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Pekka Koskela

Publications and source records attributed to Pekka Koskela.

At least 37 records · Page 2Linked to original sources
arXiv2020

Sobolev extension via reflections

We show that the extension results by Mazya and Poborchi for polynomial planar cusps can be realized via composition operators generated by reflections.

math.FA↗
arXiv2020

Sobolev homeomorphic extensions onto John domains

Given the planar unit disk as the source and a Jordan domain as the target, we study the problem of extending a given boundary homeomorphism as a Sobolev homeomorphism. For general targets, this Sobolev variant of the classical Jordan-Schoenflies theorem may admit no solution - it is possible to have a boundary homeomorphism which admits a continuous $W^{1,2}$-extension but not even a homeomorphic $W^{1,1}$-extension. We prove that if the target is assumed to be a John disk, then any boundary homeomorphism from the unit circle admits a Sobolev homeomorphic extension for all exponents $p<2$. John disks, being one sided quasidisks, are of fundamental importance in Geometric Function Theory.

math.CV↗
arXiv2018

Controlled diffeomorphic extension of homeomorphisms

Let $Ω$ be an internal chord-arc Jordan domain and $φ:\mathbb S\rightarrow\partialΩ$ be a homeomorphism. We show that $φ$ has finite dyadic energy if and only if $φ$ has a diffeomorphic extension $h: \mathbb D\rightarrow Ω$ which has finite energy.

math.CV↗
arXiv2018

Accessible Parts of Boundary for Simply Connected Domains

For a bounded simply connected domain $Ω\subset\mathbb{R}^2$, any point $z\inΩ$ and any $0<α<1$, we give a lower bound for the $α$-dimensional Hausdorff content of the set of points in the boundary of $Ω$ which can be joined to $z$ by a John curve with a suitable John constant depending only on $α$, in terms of the distance of $z$ to $\partialΩ$. In fact this set in the boundary contains the intersection $\partialΩ_z\cap\partialΩ$ of the boundary of a John sub-domain $Ω_z$ of $Ω$, centered at $z$, with the boundary of $Ω$. This may be understood as a quantitative version of a result of Makarov. This estimate is then applied to obtain the pointwise version of a weighted Hardy inequality.

math.CA↗
arXiv2017

Gradient estimates for heat kernels and harmonic functions

Let $(X,d,μ)$ be a doubling metric measure space endowed with a Dirichlet form $\E$ deriving from a "carré du champ". Assume that $(X,d,μ,\E)$ supports a scale-invariant $L^2$-Poincaré inequality. In this article, we study the following properties of harmonic functions, heat kernels and Riesz transforms for $p\in (2,\infty]$: (i) $(G_p)$: $L^p$-estimate for the gradient of the associated heat semigroup; (ii) $(RH_p)$: $L^p$-reverse Hölder inequality for the gradients of harmonic functions; (iii) $(R_p)$: $L^p$-boundedness of the Riesz transform ($p<\infty$); (iv) $(GBE)$: a generalised Bakry-Émery condition. We show that, for $p\in (2,\infty)$, (i), (ii) (iii) are equivalent, while for $p=\infty$, (i), (ii), (iv) are equivalent. Moreover, some of these equivalences still hold under weaker conditions than the $L^2$-Poincaré inequality. Our result gives a characterisation of Li-Yau's gradient estimate of heat kernels for $p=\infty$, while for $p\in (2,\infty)$ it is a substantial improvement as well as a generalisation of earlier results by Auscher-Coulhon-Duong-Hofmann [7] and Auscher-Coulhon [6]. Applications to isoperimetric inequalities and Sobolev inequalities are given. Our results apply to Riemannian and sub-Riemannian manifolds as well as to non-smooth spaces, and to degenerate elliptic/parabolic equations in these settings.

math.MG↗
arXiv2017

Traces of weighted function spaces: dyadic norms and Whitney extensions

The trace spaces of Sobolev spaces and related fractional smoothness spaces have been an active area of research since the work of Nikolskii, Aronszajn, Slobodetskii, Babich and Gagliardo among others in the 1950's. In this paper we review the literature concerning such results for a variety of weighted smoothness spaces. For this purpose, we present a characterization of the trace spaces (of fractional order of smoothness), based on integral averages on dyadic cubes, which is well adapted to extending functions using the Whitney extension operator.

