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Niko Marola

Publications and source records attributed to Niko Marola.

16 recordsLinked to original sources
arXiv2016

Characterizations of sets of finite perimeter using heat kernels in metric spaces

The overarching goal of this paper is to link the notion of sets of finite perimeter (a concept associated with $N^{1,1}$-spaces) and the theory of heat semigroups (a concept related to $N^{1,2}$-spaces) in the setting of metric measure spaces whose measure is doubling and supports a $1$-Poincaré inequality. We prove a characterization of sets of finite perimeter in terms of a short time behavior of the heat semigroup in such metric spaces. We also give a new characterization of ${\rm BV}$ functions in terms of a near-diagonal energy in this general setting.

math.AP↗
arXiv2015

A characterization of BMO self-maps of a metric measure space

This paper studies functions of bounded mean oscillation (BMO) on metric spaces equipped with a doubling measure. The main result gives characterizations for mappings that preserve BMO. This extends the corresponding Euclidean results by Gotoh to metric measure spaces. The argument is based on a generalizations Uchiyama's construction of certain extremal BMO-functions and John-Nirenberg's lemma.

math.CA↗
arXiv2015

Homeomorphisms of the Heisenberg group preserving BMO

We provide a new geometric proof of Reimann's theorem characterizing quasiconformal mappings as the ones preserving functions of bounded mean oscillation. While our proof is new already in the Euclidean spaces, it is applicable in Heisenberg groups as well as in more general stratified nilpotent Carnot groups.

math.CA↗
arXiv2015

Boundary measures, generalized Gauss-Green formulas, and mean value property in metric measure spaces

We study mean value properties of harmonic functions in metric measure spaces. The metric measure spaces we consider have a doubling measure and support a (1,1)- Poincaré inequality. The notion of harmonicity is based on the Dirichlet form defined in terms of a Cheeger differentiable structure. By studying fine properties of the Green function on balls, we characterize harmonic functions in terms of a mean value property. As a consequence, we obtain a detailed description of Poisson kernels. We shall also obtain a Gauss-Green type formula for sets of finite perimeter which posses a Minkowski content characterization of the perimeter. For the Gauss-Green formula we introduce a suitable notion of the interior normal trace of a regular ball.

math.AP↗
arXiv2015

Phragmén-Lindelöf theorem for infinity harmonic functions

We investigate a version of the Phragmén-Lindelöf theorem for solutions of the equation $Δ_\infty u=0$ in unbounded convex domains. The method of proof is to consider this infinity harmonic equation as the limit of the $p$-harmonic equation when $p$ tends to $\infty$.

math.AP↗
arXiv2014

Aspects of local to global results

We establish local to global results for a function space which is larger than the well known BMO space, and was also introduced by John and Nirenberg.

math.CA↗
arXiv2014

On the problem of unique continuation for the p-Laplace equation

We study if two different solutions of the $p$-Laplace equation $$\nabla\cdot(|\nabla u|^{p-2}\nabla u)=0,$$ where $1<p<\infty$, can coincide in an open subset of their common domain of definition. We obtain some partial results on this interesting problem.

math.AP↗
arXiv2013

On the Harnack inequality for parabolic minimizers in metric measure spaces

In this note we consider problems related to parabolic partial differential equations in geodesic metric measure spaces, that are equipped with a doubling measure and a Poincaré inequality. We prove a location and scale invariant Harnack inequality for a minimizer of a variational problem related to a doubly non-linear parabolic equation involving the p-Laplacian. Moreover, we prove the sufficiency of the Grigor'yan--Saloff-Coste theorem for general p > 1 in geodesic metric spaces. The approach used is strictly variational, and hence we are able to carry out the argument in the metric setting.

math.AP↗
arXiv2008

Sharp capacitary estimates for rings in metric spaces

We establish sharp estimates for the $p$-capacity of metric rings with unrelated radii in metric measure spaces equipped with a doubling measure and supporting a Poincaré inequality. These estimates play an essential role in the study of the local behavior of \p-harmonic Green's functions.

math.MG↗