About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 895074, 20 pages
http://dx.doi.org/10.1155/2014/895074
Research Article

Distortion of Quasiregular Mappings and Equivalent Norms on Lipschitz-Type Spaces

Faculty of Mathematics, University of Belgrade, Studentski Trg 16, 11000 Belgrade, Serbia

Received 24 January 2014; Accepted 15 May 2014; Published 21 October 2014

Academic Editor: Ljubomir B. Ćirić

Copyright © 2014 Miodrag Mateljević. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We prove a quasiconformal analogue of Koebe’s theorem related to the average Jacobian and use a normal family argument here to prove a quasiregular analogue of this result in certain domains in -dimensional space. As an application, we establish that Lipschitz-type properties are inherited by a quasiregular function from its modulo. We also prove some results of Hardy-Littlewood type for Lipschitz-type spaces in several dimensions, give the characterization of Lipschitz-type spaces for quasiregular mappings by the average Jacobian, and give a short review of the subject.

1. Introduction

The Koebe distortion theorem gives a series of bounds for a univalent function and its derivative. A result of Hardy and Littlewood relates Hölder continuity of analytic functions in the unit disk with a bound on the derivative (we refer to this result shortly as HL-result).

Astala and Gehring [1] observed that for certain distortion property of quasiconformal mappings the function , defined in Section 2, plays analogous role as when and is conformal, and they establish quasiconformal version of the well-known result due to Koebe, cited here as Lemma 4, and Hardy-Littlewood, cited here as Lemma 5.

In Section 2, we give a short proof of Lemma 4, using a version with the average Jacobian instead of , and we also characterize bi-Lipschitz mappings with respect to quasihyperbolic metrics by Jacobian and the average Jacobian; see Theorems 8, 9, and 10 and Proposition 13. Gehring and Martio [2] extended HL-result to the class of uniform domains and characterized the domains with the property that functions which satisfy a local Lipschitz condition in for some always satisfy the corresponding global condition there.

The main result of the Nolder paper [3] generalizes a quasiconformal version of a theorem, due to Astala and Gehring [4, Theorems 1.9 and 3.17] (stated here as Lemma 5) to a quasiregular version (Lemma 33) involving a somewhat larger class of moduli of continuity than , .

In the paper [5] several properties of a domain which satisfies the Hardy-Littlewood property with the inner length metric are given and also some results on the Hölder continuity are obtained.

The fact that Lipschitz-type properties are sometimes inherited by an analytic function from its modulus was first detected in [6]. Later this property was considered for different classes of functions and we will call shortly results of this type Dyk-type results. Theorem 22 yields a simple approach to Dyk-type result (the part (ii.1); see also [7]) and estimate of the average Jacobian for quasiconformal mappings in space. The characterization of Lipschitz-type spaces for quasiconformal mappings by the average Jacobian is established in Theorem 23 in space case and Theorem 24 yields Dyk-type result for quasiregular mappings in planar case.

In Section 4, we establish quasiregular versions of the well-known result due to Koebe, Theorem 39 here, and use this result to obtain an extension of Dyakonov’s theorem for quasiregular mappings in space (without Dyakonov’s hypothesis that it is a quasiregular local homeomorphism), Theorem 40. The characterization of Lipschitz-type spaces for quasiregular mappings by the average Jacobian is also established in Theorem 40.

By denote the real vector space of dimension . For a domain in with nonempty boundary, we define the distance function by , and if maps onto , in some settings, it is convenient to use short notation for . It is clear that , where is the complement of in .

Let be an open set in . A mapping is differentiable at if there is a linear mapping , called the derivative of at , such that where as . For a vector-valued function , where is a domain, we define when is differentiable at .

In Section 3, we review some results from [7, 8]. For example, in [7] under some conditions concerning a majorant , we showed the following.

Let and let be a continuous majorant such that is nonincreasing for .

Assume satisfies the following property (which we call Hardy-Littlewood -property):

Then

If, in addition, is harmonic in or, more generally, , then is equivalent to . If is a -extension domain, then is equivalent to .

