In this article we study fine regularity properties for mappings of finite distortion. Our main theorems yield strongly localized regularity results in the borderline case in the class of maps of exponentially integrable distortion. Analogues of such results were known earlier in the case of quasiconformal mappings. Moreover, we study regularity for maps whose distortion has higher exponential integrability.
We establish the optimal regularity for the distortion of inverses of mappings of finite distortion with logarithm-iterated style subexponentially integrable distortion, which generalizes the Theorem 1. of [J. Gill, Ann. Acad. Sci. Fenn. Math. 35 (2010), no. 1, 197--207].
Let $f:Ω\to\IR^2$ be a mapping of finite distortion, where $Ω\subset\IR^2 .$ Assume that the distortion function $K(x,f)$ satisfies $e^{K(\cdot, f)}\in L^p_{loc}(Ω)$ for some $p>0.$ We establish optimal regularity and area distortion estimates for $f$. Especially, we prove that $|Df|^2 \log^{β-1}(e + |Df|) \in L^1_{loc}(Ω) $ for every $β<p.$ This answers positively well known conjectures due to Iwaniec and Martin \cite{IMbook} and to Iwaniec, Koskela and Martin \cite{IKM}.
In his celebrated paper on area distortion for quasiconformal mappings, Astala showed optimal area distortion bounds and dimension distortion estimates for planar quasiconformal mappings. He asked (Question 4.4) whether a finer result held, namely absolute continuity of Hausdorff measures under push-forward by quasiconformal mappings. This was proved in one particular case relevant for removability questions, in joint work of Astala, Clop, Mateu, Orobitg and the author ("Distortion of Hausdorff measures and improved Painlev'e removability for bounded quasiregular mappings", Duke Math J., to ap…
This note addresses the quantitative aspect of the bilipschitz extension problem. The main result states that any bilipschitz embedding of $\mathbb R$ into $\mathbb R^2$ can be extended to a bilipschitz self-map of $\mathbb R^2$ with a linear bound on the distortion.
We show that every centrally symmetric bi-Lipschitz embedding of the circle into the plane can be extended to a global bi-Lipschitz map of the plane with linear bounds on the distortion. This answers a question of Daneri and Pratelli in the special case of centrally symmetric maps. For general bi-Lipschitz embeddings our distortion bound has a combination of linear and cubic growth, which improves on the prior results. The proof involves a symmetrization result for bi-Lipschitz maps which may be of independent interest.
In this paper, we first establish the Schwarz-Pick lemma of higher-order and apply it to obtain a univalency criteria for planar harmonic mappings. Then we discuss distortion theorems, Lipschitz continuity and univalency of planar harmonic mappings defined in the unit disk with linearly connected images.
We consider quasiconformal mappings of the unit disk that have a planar extension which have $p$-integrable distortion. In this paper, we establish a bound for the modulus of continuity for the inverse mapping and show sharpness of this bound. Furthermore, we also obtain bounds for the compression multifractal spectra of such mappings.
We establish two-point distortion theorems for sense-preserving planar harmonic mappings $f=h+\overline{g}$ which satisfies the univalence criteria in the unit disc such that, Becker's and Nehari`s harmonic version. In addition, we find the sharp two-point distortion theorem when $h$ is a convex function, and normalized mappings such that $h(\D)$ is a $c$-linearly connected domain. To do this, we use the order of this family.
We consider a quasi-convex planar domain Ωwith a rectifiable boundary containing an exponential cusp and show that there is no homeomorphism f: \bR^2\to\bR^2 of finite distortion with \exp(λK)\in L_{loc}^{1}(\bR^2) for some λ>0 such that f(B)=Ω. On the other hand, if we only require that K_f(x)\in L_{loc}^{p}(\bR^2), then such an f exists.
We study quasiconformal mappings of the unit disk that have planar extension with controlled distortion. For these mappings we prove a bound for the modulus of continuity of the inverse map, which somewhat surprisingly is almost as good as for global quasiconformal maps. Furthermore, we give examples which improve the known bounds for the three point property of generalized quasidisks. Finally, we establish optimal regularity of such maps when the image of the unit disk has cusp type singularities.
