Quasiregular Mappings Research Articles

We construct a quasiregular mapping in R 3 \mathbb {R}^3 that is the first to illustrate several important dynamical properties: the quasi-Fatou set contains wandering components; these quasi-Fatou components are bounded and hollow; and the Julia set has components that are genuine round spheres. The key tool in this construction is a new quasiregular interpolation in round rings in R 3 \mathbb {R}^3 between power mappings of differing degrees on the boundary components. We also exhibit the flexibility of constructions based on these interpolations by showing that we may obtain quasiregular mappings which grow as quickly, or as slowly, as desired.

Abstract We prove a Painlevé theorem for bounded quasiregular curves in Euclidean spaces extending removability results for quasiregular mappings due to Iwaniec and Martin. The theorem is proved by extending a fundamental inequality for volume forms to calibrations and proving a Caccioppoli inequality for quasiregular curves. We also establish a qualitatively sharp removability theorem for quasiregular curves whose target is a Riemannian manifold with sectional curvature bounded from above and injectivity radius bounded from below. As an application, we extend a theorem of Bonk and Heinonen for quasiregular mappings to the setting of quasiregular curves: every non-constant quasiregular $$\omega $$ ω -curve from $$\mathbb {R}^n$$ R n into $$( N, \omega )$$ ( N , ω ) , where the bounded cohomology class of $$\omega $$ ω is in the bounded Künneth ideal, has infinite energy.

Abstract In this paper, we provide an alternative approach to an expectation of Fässler et al [J. Geom. Anal. 2016] by showing that a metrically quasiregular mapping between two equiregular sub‐Riemannian manifolds of homogeneous dimension has a negligible branch set. One main new ingredient is to develop a suitable extension of the generalized Pansu differentiability theory, in spirit of earlier works by Margulis–Mostow, Karmanova, and Vodopyanov. Another new ingredient is to apply the theory of Sobolev spaces based on upper gradients developed by Heinonen, Koskela, Shanmugalingam, and Tyson to establish the necessary analytic foundations.

We study Hardy spaces H p \mathcal {H}^p , 0 > p > ∞ 0>p>\infty for quasiregular mappings on the unit ball B B in R n {\mathbb R}^n which satisfy appropriate growth and multiplicity conditions. Under these conditions we recover several classical results for analytic functions and quasiconformal mappings in H p \mathcal {H}^p . In particular, we characterize H p \mathcal {H}^p in terms of non-tangential limit functions and non-tangential maximal functions of quasiregular mappings. Among applications we show that every quasiregular map in our class belongs to H p \mathcal {H}^p for some p = p ( n , K ) p=p(n,K) . Moreover, we provide characterization of Carleson measures on B B via integral inequalities for quasiregular mappings on B B . We also discuss the Bergman spaces of quasiregular mappings and their relations to H p \mathcal {H}^p spaces and analyze correspondence between results for H p \mathcal {H}^p spaces and A \mathcal {A} -harmonic functions. A key difference between the previously known results for quasiconformal mappings in R n {\mathbb R}^n and our setting is the role of multiplicity conditions and the growth of mappings that need not be injective. Our paper extends results by Astala and Koskela, Jerison and Weitsman, Jones, Nolder, and Zinsmeister.

We prove a sharp result for the distortion of a hyperbolic-type metric under K-quasiregular mappings of the upper half plane. The proof makes use of a new kind of Bernoulli inequality and the Schwarz lemma for quasiregular mappings.

Given an analytic function f=u+iv in the unit disk D, Zygmund’s theorem gives the minimal growth restriction on u which ensures that v is in the Hardy space h1. This need not be true if f is a complex-valued harmonic function. However, we prove that Zygmund’s theorem holds if f is a harmonic K-quasiregular mapping in D. Our work makes further progress on the recent Riesz-type theorem of Liu and Zhu (Adv. Math., 2023), and the Kolmogorov-type theorem of Kalaj (J. Math. Anal. Appl., 2025), for harmonic quasiregular mappings. We also obtain a partial converse, thus showing that the proposed growth condition is the best possible. Furthermore, as an application of the classical conjugate function theorems, we establish a harmonic analogue of a well-known result of Hardy and Littlewood.

