# Plans on measures and AM-modulus

## TL;DR

This paper explores the relationship between plans on measures and the $AM$-modulus in the limiting case $p=1$, revealing new insights into their equivalence and behavior compared to traditional capacities.

## Contribution

It extends the theory of plans on measures to the case $p=1$, showing how the $AM$-modulus can be derived via plan approaches and analyzing its unique properties.

## Key findings

- $AM$-modulus can be obtained through plan approaches at $p=1$
- Unexpected behaviors of $AM$-modulus compared to usual capacities
- Relations between $M_1$-modulus and $AM$-modulus are established

## 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 Savar\'e 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.