On the duality between p-Modulus and probability measures

Abstract

Motivated by recent developments on calculus in metric measure spaces $(X,\mathsf{d},\mathfrak{m})$, we prove a general duality principle between Fuglede's notion of $p$-modulus for families of finite Borel measures in $(X,\mathsf{d})$ and probability measures with barycenter in $L^q(X,\mathfrak{m})$, with $q$ dual exponent of $p\in (1,\infty )$. 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 $p$-Modulus (Koskela-MacManus '98, Shanmugalingam '00) and suitable probability measures in the space of curves (Ambrosio-Gigli-Savare '11)

Minor corrections, typos fixed

References in corpus (2)

• Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces
• Structure of measures in Lipschitz differentiability spaces

Cited by in corpus (7)

• Sobolev and BV spaces on metric measure spaces via derivations and integration by parts
• Curvewise characterizations of minimal upper gradients and the construction of a Sobolev differential
• Almost uniform domains and Poincaré inequalities
• Tensorization of Cheeger energies, the space $H^{1,1}$ and the area formula for graphs
• Plans on measures and AM-modulus
• AM-modulus and Hausdorff measure of codimension one in metric measure spaces
• Metric Sobolev spaces I: equivalence of definitions