On the duality between $p$-modulus and probability measures

Abstract

Motivated by recent developments on calculus in metric measure spaces (X, d, m), we prove a general duality principle between Fuglede's notion [15] of p-modulus for families of finite Borel measures in (X, d) and probability measures with barycenter in L q (X, m), with q dual exponent of p ∈ (1, ∞). We apply this general duality principle to study null sets for families of parametric and non-parametric curves in X. In the final part of the paper we provide a new proof, independent of optimal transportation, of the equivalence of notions of weak upper gradient based on ppodulus ([21], [23]) and suitable probability measures in the space of curves ([6], [7]).

Three superposition principles: Currents, continuity equations and curves of measures

“…in the theory of Sobolev and BV spaces over metric measure spaces (see, e.g. [19,3,4,5,9,8,7,1]). In the present paper we extend the original result of S. Lisini showing additionally the relationship between curves of measures and continuity equations for general metric space setting, and show that in fact this result is a corollary of the superposition principle for solutions of continuity equations (i.e.…”

The Poincaré Inequality Does Not Improve with Blow-Up

“…The following Theorem summarizes a geometric characterization of (1, P )-Poincaré inequalities. It combines results of Heinonen-Koskela [HK98], Haj lasz-Koskela [HK95], Keith [Kei03], and Ambrosio, Di Marino and Savaré [ADS13], and the proof is included just for the sake of completeness. Note that we will take Theorem 4.7 as the working definition of the Poincaré inequality, and so we will not need to recall the usual definition of the Poincaré inequality.…”

“…Applying the main result of [ADS13] we get a probability π on Σ good such that, denoting by ν = Σ good η dπ(η), we get:…”

Tensorization of Cheeger energies, the space H1,1 and the area formula for graphs

“…See [4] for a much more detailed comparison between notions of negligibility for families of curves, both parametric and non-parametric. The next theorem is one of the main results of [1].…”