Poincaré inequality for abstract spaces

Abstract

POINCARE INEQUALITY FOR ABSTRACT SPACES ALIREZA RANJBAR-MOTLAGHThe Poincare inequality is generalised to metric-measure spaces which support a strong version of the doubling condition. This generalises the Poincare inequality for manifolds whose Ricci curvature is bounded from below and metric-measure spaces which satisfy the measure contraction property.

Heat Flow on Alexandrov Spaces

“…Therefore E also coincides with Cheeger's energy functional ( [9]), because the local Lipschitz constant is the minimal generalized upper gradient ([9, Theorem 6.1]) and Lipschitz functions are dense in both Sobolev spaces (thanks to the weak Poincaré inequality for upper gradients and the volume doubling condition, [9,Theorem 4.24]). Indeed, in our framework, the volume doubling condition directly follows from the Bishop-Gromov volume comparison theorem and the Poincaré inequality is a consequence of [21] and [36], for instance.…”

“…with some constant C > 0 being independent of t, x, y, for small t. It gives a bound for T t L 1 →L ∞ . By a general argument, (4.8) follows from the local Poincaré inequality and the volume doubling condition, both of which depend only on the dimension n of X and a lower curvature bound (see [20,21,36,39] for instance). However, we should be careful if we want to know whether C in (4.8) depends on the diameter and/or the volume of X.…”

A topological splitting theorem for weighted Alexandrov spaces

“…We assume that µ satisfies BG(κ, N) on some neighborhood Ω ′ of Ω for two real numbers N ≥ 1 and κ. According to the result of Ranjbar-Motlagh [37], we have a (1, 1)-Poincaré inequality on any ball in Ω ′ in the sense of upper gradient, which implies a (1, p)-Poincaré inequality on any ball in Ω ′ for any p ≥ 1. Since the infinitesimal Bishop-Gromov condition implies the volume doubling condition for balls in Ω ′ , we can apply Cheeger's theory [5] of Sobolev spaces on the metric measure space (Ω, d, µ).…”

“…For an n-dimensional complete Riemannian manifold, the Riemannian volume measure satisfies BG(κ, n) if and only if the Ricci curvature satisfies Ric ≥ (n − 1)κ (see [27,Theorem 3.2] for the 'only if' part). We see some studies on similar (or same) conditions to BG(κ, N) in [6,13,39,18,19,37,27,45,21] etc. BG(κ, N) is sometimes called the Measure Contraction Property and is weaker than the curvaturedimension (or lower N-Ricci curvature) condition CD((N − 1)κ, N) introduced by Sturm [40] and Lott-Villani [24] in terms of mass transportation.…”

Infinitesimal Bishop–Gromov condition for Alexandrov spaces

“…For an n-dimensional complete Riemannian manifold, the Riemannian volume measure satisfies BG(κ, n) if and only if the Ricci curvature satisfies Ric ≥ (n − 1)κ (see Theorem 3.2 of [10] for the 'only if' part). We see some studies on similar (or same) conditions to BG(κ, n) in [2, 18,6,7,15,10,21] etc. BG(κ, n) is sometimes called the Measure Contraction Property and is weaker than the curvature-dimension (or lower n-Ricci curvature) condition, CD((n−1)κ, n), introduced by Sturm [19,20] and Lott-Villani [9] in terms of mass transportation.…”