arXiv subjects

Olli Martio

Publications and source records attributed to Olli Martio.

7 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↗

arXiv2011

Metrization of weighted graphs

We find a set of necessary and sufficient conditions under which the weight $w:E\to\mathbb R^+$ on the graph $G=(V,E)$ can be extended to a pseudometric $d:V\times V\to\mathbb R^+$. If these conditions hold and $G$ is a connected graph, then the set $\mathfrak M_w$ of all such extensions is nonvoid and the shortest-path pseudometric $d_w$ is the greatest element of $\mathfrak M_w$ with respect to the partial ordering $d_1 \leqslant d_2$ if and only if $d_1(u,v) \leqslant d_2(u,v)$ for all $u,v\in V$. It is shown that every nonvoid poset $(\mathfrak M_w,\leqslant)$ contains the least element $ρ_{0,w}$ if and only if $G$ is a complete $k$-partite graph with $k\geqslant 2$ and in this case the explicit formula for computation of $ρ_{0,w}$ is obtained.

math.CO↗

arXiv2010

Wiman and Arima theorems for quasiregular mappings

Generalizations of the theorems of Wiman and of Arima on entire functions are proved for spatial quasiregular mappings.

math.CA↗

arXiv2009

Products of metric spaces, covering numbers, packing numbers and characterizations of ultrametric spaces

We describe some Cartesian products of metric spaces and find conditions under which products of ultrametric spaces are ultrametric.

math.MG↗

arXiv2007

Ahlfors theorems for differential forms

Some counterparts of theorems of Phragmén-Lindelöf and of Ahlfors are proved for differential forms of ${\cal WT}$--classes.}

math.AP↗

arXiv2007

On quasiplanes in Euclidean spaces

A variational inequality for the images of $k$-dimensional hyperplanes under quasiconformal maps of the $n$-dimensional Euclidean space is proved when $1\le k\le n-2 .$

math.CA↗