Plans on measures and AM-modulus

Abstract

For measuring families of curves, or, more generally, of measures, $M_p$-modulus is traditionally used. More recent studies use so-called plans on measures. In their fundamental paper \cite{ADS}, Ambrosio, Di Marino and Savaré proved that these two approaches are in some sense equivalent within $1<p<\infty$. We consider the limiting case $p=1$ and show that the $AM$-modulus can be obtained alternatively by the plan approach. On the way, we demonstrate unexpected behavior of the $AM$-modulus in comparison with usual capacities and consider the relations between the $M_1$--modulus and the $AM$--modulus.

References in corpus (3)

• Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below
• Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces
• On the duality between p-Modulus and probability measures

Cited by in corpus (1)

• Curvewise characterizations of minimal upper gradients and the construction of a Sobolev differential