Plans on measures and AM-modulus

Vendula Honzlová Exnerová, Ondřej F. K. Kalenda, Jan Malý, Olli Martio

Published 2019-04-09Version 1

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 [4], 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.