arXiv subjects

Vendula Honzlová Exnerová

Publications and source records attributed to Vendula Honzlová Exnerová.

2 recordsLinked to original sources

arXiv2020

Plans on measures and AM-modulus

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.

math.FA↗

arXiv2019

AM-modulus and Hausdorff measure of codimension one in metric measure spaces

Let $Γ(E)$ be the family of all paths which meet a set $E$ in the metric measure space $X$. The set function $E \mapsto AM(Γ(E))$ defines the $AM$--modulus measure in $X$ where $AM$ refers to the approximation modulus. We compare $AM(Γ(E))$ to the Hausdorff measure $co\mathcal H^1(E)$ of codimension one in $X$ and show that $$co\mathcal H^1(E) \approx AM(Γ(E))$$ for Suslin sets $E$ in $X$. This leads to a new characterization of sets of finite perimeter in $X$ in terms of the $AM$--modulus. We also study the level sets of $BV$ functions and show that for a.e. $t$ these sets have finite $co\mathcal H^1$--measure. Most of the results are new also in $\mathbb R^n$.

math.FA↗