Mapping problems for quasiregular mappings

Quasiregular mappings are a natural generalization of analytic functions to higher dimensions. Quasiregular mappings have many properties. Our work in this paper is to prove the following theorem: If f  a b is a quasiregular mapping which maps the plane onto the plane, then f is a bijection. We do this by finding the connection between quasiregular and quasiconformal mappings.

Two theorems are given regarding the means of quasiconformal and quasiregular mappings. Together they show that the principle of subordination for means of analytic functions has no analog, at least in the case of plane quasiregular mappings.

We develop a new tool based on quasiconformal dynamics and conformal dynamics of discrete group actions in 3-geometries to construct new types of quasiregular and quasisymmetric mappings in space. This tool has close relations to new effects in Teichmüller spaces of conformally flat structures on closed hyperbolic 3-manifolds/orbifolds and non-trivial hyperbolic 4-cobordisms, to the hyperbolic and conformal interbreedings as well as to non-faithful discrete representations of uniform hyperbolic 3-lattices. We demonstrate several applications of this tool and new types of quasiregular mappings in space. Leaving such applications to geometry and topology of manifolds to another our papers [10, 11], here we continue a series of applications of our constructions to long standing problems for quasiregular mappings in space, including M.A. Lavrentiev surjectivity problem, Pierre Fatou problem on radial limits and Matti Vuorinen injectivity and asymptotics problems for bounded quasiregular mappings in the unit 3-ball (cf. [4, 7-9]).

We develop a new tool based on quasiconformal dynamics and conformal dynamics of discrete group actions in 3-geometries to construct new types of quasiregular and quasisymmetric mappings in space. This tool has close relations to new effects in Teichmüller spaces of conformally flat structures on closed hyperbolic 3-manifolds/orbifolds and non-trivial hyperbolic 4-cobordisms, to the hyperbolic and conformal interbreedings as well as to non-faithful discrete representations of uniform hyperbolic 3-lattices.We demonstrate several applications of this tool and new types of quasiregular mappings in space. Leaving such applications to geometry and topology of manifolds to another our papers [10, 11], here we continue a series of applications of our constructions to long standing problems for quasiregular mappings in space, including M.A. Lavrentiev surjectivity problem, Pierre Fatou problem on radial limits and Matti Vuorinen injectivity and asymptotics problems for bounded quasiregular mappings in the unit 3-ball (cf. [4, 7–9]).

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 define Hardy spaces {mathcal {H}}^p for quasiregular mappings in the plane, and show that for a particular class of these mappings many of the classical properties that hold in the classical setting of analytic mappings still hold. This particular class of quasiregular mappings can be characterised in terms of composition operators when the symbol is quasiconformal. Relations between Carleson measures and Hardy spaces play an important role in the discussion. This program was initiated and developed for Hardy spaces of quasiconformal mappings by Astala and Koskela in 2011 in their paper {mathcal {H}}^p-theory for Quasiconformal Mappings (Pure Appl Math Q 7(1):19–50, 2011).

1. A quasiconformal mapping as originally envisaged by Grdtzsch [4] is in its simplest form a continuously differentiable mapping from a plane domain onto another plane domain or onto a Riemann covering surface. The condition of quasiconformality is then expressed roughly by saying that apart from branch points an infinitesimal circle goes into an infinitesimal ellipse the ratio of whose principal axes is uniformly bounded and uniformly bounded from zero. Grbtzsch himself recognized that the essential properties of quasiconformal mappings were retained under less stringent conditions and Teichmuller indicated the admissibility of quite general exceptional points and curves. It was early realized that almost all proofs of properties of quasiconformal mappings rely on some form of the method of the extremal metric. Consequently, several authors [1, 6] have suggested revising the definition of a quasiconformal mapping, dropping all assumption of differentiability and using the notion of the conformal module of a quadrangle. We will give now this definition confining ourselves for simplicity to homeomorphic sense-preserving mappings from one plane domain to another. However by appropriate use of local uniformizing parameters, our results can be extended to the most general quasiconformal mappings. By a quadrangle Q we mean a simply-connected domain of hyperbolic type with four distinguished boundary elements called vertices. These divide the boundary elements in their natural cyclic order into four sides. The quadrangle admits a conformal mapping onto a rectangle whose pairs of opposite sides have lengths, say a and b, with the vertices of the quadrangle corresponding to the corners of the rectangle. The ratio a/b is a conformal invariant of the quadrangle and it and its reciprocal may be called modules of the quadrangle. They may be obtained directly without reference to the mapping on the rectangle by means of the following module problem. Let r denote the class of open arcs on Q joining a pair of opposite sides of the quadrangle and locally rectifiable in the sense that every closed subarc on them is rectifiable. Suppose that Q lies in the z-plane (z = x + iy) and let p(x, y) be a real valued non-negative func-

Suppose that μ \mu is a finite positive measure on the unit disk. Carleson showed that the L 2 ( μ ) {L^2}(\mu ) -norm is bounded by the H 2 {H^2} -norm uniformly over the space of analytic functions on the unit disk if and only if μ \mu is a Carleson measure. Analogues of this result exist for Bergmann spaces of analytic functions in the disk and in the unit ball of C n {C^n} . We prove here real variable analogues of certain Bergmann space results using quasiconformal and quasiregular mappings.