We consider Sobolev mappings $f\in W^{1,q}(Ω,\IC)$, $1<q<\infty$, between planar domains $Ω\subset \IC$. We analyse the Radon-Riesz property for convex functionals of the form \[f\mapsto \int_ΩΦ(|Df(z)|,J(z,f)) \; dz \] and show that under certain criteria, which hold in important cases, weak convergence in $W_{loc}^{1,q}(Ω)$ of (for instance) a minimising sequence can be improved to strong convergence. This finds important applications in the minimisation problems for mappings of finite distortion and the $L^p$ and $Exp$\,-Teichmüller theories.
We extend a well-known theorem by Jones and Makarov [JM] on the singularity of boundary distortion of planar conformal mappings. We use a different technique to recover the previous result and, moreover, generalize the result for quasiconformal mappings of the unit ball $\B^n\subset \mathbb{R}^n$, $n\ge 2$. We also establish an estimate on the Hausdorff (gauge) dimension of the boundary of the image domain outside an exceptional set of given size on the sphere $\partial \B^n$. Furthermore, we show that this estimate is essentially sharp. [JM] P. W. Jones and N. Makarov: Density properties of h…
In this article, we consider the Hölder continuity of injective maps in Orlicz-Sobolev classes defined on the unit ball. Under certain conditions on the growth of dilatations, we obtain the Hölder continuity of the indicated class of mappings. In particular, under certain special restrictions, we show that Lipschitz continuity of mappings holds. We also consider Hölder and Lipschitz continuity of harmonic mappings and in particular of harmonic mappings in Orlicz-Sobolev classes. In addition in planar case, we show in some situations that the map is bi-Lipschitzian if Beltrami coefficient is Hö…
In this paper, we establish a three circles type theorem, involving the harmonic area function, for harmonic mappings. Also, we give bounds for length and area distortion for harmonic quasiconformal mappings. Finally, we will study certain Lipschitz-type spaces on harmonic mappings.
The conformal mapping $f(z)=(z+1)^2 $ from $\mathbb{D}$ onto the standard cardioid has a homeomorphic extension of finite distortion to entire $\mathbb{R}^2 .$ We study the optimal regularity of such extensions, in terms of the integrability degree of the distortion and of the derivatives, and these for the inverse. We generalize all outcomes to the case of conformal mappings from $\mathbb{D}$ onto cardioid-type domains.
We study minimisers of the $p$-conformal energy functionals, \[ \mathsf{E}_p(f):=\int_\ID \IK^p(z,f)\,dz,\quad f|_\IS=f_0|_\IS, \] defined for self mappings $f:\ID\to\ID$ with finite distortion and prescribed boundary values $f_0$. Here \[ \IK(z,f) = \frac{\|Df(z)\|^2}{J(z,f)} = \frac{1+|μ_f(z)|^2}{1-|μ_f(z)|^2}\] is the pointwise distortion functional and $μ_f(z)$ is the Beltrami coefficient of $f$. We show that for quasisymmetric boundary data the limiting regimes $p\to\infty$ recover the classical Teichmüller theory of extremal quasiconformal mappings (in part a result of Ahlfors), and for …
We prove estimates in Hölder spaces for some Cauchy-type integral operators representing holomorphic functions in Cartesian and symmetric products of planar domains. As a consequence, we obtain information on the boundary regularity in Hölder spaces of proper holomorphic maps between symmetric products of planar domains.
This paper is about hyperbolic properties on planar graphs. First, we study the relations among various kinds of strong isoperimetric inequalities on planar graphs and their duals. In particular, we show that a planar graph satisfies a strong isoperimetric inequality if and only if its dual has the same property, if the graph satisfies some minor regularity conditions and we choose an appropriate notion of strong isoperimetric inequalities. Second, we consider planar graphs where negative combinatorial curvatures dominate, and use the outcomes of the first part to strengthen the results of Hig…
The classical Painlevé theorem tells that sets of zero length are removable for bounded analytic functions, while (some) sets of positive length are not. For general $K$-quasiregular mappings in planar domains the corresponding critical dimension is $\frac{2}{K+1}$. We show that when $K>1$, unexpectedly one has improved removability. More precisely, we prove that sets $E$ of $σ$-finite Hausdorff $\frac{2}{K+1}$-measure are removable for bounded $K$-quasiregular mappings. On the other hand, $\dim(E) = \frac{2}{K+1}$ is not enough to guarantee this property. We also study absolute continuity pro…
