The Poincaré inequality is an open ended condition (original) (raw)

First order Poincaré inequalities in metric measure spaces

Annales Academiae Scientiarum Fennicae Mathematica, 2013

We study a generalization of classical Poincaré inequalities, and study conditions that link such an inequality with the first order calculus of functions in the metric measure space setting when the measure is doubling and the metric is complete. The first order calculus considered in this paper is based on the approach of the upper gradient notion of Heinonen and Koskela [HeKo]. We show that under a Vitali type condition on the BMO-Poincaré type inequality of Franchi, Pérez and Wheeden [FPW], the metric measure space should also support a p-Poincaré inequality for some 1 ≤ p < ∞, and that under weaker assumptions, the metric measure space supports an ∞-Poincaré inequality in the sense of [DJS].

Uniform Poincaré inequalities on measured metric spaces

manuscripta mathematica

Consider a proper geodesic metric space (X, d) equipped with a Borel measure µ. We establish a family of uniform Poincaré inequalities on (X, d, µ) if it satisfies a local Poincaré inequality (P loc) and a condition on growth of volume. Consequently if µ is doubling and supports (P loc) then it satisfies a (σ, β, σ)-Poincaré inequality. If (X, d, µ) is a δ-hyperbolic space then using the volume comparison theorem in [3] we obtain a uniform Poincaré inequality with exponential growth of the Poincaré constant. Next we relate growth of Poincaré constants to growth of discrete subgroups of isometries of X which act on it properly. We show that if X is the universal cover of a compact CD(K, ∞) space then it supports a uniform Poincaré inequality and the Poincaré constant depends on the growth of the fundamental group.

Poincaré inequality meets Brezis–Van Schaftingen–Yung formula on metric measure spaces

Journal of Functional Analysis

Let (X, ρ, µ) be a metric measure space of homogeneous type which supports a certain Poincaré inequality. Denote by the symbol C * c (X) the space of all continuous functions f with compact support satisfying that Lip f := lim sup r→0 sup y∈B(•,r) | f (•) − f (y)|/r is also a continuous function with compact support and Lip f = lim r→0 sup y∈B(•,r) | f (•) − f (y)|/r converges uniformly. Let p ∈ [1, ∞). In this article, the authors prove that, for any f ∈ C * c (X), sup λ∈(0,∞) λ p X µ y ∈ X : | f (x) − f (y)| > λρ(x, y)[V(x, y)] 1 p dµ(x) ∼ X [Lip f (x)] p dµ(x) with the positive equivalence constants independent of f , where V(x, y) := µ(B(x, ρ(x, y))). This generalizes a recent surprising formula of H. Brezis, J. Van Schaftingen, and P.-L. Yung from the n-dimensional Euclidean space R n to X. Applying this generalization, the authors establish new fractional Sobolev and Gagliardo-Nirenberg inequalities in X. All these results have a wide range of applications. Particularly, when applied to two concrete examples, namely, R n with weighted Lebesgue measure and the complete n-dimensional Riemannian manifold with non-negative Ricci curvature, all these results are new. The proofs of these results strongly depend on the geometrical relation of differences and derivatives in the metric measure space and the Poincaré inequality.

Lebesgue points via the Poincaré inequality

Science China Mathematics, 2015

In this article, we show that in a Q-doubling space (X, d, µ), Q > 1, which satisfies a chain condition, if we have a Q-Poincaré inequality for a pair of functions (u, g) where g ∈ L Q (X), then u has Lebesgue points H h -a.e. for h(t) = log 1−Q−ǫ (1/t).

A maximal Function Approach to Two-Measure Poincaré Inequalities

The Journal of Geometric Analysis

This paper extends the self-improvement result of Keith and Zhong in [16] to the two-measure case. Our main result shows that a two-measure (p, p)-Poincaré inequality for 1 < p < ∞ improves to a (p, p − ε)-Poincaré inequality for some ε > 0 under a balance condition on the measures. The corresponding result for a maximal Poincaré inequality is also considered. In this case the left-hand side in the Poincaré inequality is replaced with an integral of a sharp maximal function and the results hold without a balance condition. Moreover, validity of maximal Poincaré inequalities is used to characterize the self-improvement of two-measure Poincaré inequalities. Examples are constructed to illustrate the role of the assumptions. Harmonic analysis and PDE techniques are used extensively in the arguments.

A simple proof of the Poincaré inequality for a large class of probability measures

Electronic Communications in Probability, 2008

We give a simple and direct proof of the existence of a spectral gap under some Lyapunov type condition which is satisfied in particular by log-concave probability measures on Rn. The proof is based on arguments introduced in (2), but for the sake of completeness, all details are provided.