arXiv research

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.

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.

We prove that for combinatorial graphs with non-negative Ollivier curvature, one has \[ \|P_t μ- P_t ν\|_1 \leq \frac{W_1(μ,ν)}{\sqrt{t}} \] for all probability measures $μ,ν$ where $P_t$ is the heat semigroup and $W_1$ is the $\ell_1$-Wasserstein distance. This turns out to be an equivalent formulation of a version of…

We continue our study of geometric analysis on (possibly non-reversible) Finsler manifolds, based on the Bochner inequality established by the author and Sturm. Following the approach of the $Γ$-calculus a la Bakry et al, we show the dimensional versions of the Poincare--Lichnerowicz inequality, the logarithmic Sobolev…

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…

A reverse Riesz estimate and spectral gap imply a Poincaré inequality.

Let $\mathbb{M}$ be a smooth connected manifold endowed with a smooth measure $μ$ and a smooth locally subelliptic diffusion operator $L$ satisfying $L1=0$, and which is symmetric with respect to $μ$. We show that if $L$ satisfies, with a non negative curvature parameter, the generalized curvature inequality introduced…

The paper proves Hardy inequalities on Finsler manifolds using superharmonicity.

We establish various $L^{p}$ estimates for the Schrödinger operator $-Δ+V$ on Riemannian manifolds satisfying the doubling property and a Poincaré inequality, where $Δ$ is the Laplace-Beltrami operator and $V$ belongs to a reverse Hölder class. At the end of this paper we apply our result on Lie groups with polynomial …

Uniform Poincaré inequalities established for various metric spaces.

Sharp Gaussian isoperimetry proven along Ricci flow.

We establish a quantitative isoperimetric inequality for weighted Riemannian manifolds with $\mathrm{Ric}_{\infty} \ge 1$. Precisely, we give an upper bound of the volume of the symmetric difference between a Borel set and a sub-level (or super-level) set of the associated guiding function (arising from the needle deco…

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

In my previaou paper of K. Horihata, we have proposed a Ginzburg-Landau system with a time-dependent parameter and then passing to the limit we have constructed a harmonic heat flow into spheres. Thanks to this scheme, we establish a few energy inequalities of our flow: (i) monotonical inequalities and (ii) a reverse P…

Sharp stability results for reverse isoperimetric inequalities in 2D.

Let $(X,d,μ)$ be a doubling metric measure space endowed with a Dirichlet form $\E$ deriving from a "carré du champ". Assume that $(X,d,μ,\E)$ supports a scale-invariant $L^2$-Poincaré inequality. In this article, we study the following properties of harmonic functions, heat kernels and Riesz transforms for $p\in (2,\i…

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.

Lorentz-Finsler geometry reveals new and old inequalities.

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.

Proof of reverse isoperimetric inequality for black holes.

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.

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.

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.

Sharp inequality for compactifying Poincaré-Einstein manifolds.

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.

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

Sharp lower bound found for geodesic ball eigenvalues.

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

The paper proves a reverse isoperimetric inequality and applies it to analyze surface flows.

The paper derives inequalities and formulas for generalized Ricci flow.

Established a Hardy inequality on Finsler manifolds.

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