Modulus and the Poincar� inequality on metric measure spaces

Abstract

The purpose of this paper is to develop the understanding of modulus and the Poincaré inequality, as defined on metric measure spaces. Various definitions for modulus and capacity are shown to coincide for general collections of metric measure spaces. Consequently, modulus is shown to be upper semi-continuous with respect to the limit of a sequence of curve families contained in a converging sequence of metric measure spaces. Moreover, several competing definitions for the Poincaré inequality are shown to coincide, if the underlying measure is doubling. One such characterization considers only continuous functions and their continuous upper gradients, and extends work of Heinonen and Koskela. Applications include showing that the p-Poincaré inequality (with a doubling measure), for p ≥ 1, persists through to the limit of a sequence of converging pointed metric measure spaces -this extends results of Cheeger. A further application is the construction of new doubling measures in Euclidean space which admit a 1-Poincaré inequality.

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

“…Poincaré inequalities and moduli of curve families. The following result of Keith [23,Theorem 2] will be of great importance in this paper.…”

“…We state here a version of this result due to Keith [23], in a form which is suitable for our setting. For similar results, see Koskela [27] and Cheeger [10, §9].…”

“…By Keith [23,Theorem 1], it suffices to construct curve families Γ ′ m contained in S m for each m with p-modulus uniformly bounded from below. Since modulus is upper semicontinuous, this bound will continue to the limit.…”

The Lip-lip equality is stable under blow-up

“…Moreover, (2.35) holds with K = 1; (Keith): [Kei02,Kei04]; if (X, µ) is a doubling metric measure space which satisfies the Lip-lip inequality (2.35), then (X, µ) is a differentiability space whose analytic dimension is bounded by an expression that depends only on C µ and K. Moreover [Kei03], the Poincaré inequality is stable under measured Gromov-Hausdorff convergence provided all the relevant constants are uniformly bounded; for example, blow-ups of PI-spaces are PI-spaces (with the same PI-exponent); (Bate-Speight): [BS13]; if (X, µ) is a differentiability space then µ is asymptotically doubling in the sense that for µ-a.e. x there are (C x , r x ) ∈ (0, ∞) 2 such that:…”

“…This question should be compared with the easier case where (X, µ) is a PI-space: then any blow-up is still a PI-space. This follows from the stability of the Poincaré inequality (with uniform constants) under measured Gromov-Hausdorff convergence, which can be seen using Keith's elegant characterization of PI-spaces in terms of moduli of families of curves [Kei03]. Essentially, the argument reduces to the uppersemicontinuity of modulus (which is dual to the lower-semicontinuity of length).…”