arXiv subjects

Jana Bjorn

Publications and source records attributed to Jana Bjorn.

7 recordsLinked to original sources

arXiv2021

Classification of metric measure spaces and their ends using $p$-harmonic functions

By seeing whether a Liouville type theorem holds for positive, bounded, and/or finite energy $p$-harmonic and $p$-quasiharmonic functions, we classify proper metric spaces equipped with a locally doubling measure supporting a local $p$-Poincaré inequality. Similar classifications have earlier been obtained for Riemann surfaces and Riemannian manifolds. We also study the inclusions between these classes of metric measure spaces, and their relationship to the $p$-hyperbolicity of the metric space and its ends. In particular, we characterize spaces that carry nonconstant $p$-harmonic functions with finite energy as spaces having at least two well-separated $p$-hyperbolic sequences. We also show that every such space $X$ has a function $f \notin L^p(X) + \mathbb{R} $ with finite $p$-energy.

math.MG↗

arXiv2021

Extension and trace results for doubling metric measure spaces and their hyperbolic fillings

In this paper we study connections between Besov spaces of functions on a compact metric space $Z$, equipped with a doubling measure, and the Newton--Sobolev space of functions on a uniform domain $X_\varepsilon$. This uniform domain is obtained as a uniformization of a (Gromov) hyperbolic filling of $Z$. To do so, we construct a family of hyperbolic fillings in the style of the work of Bonk and Kleiner and the work of Bourdon and Pajot. Then for each parameter $β>0$ we construct a lift $μ_β$ of the doubling measure $ν$ on $Z$ to $X_\varepsilon$, and show that $μ_β$ is doubling and supports a $1$-Poincaré inequality. We then show that for each $θ$ with $0<θ<1$ and $p\ge 1$ there is a choice of $β=p(1-θ)\logα$ such that the Besov space $B^θ_{p,p}(Z)$ is the trace space of the Newton--Sobolev space $N^{1,p}(X_\varepsilon,μ_β)$ when $\varepsilon=\logα$. Finally, we exploit the tools of potential theory on $X_\varepsilon$ to obtain fine properties of functions in $B^θ_{p,p}(Z)$, such as their quasicontinuity and quasieverywhere existence of $L^q$-Lebesgue points with $q=s_νp/(s_ν-pθ)$, where $s_ν$ is a doubling dimension associated with the measure $ν$ on $Z$. Applying this to compact subsets of Euclidean spaces improves upon a result of Netrusov in $\mathbb{R}^n$.

math.MG↗

arXiv2019

Bounded geometry and $p$-harmonic functions under uniformization and hyperbolization

The uniformization and hyperbolization transformations formulated by Bonk, Heinonen and Koskela in \emph{"Uniformizing Gromov Hyperbolic Spaces"}, Astérisque {\bf 270} (2001), dealt with geometric properties of metric spaces. In this paper we consider metric measure spaces and construct a parallel transformation of measures under the uniformization and hyperbolization procedures. We show that if a locally compact roughly starlike Gromov hyperbolic space is equipped with a measure that is uniformly locally doubling and supports a uniformly local $p$-Poincaré inequality, then the transformed measure is globally doubling and supports a global $p$-Poincaré inequality on the corresponding uniformized space. In the opposite direction, we show that such global properties on bounded locally compact uniform spaces yield similar uniformly local properties for the transformed measures on the corresponding hyperbolized spaces. We use the above results on uniformization of measures to characterize when a Gromov hyperbolic space, equipped with a uniformly locally doubling measure supporting a uniformly local $p$-Poincaré inequality, carries nonconstant globally defined $p$-harmonic functions with finite $p$-energy. We also study some geometric properties of Gromov hyperbolic and uniform spaces. While the Cartesian product of two Gromov hyperbolic spaces need not be Gromov hyperbolic, we construct an indirect product of such spaces that does result in a Gromov hyperbolic space. This is done by first showing that the Cartesian product of two bounded uniform domains is a uniform domain.

math.MG↗

arXiv2018

Sphericalization and p-harmonic functions on unbounded domains in Ahlfors regular metric spaces

We use sphericalization to study the Dirichlet problem, Perron solutions and boundary regularity for p-harmonic functions on unbounded sets in Ahlfors regular metric spaces. Boundary regularity for the point at infinity is given special attention. In particular, we allow for several "approach directions" towards infinity and take into account the massiveness of their complements. In 2005, Llorente-Manfredi-Wu showed that the p-harmonic measure on the upper half space $R^n_+, n \ge 2$, is not subadditive on null sets when $p \neq 2$. Using their result and spherical inversion, we create similar bounded examples in the unit ball $B \subset R^n$ showing that the n-harmonic measure is not subadditive on null sets when $n \ge 3$, and neither are the p-harmonic measures in $B$ generated by certain weights depending on $p\neq 2$ and $n \ge 2$.

math.AP↗

arXiv2018

Locally $p$-admissible measures on $\mathbb{R}$

In this note we show that locally $p$-admissible measures on $\mathbb{R}$ necessarily come from local Muckenhoupt $A_p$ weights. In the proof we employ the corresponding characterization of global $p$-admissible measures on $\mathbb{R}$ in terms of global $A_p$ weights due to Björn, Buckley and Keith, together with tools from analysis in metric spaces, more specifically preservation of the doubling condition and Poincaré inequalities under flattening, due to Durand-Cartagena and Li. As a consequence, the class of locally $p$-admissible weights on $\mathbb{R}$ is invariant under addition and satisfies the lattice property. We also show that measures that are $p$-admissible on an interval can be partially extended by periodical reflections to global $p$-admissible measures. Surprisingly, the $p$-admissibility has to hold on a larger interval than the reflected one, and an example shows that this is necessary.

math.MG↗

arXiv2013

The Dirichlet problem for p-harmonic functions with respect to the Mazurkiewicz boundary

In this paper we develop the Perron method for solving the Dirichlet problem for the analog of the p-Laplacian, i.e. for p-harmonic functions, with Mazurkiewicz boundary values. The setting considered here is that of metric spaces, where the boundary of the domain in question is replaced with the Mazurkiewicz boundary. Resolutivity for Sobolev and continuous functions, as well as invariance results for perturbations on small sets, are obtained. We use these results to improve the known resolutivity and invariance results for functions on the standard (metric) boundary. We also illustrate the results of this paper by discussing several examples.

math.MG↗

arXiv2012

Prime ends for domains in metric spaces

In this paper we propose a new definition of prime ends for domains in metric spaces under rather general assumptions. We compare our prime ends to those of Carathéodory and Näkki. Modulus ends and prime ends, defined by means of the \p-modulus of curve families, are also discussed and related to the prime ends. We provide characterizations of singleton prime ends and relate them to the notion of accessibility of boundary points, and introduce a topology on the prime end boundary. We also study relations between the prime end boundary and the Mazurkiewicz boundary. Generalizing the notion of John domains, we introduce almost John domains, and we investigate prime ends in the settings of John domains, almost John domains and domains which are finitely connected at the boundary.

math.MG↗