math.CA↗
arXiv2016

A quasiconformal composition problem for the Q-spaces

Given a quasiconformal mapping $f:\mathbb R^n\to\mathbb R^n$ with $n\ge2$, we show that (un-)boundedness of the composition operator ${\bf C}_f$ on the spaces $Q_α(\mathbb R^n)$ depends on the index $α$ and the degeneracy set of the Jacobian $J_f$. We establish sharp results in terms of the index $α$ and the local/global self-similar Minkowski dimension of the degeneracy set of $J_f$. This gives a solution to [Problem 8.4, 3] and also reveals a completely new phenomenon, which is totally different from the known results for Sobolev, BMO, Triebel-Lizorkin and Besov spaces. Consequently, Tukia-Väisälä's quasiconformal extension $f:\mathbb R^n\to\mathbb R^n$ of an arbitrary quasisymmetric mapping $g:\mathbb R^{n-p}\to \mathbb R^{n-p}$ is shown to preserve $Q_α (\mathbb R^n)$ for any $(α,p)\in (0,1)\times[2,n)\cup(0,1/2)\times\{1\}$. Moreover, $Q_α(\mathbb R^n)$ is shown to be invariant under inversions for all $0<α<1$.

math.FA↗
arXiv2016

A density problem for Sobolev spaces on Gromov hyperbolic domains

We prove that for a bounded domain $Ω\subset \mathbb R^n$ which is Gromov hyperbolic with respect to the quasihyperbolic metric, especially when $Ω$ is a finitely connected planar domain, the Sobolev space $W^{1,\,\infty}(Ω)$ is dense in $W^{1,\,p}(Ω)$ for any $1\le p<\infty$. Moreover if $Ω$ is also Jordan or quasiconvex, then $C^{\infty}(\mathbb R^n)$ is dense in $W^{1,\,p}(Ω)$ for $1\le p<\infty$.

math.FA↗
arXiv2015

A density problem for Sobolev spaces on planar domains

We prove that for a bounded simply connected domain $Ω\subset \mathbb R^2$, the Sobolev space $W^{1,\,\infty}(Ω)$ is dense in $W^{1,\,p}(Ω)$ for any $1\le p<\infty$. Moreover, we show that if $Ω$ is Jordan, then $C^{\infty}(\mathbb R^2)$ is dense in $W^{1,\,p}(Ω)$ for $1\le p<\infty$.

math.CA↗
arXiv2015

Approximation and quasicontinuity of Besov and Triebel-Lizorkin functions

We show that, for $0<s<1$, $0<p<\infty$, $0<q<\infty$, Hajłasz-Besov and Hajłasz-Triebel-Lizorkin functions can be approximated in the norm by discrete median convolutions. This allows us to show that, for these functions, the limit of medians, \[ \lim_{r\to 0}m_u^γ(B(x,r))=u^*(x), \] exists quasieverywhere and defines a quasicontinuous representative of $u$. The above limit exists quasieverywhere also for Hajłasz functions $u\in M^{s,p}$, $0<s\le 1$, $0<p<\infty$, but approximation of $u$ in $M^{s,p}$ by discrete (median) convolutions is not in general possible.

math.FA↗
arXiv2015

Bounded compositions on scaling invariant Besov spaces

For $0 < s < 1 < q < \infty$, we characterize the homeomorphisms $φ: \real^n \to \real^n$ for which the composition operator $f \mapsto f \circ φ$ is bounded on the homogeneous, scaling invariant Besov space $\dot{B}^s_{n/s,q}(\real^n)$, where the emphasis is on the case $q\not=n/s$, left open in the previous literature. We also establish an analogous result for Besov-type function spaces on a wide class of metric measure spaces as well, and make some new remarks considering the scaling invariant Triebel-Lizorkin spaces $\dot{F}^s_{n/s,q}(\real^n)$ with $0 < s < 1$ and $0 < q \leq \infty$.

math.CA↗
arXiv2014

Intrinsic Hardy-Orlicz spaces of conformal mappings

We define a new type of Hardy-Orlicz spaces of conformal mappings on the unit disk where in place of the value |f(x)| we consider the intrinsic path distance between f(x) and f(0) in the image domain. We show that if the Orlicz function is doubling then these two spaces are actually the same, and we give an example when the intrinsic Hardy-Orlicz space is strictly smaller.

math.CV↗