The distortion of six different intrinsic metrics and quasi-metrics under conformal and quasiregular mappings is studied in a few simple domains $G\subsetneq\mathbb{R}^n$. The already known inequalities between the hyperbolic metric and these intrinsic metrics for points $x,y$ in the unit ball $\mathbb{B}^n$ are improved by limiting the absolute values of the points $x,y$ and the new results are then used to study the conformal distortion of the intrinsic metrics. For the triangular ratio metric between two points $x,y\in\mathbb{B}^n$, the conformal distortion is bounded in terms of the hyperbolic midpoint and the hyperbolic distance of $x,y$. Furthermore, quasiregular and quasiconformal mappings are studied, and new sharp versions of the Schwarz lemma are introduced.

Introduction. The general trend of the geometric function theory in R is to generalize certain topological aspects of the analytic functions of one complex variable. The category of mappings that one usually considers in higher dimensions are the mappings with finite distortion, thus, in particular, quasiconformal and quasiregular mappings. This program, whose origin can be traced back to the works of M. A. Lavrentiev (1938), L. V. Ahlfors (1954), F. W. Gehring (1961), J. Vaisala (1961) and Y. Reshetnyak (1966), was held by an important school of Finnish geometers in the 1970’s, led by O. Martio, S. Rickman and J. Vaisala. For a recent account see [Ri] and [Vu]. But in this drive towards generalizations of analytic functions, one aspect has been quite neglected. This is the fact that the mappings in question solve important first order systems of PDEs analogous in many respects to the Cauchy-Riemann equations.

In this chapter we build the foundation for the work that comes in the rest of the book. We begin with the definition of two conformal invariants, the modulus of a curve family and the capacity of a condenser, which are two closely related notions. These tools enable us to define quasiconformal and quasiregular mappings which are the basic classes of mappings to be studied. Several examples of quasiconformal mappings are given illustrating the importance of this class of functions and their role in Geometric Function Theory. Moduli of continuity of harmonic mappings, which are either quasiconformal or quasiregular at the same time, are considered and some sharp estimates are given for all dimensions n ≥ 2. In particular, we study the case of Lipschitz continuity of mappings defined in the unit ball.

The authors study mappings that satisfy some estimate of the distortion of the modulus of families of paths. Under certain conditions on the domains between which the mappings act, we established that, these mappings are Hölder logarithmic continuous at the boundary points. It is known that, the Hölder continuity is established for many classes of mappings, say quasiconformal and quasiregular mappings. In this regard, it is possible to point to the classical distortion estimates by Martio-Rickman-Väisälä type, as well as the estimates related to the modern classes of mappings with finite distortion. In particular, V.I. Ryazanov together with R.R. Salimov and E.O. Sevost'yanov established local distortion estimates for plane and spatial mappings under FMO condition, or under the Lehto-type integral condition. Recently, the second co-author have obtained Hölder logarithmic continuity for the studied class at points of the unit sphere. This article considers the situation of similar mappings of different domains, not only the unit sphere. Namely, we consider mappings between quasiextremal distance domains (QED-domains) and convex domains. Note that, quasiextremal distance domains introduced by Gehring and Martio are structures in which the modulus of families of paths is metrically related to the diameter of sets. Also, convex domains are involved in the formulation of the main result; we consider mappings that surjectively act onto them. In addition, the article contains the formulations and proofs for some other results on this topic. We consider several more cases in detail, in particular when: 1) the definition domain is a domain with a locally quasiconformal boundary, and the image domain is a bounded convex domain; 2) the definition domain is a regular domain in the sense of prime ends, and the image domain is a bounded convex domain; 3) the mapping acts between the QED-domain and the bounded convex domain and has a fixed point. In all three cases, the mapping is Hölder logarithmic continuous; moreover, in case 2), which refers to prime ends, logarithmic continuity should also be understood in terms of prime ends.

In this chapter we introduce quasiregular and quasiconformal mappings and their basic properties.

The theorems of TWB (Teichmuller-Wittich-Belinskii) type imply the local conformality (or weaker properties) of quasiconformal mappings at a prescribed point under assumptions of the finiteness of appropriate integral averages of the quantity K μ (z) − 1, where K μ (z) stands for the real dilatation coefficient. We establish the extremal bounds for distortions of the moduli of annuli in terms of integrals in TWB theorems under quasiconformal and quasiregular mappings and illustrate their sharpness by several examples. Some local conditions weaker than the conformality are also discussed.

Beardon and Minda gave a characterization of normal families of holomorphic and meromorphic functions in terms of a locally uniform Lipschitz condition. Here, we generalize this viewpoint to families of mappings in higher dimensions that are locally uniformly continuous with respect to a given modulus of continuity. Our main application is to the normality of families of quasiregular mappings through a locally uniform Hölder condition. This provides a unified framework in which to consider families of quasiregular mappings, both recovering known results of Miniowitz, Vuorinen and others and yielding new results. In particular, normal quasimeromorphic mappings, Yosida quasiregular mappings and Bloch quasiregular mappings can be viewed as classes of quasiregular mappings which arise through consideration of various metric spaces for the domain and range. We give several characterizations of these classes and obtain upper bounds on the rate of growth in each class.