The Poincaré inequality is an open ended condition

Abstract

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.

Modulus and Poincaré Inequalities on Non-Self-Similar Sierpiński Carpets

“…Metric spaces equipped with doubling measures that support Poincaré inequalities (also known as PI spaces) are ideal environments for first-order analysis and differential geometry [20], [10], [21], [24], [26]. Extending the scope of this theory by verifying Poincaré inequalities on new classes of spaces is a problem of high interest and relevance.…”

“…A significant recent result of Keith and Zhong [26] asserts that the set of values of p for which a given complete PI space supports a p-Poincaré inequality, is necessarily an open subset of [1, +∞). Theorems 1.5 and 1.6 have a number of interesting consequences which we now enunciate.…”

Adams inequality on metric measure spaces

“…In Theorem 1.4, we assume that weak (1, t)-Poincaré inequality holds for some 1 ≤ t < p. This better Poincaré inequality follows from the weak (1, p)-Poincaré inequality by the result in [13]. The case p = s is not included in the Theorem 1.4 when the weak (1, 1)-Poincaré inequality holds.…”

Real Interpolation of Sobolev Spaces Associated to a Weight

“…Let (X, d, µ) be a complete metric-measure space with µ locally doubling and admitting a local Poincaré inequality (P qloc ), for some 1 < q < ∞. Then there exists ǫ > 0 such that (X, d, µ) admits (P ploc ) for every p > q − ǫ (see [20] and section 4 in [6]).…”

“…Thus, if the set of q such that (P qloc ) holds is not empty, then it is an interval unbounded on the right. A recent result from Keith-Zhong [20] asserts that this interval is open in [1, +∞[ in the following sense:…”