arXiv subjects
Publications and source records attributed to Pekka Koskela.
In this article, we show that in a $Q$-doubling space $(X,d,μ),$ $Q>1,$ that supports a $Q$-Poincaré inequality and satisfies a chain condition, sets of $Q$-capacity zero have generalized Hausdorff $h$-measure zero for $h(t)=\log^{1-Q-ε}(1/t).$
In this article, we show that in a $Q$-doubling space $(X,d,μ),$ $Q>1,$ which satisfies a chain condition, if we have a $Q$-Poincaré inequality for a pair of functions $(u,g)$ where $g\in L^Q(X),$ then $u$ has Lebesgue points $H^h$-a.e. for $h(t)=\log^{1-Q-ε}(1/t).$ We also discuss how the existence of Lebesgue points follows for $u\in W^{1,Q}(X)$ where $(X,d,μ)$ is a complete $Q$-doubling space supporting a $Q$-Poincaré inequality for $Q>1.$
We provide new conditions that ensure that two metric measure spaces are not quasiconformally equivalent. As an application we deduce that there exists no quasiconformal map between the sub-Riemannian Heisenberg and roto-translation groups.
In this paper, we establish Luzin's condition (N) for mappings in certain Sobolev-Orlicz spaces with certain moduli of continuity. Further, given a mapping in these Sobolev-Orlicz spaces, we give bounds on the size of the exceptional set where Luzin's condition (N) may fail. If a mapping violates Luzin's condition (N), we show that there is a Cantor set of measure zero that is mapped to a set of positive measure.
Let $Ω\subset \rr^2$ be a bounded simply connected domain. We show that, for a fixed (every) $p\in (1,\fz),$ the divergence equation $\mathrm{div}\,\mathbf{v}=f$ is solvable in $W^{1,p}_0(Ω)^2$ for every $f\in L^p_0(Ω)$, if and only if $Ω$ is a John domain, if and only if the weighted Poincaré inequality $$\int_Ω|u(x)-u_Ω|^q\,dx\le C\int_Ω|\nabla u(x)|^q\dist(x,\partial Ω)^q\,dx$$ holds for some (every) $q\in [1,\fz)$. In higher dimensions similar results are proved under some additional assumptions on the domain in question.
The aim of this paper is two-fold: First, we obtain a better understanding of the intrinsic distance of diffusion processes. Precisely, (i) for all $n\ge1$, the diffusion matrix $A$ is weak upper semicontinuous on $Ω$ if and only if the intrinsic differential and the local intrinsic distance structures coincide; (ii) if $n=1$, or if $n\ge2$ and $A$ is weak upper semicontinuous on $Ω$, the intrinsic distance and differential structures always coincide; (iii) if $n\ge2$ and $A$ fails to be weak upper semicontinuous on $Ω$, the (non-) coincidence of the intrinsic distance and differential structures depend on the geometry of the non-weak-upper-semicontinuity set of $A$. Second, for an arbitrary diffusion matrix $A$, we show that the intrinsic distance completely determines the absolute minimizer of the corresponding $L^\infty$-variational problem, and then obtain the existence and uniqueness for given boundary data. We also give an example of a diffusion matrix $A$ for which there is an absolute minimizer that is not of class $C^1$. When $A$ is continuous, we also obtain the linear approximation property of the absolute minimizer.
We prove that for mappings $W^{1,n}(B^n, \R^n),$ continuous up to the boundary, with modulus of continuity satisfying certain divergence condition, the image of the boundary of the unit ball has zero $n$-Hausdorff measure. For Hölder continuous mappings we also prove an essentially sharp generalized Hausdorff dimension estimate.
Let $ \mathscr E $ be a regular, strongly local Dirichlet form on $L^2(X, m)$ and $d$ the associated intrinsic distance. Assume that the topology induced by $d$ coincides with the original topology on $ X$, and that $X$ is compact, satisfies a doubling property and supports a weak $(1, 2)$-Poincaré inequality. We first discuss the (non-)coincidence of the intrinsic length structure and the gradient structure. Under the further assumption that the Ricci curvature of $X$ is bounded from below in the sense of Lott-Sturm-Villani, the following are shown to be equivalent: (i) the heat flow of $\mathscr E$ gives the unique gradient flow of $\mathscr U_\infty$, (ii) $\mathscr E$ satisfies the Newtonian property, (iii) the intrinsic length structure coincides with the gradient structure. Moreover, for the standard (resistance) Dirichlet form on the Sierpinski gasket equipped with the Kusuoka measure, we identify the intrinsic length structure with the measurable Riemannian and the gradient structures. We also apply the above results to the (coarse) Ricci curvatures and asymptotics of the gradient of the heat kernel.
We characterize Gromov hyperbolicity of the quasihyperbolic metric space (Ω,k) by geometric properties of the Ahlfors regular length metric measure space (Ω,d,μ). The characterizing properties are called the Gehring--Hayman condition and the ball--separation condition.
On a metric measure space satisfying the doubling property, we establish several optimal characterizations of Besov and Triebel-Lizorkin spaces, including a pointwise characterization. Moreover, we discuss their (non)triviality under a Poincaré inequality.
We prove that the composition of a quasi-nearly subharmonic function and a quasiregular mappings of bounded multiplicity is quasi-nearly subharmonic. Also, we prove that if $u\circ f$ is quasi-nearly subharmonic for all quasi-nearly subharmonic $u$ and $f$ satisfies some additional conditions, then $f$ is quasiconformal. Similar results are further established for the class of regularly oscillating functions.
In this paper, the authors characterize, in terms of pointwise inequalities, the classical Besov spaces $\dot B^s_{p,\,q}$ and Triebel-Lizorkin spaces $\dot F^s_{p,\,q}$ for all $s\in(0,\,1)$ and $p,\,q\in(n/(n+s),\,\infty],$ both in ${\mathbb R}^n$ and in the metric measure spaces enjoying the doubling and reverse doubling properties. Applying this characterization, the authors prove that quasiconformal mappings preserve $\dot F^s_{n/s,\,q}$ on $\rn$ for all $s\in(0,\,1)$ and $q\in(n/(n+s),\,\infty]$. A metric measure space version of the above morphism property is also established.
In this paper, we establish the equivalence between the Hajłasz-Sobolev spaces or classical Triebel-Lizorkin spaces and a class of grand Triebel-Lizorkin spaces on Euclidean spaces and also on metric spaces that are both doubling and reverse doubling. In particular, when $p\in(n/(n+1),\infty)$, we give a new characterization of the Hajłasz-Sobolev spaces $\dot M^{1, p}({\mathbb R}^n)$ via a grand Littlewood-Paley function.
We establish simple pointwise characterizations of functions in the Hardy-Sobolev spaces within the range n/(n+1)<p <=1. In addition, classical Hardy inequalities are extended to the case p <= 1.