABP estimate, but also provides an effective substitution of Jacobi fields computation in the non-smooth framework, which has potential applications to many problems in non-smooth geometric analysis.

math.MG↗
arXiv2024

On the asymptotic behaviour of the fractional Sobolev seminorms in metric measure spaces: asymptotic volume ratio, volume entropy and rigidity

We study the asymptotic behaviour of suitably defined seminorms in general metric measure spaces. As a particular case we provide new and shorter proofs of the Maz'ya-Shaposhnikova's theorem on the asymptotic behaviour of the fractional Sobolev $s$-seminorm, in the setting of metric measure spaces and with general mollifiers, as well as of the Ludwig's result on finite dimensional Banach spaces. Our result also provides new spaces satisfying an asymptotic formula and it also builds a link between the asymptotic formula for functions and the asymptotic volume ratio of a metric measure space. In addition, we prove two related rigidity results for metric measure spaces with synthetic Ricci curvature bound which are new even in the smooth setting.

math.FA↗
arXiv2024

Maz'ya-Shaposhnikova meet Bishop-Gromov

We find a surprising link between Maz'ya-Shaposhnikova's well-known asymptotic formula concerning fractional Sobolev seminorms and the generalized Bishop-Gromov inequality. In the setting of abstract metric measure spaces we prove the validity of a large family of asymptotic formulas concerning non-local energies. Important examples which are covered by our approach are for instance Carnot groups, Riemannian manifolds with Ricci curvature bounded from below and non-collapsed RCD spaces. We also extend the classical Maz'ya-Shaposhnikova's formula on Euclidean spaces to a wider class of mollifiers.

math.MG↗
arXiv2023

Sharp uncertainty principles on metric measure spaces

We study the Heisenberg-Pauli-Weyl uncertainty principle and the Caffarelli-Kohn-Nirenberg interpolation inequalities, on metric measure spaces satisfying measure contraction property. Using localization techniques, we show that these inequalities are valid only on volume cones.

math.MG↗
arXiv2021

On the asymptotic behaviour of the fractional Sobolev seminorms in metric measure spaces: Bourgain-Brezis-Mironescu's theorem revisited

We generalize Bourgain-Brezis-Mironescu's asymptotic formula for fractional Sobolev functions, in the setting of abstract metric measure spaces, under the assumption that at almost every point the tangent space in the measured Gromov-Hausdorff sense is a finite dimensional Banach space or a Carnot group. Our result not only covers the known results concerning Euclidean spaces, weighted Riemannian manifolds and finite dimensional Banach spaces, but also extends Bourgain-Brezis-Mironescu's formula to $\rcdkn$ spaces and sub-Riemannian manifolds.

math.FA↗
arXiv2021

Rigidity of some functional inequalities on RCD spaces

We study the cases of equality and prove a rigidity theorem concerning the 1-Bakry-Émery inequality. As an application, we prove the rigidity of the Gaussian isoperimetric inequality, the logarithmic Sobolev inequality and the Poincaré inequality in the setting of ${\rm RCD}(K, \infty)$ metric measure spaces. This unifies and extends to the non-smooth setting the results of Carlen-Kerce, Morgan, Bouyrie, Ohta-Takatsu, Cheng-Zhou. Examples of non-smooth spaces fitting our setting are measured-Gromov Hausdorff limits of Riemannian manifolds with uniform Ricci curvature lower bound, and Alexandrov spaces with curvature lower bound. Some results including the rigidity of $Φ$-entropy inequalities, the rigidity of the 1-Bakry-Émery inequality are of independent interest even in the smooth setting.

math.FA↗
arXiv2020

Sharp Poincaré inequalities under Measure Contraction Property

We prove a sharp Poincaré inequality for subsets $Ω$ of (essentially non-branching) metric measure spaces satisfying the Measure Contraction Property $\textrm{MCP}(K,N)$, whose diameter is bounded above by $D$. This is achieved by identifying the corresponding one-dimensional model densities and a localization argument, ensuring that the Poincaré constant we obtain is best possible as a function of $K$, $N$ and $D$. Another new feature of our work is that we do not need to assume that $Ω$ is geodesically convex, by employing the geodesic hull of $Ω$ on the energy side of the Poincaré inequality. In particular, our results apply to geodesic balls in ideal sub-Riemannian manifolds, such as the Heisenberg group.

math.MG↗
arXiv2019

Curvature-dimension conditions for diffusions under time change

We derive precise transformation formulas for synthetic lower Ricci bounds under time change. More precisely, for local Dirichlet forms we study how the curvature-dimension condition in the sense of Bakry-Emery will transform under time change. Similarly, for metric measure spaces we study how the curvature-dimension condition in the sense of Lott-Sturm-Villani will transform under time change.

math.FA↗
arXiv2018

Characterizations of monotonicity of vector fields on metric measure spaces

We characterize the convexity of functions and the monotonicity of vector fields on metric measure spaces with Riemannian Ricci curvature bounded from below. Our result offers a new approach to deal with some rigidity theorems such as `splitting theorem' and `volume cone implies metric cone theorem' in non-smooth context.

math.FA↗
arXiv2018

Conformal transformation on metric measure spaces

We study several problems concerning conformal transformation on metric measure spaces, including the Sobolev space, the differential structure and the curvature-dimension condition under conformal transformations. This is the first result about preservation of lower curvature bounds under perturbation, which is new even on Alexandrov spaces.

math.MG↗
arXiv2018

Ricci tensor on ${\rm RCD}^*(K,N)$ spaces

We obtain an improved Bochner inequality based on the curvature-dimension condition ${\rm RCD}^*(K,N)$ and propose a definition of $N$-dimensional Ricci tensor on metric measure spaces.

math.MG↗
arXiv2018

New characterizations of Ricci curvature on RCD metric measure spaces

We prove that on a large family of metric measure spaces, if the $L^p$-gradient estimate for heat flows holds for some $p>2$, then the $L^1$-gradient estimate also holds. This result extends Savaré's result on metric measure spaces, and provides a new proof to von Renesse-Sturm theorem on smooth metric measure spaces. As a consequence, we propose a new analysis object based on Gigli's measure-valued Ricci tensor, to characterize the Ricci curvature of ${\rm RCD}$ space in a local way. The argument is a new iteration technique based on non-smooth Bakry-Émery theory, which is a new method to study the curvature dimension condition of metric measure spaces.

math.FA↗
arXiv2017

Ricci tensor on smooth metric measure space with boundary

The aim of this note is to study the measure-valued Ricci tensor on smooth metric measure space with boundary, which is a generalization of Bakry-Emery's modified Ricci tensor on weighted Riemannian manifold. As an application, we offer a new approach to study curvature-dimension condition of smooth metric measure space with boundary.

math.MG↗
arXiv2017

Sobolev spaces on warped products

We study the structure of Sobolev spaces on the cartesian/warped products of a given metric measure space and an interval. Our main results are: - the characterization of the Sobolev spaces in such products - the proof that, under natural assumptions, the warped products possess the Sobolev-to-Lipschitz property, which is key for geometric applications. The results of this paper have been needed in the recent proof of the `volume-cone-to-metric-cone' property of ${\sf RCD}$ spaces obtained by the first author and De Philippis.

math.FA↗