arXiv research

We show that any $d$-Ahlfors regular subset of $\mathbb{R}^{n}$ supporting a weak $(1,d)$-Poincaré inequality with respect to surface measure is uniformly rectifiable.

LMC algorithm receives first convergence guarantees under weak smoothness conditions.

This paper extends the convergence analysis of Langevin Monte Carlo beyond Poincaré inequalities.

For each $β>1$ we construct a family $F_β$ of metric measure spaces which is closed under the operation of taking weak-tangents (i.e.~blow-ups), and such that each element of $F_β$ admits a $(1,P)$-Poincaré inequality if and only if $P>β$.

The paper proves optimizability implies inequalities for sampling.

It is shown that curvature-dimension bounds CD(N, k) for a metric measure space (X,d,m) in the sense of Sturm imply a weak L^1- Poincare-inequality under some symmetry assumption on the choice of transport rays in the cut locus of (X,d). This condition is satisfied if (X,d) has m-almost surely no branching points.

We give sufficient conditions for a measured length space (X,d,m) to admit local and global Poincare inequalities. We first introduce a condition DM on (X,d,m), defined in terms of transport of measures. We show that DM, along with a doubling condition on m, implies a scale-invariant local Poincare inequality. We show …

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 Poincaré inequality and the measure contraction property follow from t…

New method samples from non-log-concave distributions with weak dissipativity.

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 …

Uniform Poincaré inequalities established for various metric spaces.

Study non-asymptotic Langevin Monte Carlo for Gibbs distributions.

The study improves Poincaré and log-Sobolev inequalities on hyperbolic spaces.

Let $(X,d,μ)$ be a complete metric measure space, with $μ$ a locally doubling measure, that supports a local weak $L^2$-Poincaré 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 func…

New proof shows inequality without restrictions.

We prove a Poincare type inequality for differential forms on compact manifolds by means of a constructive 'globalization' of a local Poincare inequality on convex sets.

Study Poincaré inequality in metric spaces via separating sets.

We investigate the possibility of improving the $p$-Poincaré inequality $\|\nabla_{\mathbb{H}^N} u\|_p \ge Λ_p \|u\|_p$ on the hyperbolic space, where $p>2$ and $Λ_p:=[(N-1)/p]^{p}$ is the best constant for which such inequality holds. We prove several different, and independent, improved inequalities, one of which is …

Using an inverse system of metric graphs as in: J. Cheeger and B. Kleiner, "Inverse limit spaces satisfying a Poincaré inequality", we provide a simple example of a metric space $X$ that admits Poincaré inequalities for a continuum of mutually singular measures.

New Poincaré inequality for differential forms on manifolds.

Proves inequality linking function deviation to gradient norm on compact manifolds.

The paper proves new inequalities for convex hypersurfaces using centro-affine geometry.

The paper derives new inequalities on manifolds and applies them to convex hypersurfaces.

We prove that complete Riemannian manifolds with polynomial growth and Ricci curvature bounded from below, admit uniform Poincaré inequalities. A global, uniform Poincaré inequality for horospheres in the universal cover of a closed, $n$-dimensional Riemannian manifold with pinched negative sectional curvature follows …

Sharp inequality linking interior and boundary Yamabe invariants on specific manifolds.

We study the validity of the $L^p$ inequality for the Riesz transform when $p>2$ and of its reverse inequality when $p<2$ on complete Riemannian manifolds under the doubling property and some Poincaré inequalities.

The paper proves Hardy inequalities on Finsler manifolds using superharmonicity.

The article proves a Poincaré inequality for hypersurfaces and applies it to rigidity results.

The paper proves inequalities on Finsler manifolds under Ricci curvature bounds.

Study complete manifolds with weighted Poincaré inequality and Ricci curvature bounds.

The paper introduces boundary operators for Poincaré-Einstein manifolds and proves higher order trace inequalities.

We study the isoperimetric, functional and concentration properties of $n$-dimensional weighted Riemannian manifolds satisfying the Curvature-Dimension condition, when the generalized dimension $N$ is negative, and more generally, is in the range $N \in (-\infty,1)$, extending the scope from the traditional range $N \i…

The development of global sensitivity analysis of numerical model outputs has recently raised new issues on 1-dimensional Poincaré inequalities. Typically two kind of sensitivity indices are linked by a Poincaré type inequality, which provide upper bounds of the most interpretable index by using the other one, cheaper …

A carpet is a metric space homeomorphic to the Sierpinski carpet. We characterize, within a certain class of examples, non-self-similar carpets supporting curve families of nontrivial modulus and supporting Poincaré inequalities. Our results yield new examples of compact doubling metric measure spaces supporting Poinca…

Short note proves Poincaré inequality for 4-manifold forms.

Study on functional inequalities on simple edge spaces.

We prove structure theorems for complete manifolds satisfying both the Ricci curvature lower bound and the weighted Poincaré inequality. In the process, a sharp decay estimate for the minimal positive Green's function is obtained. This estimate only depends on the weight function of the Poincaré inequality, and yields …

New proof of surface group theorem for 2D Poincaré duality groups.

The paper finds the Finsler structure of Apollonian weak metric on unit disc.

Sharp inequality for compactifying Poincaré-Einstein manifolds.

Sharp lower bound found for geodesic ball eigenvalues.

The study proves topological rigidity for certain geometric shapes using Poincaré inequalities.

The paper derives inequalities and formulas for generalized Ricci flow.

Graphs with nonnegative Bakry-Émery curvature have volume doubling and Poincaré inequalities.

The study improves Bochner inequality on Finsler manifolds to derive important inequalities.

In this paper we prove $L^{\infty}$ type gap theorems in Yang-Mills theory for complete four-dimensional manifolds with a weighted Poincaré inequality. We apply the theorems to a broad class of complete manifolds satisfying weighted Poincaré inequalities. In particular, we obtain a gap theorem on the Euclidean space wi…

We study some equivalent properties of the curvature-dimension conditions $CD(n,K)$ inequality on infinite, but locally finite graph. These equivalences are gradient estimate, Poincaré type inequalities and reverse Poincaré inequalities. And we also obtain one equivalent property of gradient estimate for a new notion o…

We prove an optimal reverse Poincaré inequality for the heat semigroup generated by the sub-Laplacian on a Carnot group of any step. As an application we give new proofs of the isoperimetric inequality and of the boundedness of the Riesz transform in Carnot groups.