A note on rearrangement Poincar'e inequalities and the doubling 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].

Sobolev type inequalities for rearrangement invariant spaces

Positivity, 2010

In the setting of rearrangement invariant spaces, optimal Sobolev inequalities (via the gradient) are well understood. By means of an alternative functional, we obtain new Sobolev inequalities which are finer than (and not necessarily equivalent to) the ones mentioned above.

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.

The Poincaré inequality is an open ended condition

Annals of Mathematics, 2008

Let p > 1 and let (X, d, μ) be a complete metric measure space with μ Borel and doubling that admits a (1, p)-Poincaré inequality. Then there exists ε > 0 such that (X, d, μ) admits a (1, q)-Poincaré inequality for every q > p−ε, quantitatively.

Characterizations of weak reverse H\"older inequalities on metric measure spaces

2021

We present ten different characterizations of functions satisfying a weak reverse Hölder inequality on an open subset of a metric measure space with a doubling measure. Among others, we describe these functions as a class of weak A∞ weights, which is a generalization of Muckenhoupt weights that allows for nondoubling weights. Although our main results are modeled after conditions that hold true for Muckenhoupt weights, we also discuss two conditions for Muckenhoupt A∞ weights that fail to hold for weak A∞ weights.

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.