Removable Singularities for Quasiregular Mappings

As it is known, conformal mappings are locally Lipschitz at inner points of a domain, and quasiconformal (quasiregular) mappings are locally H ̈older continuous. As for estimates of the distortion of mappings at boundary points of the domain, this problem has not been studied sufficiently even for these classes. We partially fill this gap by considering in this manuscript not even local behavior at the boundary points, but global behavior in the domain of one class of mappings. The paper is devoted to studying mappings with finite distortion. The goal of our investigation is obtaining the distance distortion for mappings at inner and boundary points. Here we study mappings satisfying Poletsky’s inequality in the inverse direction. We obtain conditions under which these mappings are either logarithmic H ̈older continuous or H ̈older continuous in the closure of a domain. We consider several important cases in the manuscript, studying separately bounded convex domains and domains with locally quasiconformal boundaries, as well as domains of more complex structure in which the corresponding distortion estimates must be understood in terms of prime ends. In all the above situations we show that the maps are logarithmically H ̈older continuous, which is somewhat weaker than the usual H ̈older continuity. However, in the last section we consider the case where the maps are still H ̈older continuous in the usual sense. The research technique is associated with the use of the method of moduli and the method of paths liftings. A key role is also played by the lower bounds of the Loewner type for the modulus of families of paths, which are valid only in domains with a special geometry, in particular, bounded convex domains. Another important fact which is also valid for domains of the indicated type, is the possibility of joining pairs of different points in a domain by paths lying (up to a constant) at a distance no closer than a distance between above points.

We study a new hyperbolic type metric recently introduced by Song and Wang. We present formulas for it in the upper half-space and the unit ball domains and find its sharp inequalities with the hyperbolic metric and the triangular ratio metric. We also improve existing ball inclusion results and give bounds for the distortion of this new metric under conformal and quasiregular mappings.

Zygmund theorem for harmonic quasiregular mappings

We investigate the distortion of the Assouad dimension and (regularized) spectrum of sets under planar quasiregular maps. The respective results for the Hausdorff and upper box-counting dimension follow immediately from their quasiconformal counterparts by employing elementary properties of these dimension notions (e.g. countable stability and Lipschitz stability). However, the Assouad dimension and spectrum do not share such properties. We obtain upper bounds on the Assouad dimension and spectrum of images of compact sets under holomorphic and planar quasiregular maps by studying their behavior around their critical points. As an application, the invariance of porosity of compact subsets of the plane under quasiregular maps is established.

We introduce an integrability condition for the reciprocal of the Jacobian determinant which guarantees the boundedness of the local index of quasiregular mappings. We also study the uniform limits of quasiregular mappings for which such an integrability condition of the reciprocal of the Jacobian holds uniformly.

The Ptolemaic Characteristic of Tetrads and Quasiregular Mappings

Riesz and Kolmogorov inequality for harmonic quasiregular mappings

In this paper, we generalize the Riesz theorem for harmonic quasiregular mappings for a special case (when p = 2) in the unit disc. Our results improve similar results in this field and are proved with milder conditions. Moreover, we prove another variant forms of Riesz inequality for harmonic quasiregular functions.

This article is devoted to the study of mappings with bounded andfinite distortion defined in some domain of the Euclidean space. Weconsider mappings that satisfy some upper estimates for thedistortion of the modulus of families of paths, where the order ofthe modulus equals to $p,$ $n-1<p\leqslant n.$ The main problemstudied in the manuscript is the investigation of the boundarybehavior of such mappings, more precisely, the distortion of thedistance under mappings near boundary points. The publication isprimarily devoted to definition domains with ``bad boundaries'', inwhich the mappings not even have a continuous extension to theboundary in the Euclidean sense. However, we introduce the conceptof a quasiconformal regular domain in which the specified continuousextension is valid and the corresponding distance distortionestimates are satisfied; however, both must be understood in thesense of the so-called prime ends. More precisely, such estimateshold in the case when the mapping acts from a quasiconformal regulardomain to an Ahlfors regular domain with the Poincar\'e inequality.The consideration of domains that are Ahlfors regular and satisfythe Poincar\'e inequality is due to the fact that, lower estimates forthe modulus of families of paths through the diameter of thecorresponding sets hold in these domains. (There are the so-calledLoewner-type estimates). We consider homeomorphisms and mappingswith branching separately. The main analytical condition under whichthe results of the paper were obtained is the finiteness of theintegral averages of some majorant involved in the defining modulusinequality under infinitesimal balls. This condition includes thesituation of quasiconformal and quasiregular mappings, because forthem the specified majorant is itself bounded in a definitiondomain. Also, the results of the article are valid for more generalclasses for which Poletsky-type upper moduli inequalities aresatisfied, for example, for mappings with finite length distortion.

Nonlinear Dirichlet Forms Associated with Quasiregular Mappings

On angular limits of quasiregular mappings

Injectivity criteria of linear combinations of harmonic quasiregular mappings

Landau-Bloch type theorem for elliptic and quasiregular harmonic mappings