In Section 3, we also consider Lipschitz-type spaces of pluriharmonic mappings and extend some results from [9].

In Appendices A and B we discuss briefly distortion of harmonic qc maps, background of the subject, and basic property of qr mappings, respectively. For more details on related qr mappings we refer the interested reader to [10].

2. Quasiconformal Analogue of Koebe’s Theorem and Applications

Throughout the paper we denote by , , and open subset of , .

Let , (abbreviated ) and let , stand for the unit ball and the unit sphere in , respectively. Sometimes we write and instead of and , respectively. For a domain let be a continuous function. We say that is a weight function or a metric density if, for every locally rectifiable curve in , the integral exists. In this case we call the -length of . A metric density defines a metric as follows. For , let where the infimum is taken over all locally rectifiable curves in joining and .

For the modern mapping theory, which also considers dimensions , we do not have a Riemann mapping theorem and therefore it is natural to look for counterparts of the hyperbolic metric. So-called hyperbolic type metrics have been the subject of many recent papers. Perhaps the most important metrics of these metrics are the quasihyperbolic metric and the distance ratio metric of a domain (see [11, 12]). The quasihyperbolic metric of is a particular case of the metric when (see [11, 12]).

Given a subset of or , a function (or, more generally, a mapping from into or ) is said to belong to the Lipschitz space if there is a constant , which we call Lipschitz constant, such that for all . The norm is defined as the smallest in (7).

There has been much work on Lipschitz-type properties of quasiconformal mappings. This topic was treated, among other places, in [15, 7, 1322].

As in most of those papers, we will currently restrict ourselves to the simplest majorants . The classes with will be denoted by (or by ). and are called, respectively, Lipschitz constant and exponent (of on ). We say that a domain is uniform if there are constants and such that each pair of points can be joined by rectifiable arc in for which for each ; here denotes the length of and the components of . We define . The smallest for which the previous inequalities hold is called the uniformity constant of and we denote it by . Following [2, 17], we say that a function belongs to the local Lipschitz space if (7) holds, with a fixed , whenever and . We say that is a -extension domain if . In particular if , we say that is a -extension domain; this class includes the uniform domains mentioned above.

Suppose that is a curve family in . We denote by the family whose elements are nonnegative Borel-measurable functions which satisfy the condition for every locally rectifiable curve , where denotes the arc length element. For , with the notation where denotes the Euclidean volume element , we define the -modulus of by

We will denote simply by and call it the modulus of .

Suppose that is a homeomorphism. Consider a path family in and its image family . We introduce the quantities where the suprema are taken over all path families in such that and are not simultaneously or .

Definition 1. Suppose that is a homeomorphism; we call the inner dilatation and the outer dilatation of . The maximal dilatation of is . If , we say is -quasiconformal (abbreviated qc).

Suppose that is a homeomorphism and , , and .

For each such that we set , .

Definition 2. The linear dilatation of at is

Theorem 3 (the metric definition of quasiconformality). A homeomorphism is qc if and only if is bounded on .

Let be a domain in and let be continuous. We say that is quasiregular (abbreviated qr) if(1)belongs to Sobolev space ,(2)there exists , , such that

The smallest in (13) is called the outer dilatation . A qr mapping is a qc if and only if it is a homeomorphism. First we need Gehring’s result on the distortion property of qc (see [23, page 383], [24, page 63]).

Gehring’s Theorem. For every and , there exists a function with the following properties:(1) is increasing,(2),(3),(4)Let and be proper subdomains of and let be a -qc. If and are points in such that , then

Introduce the quantity, mentioned in the introduction, associated with a quasiconformal mapping ; here is the Jacobian of , while stands for the ball and for its volume.

Lemma 4 (see [4]). Suppose that and are domains in : If is -quasiconformal, then where is a constant which depends only on and .

Set .

Lemma 5 (see [4]). Suppose that is a uniform domain in and that and are constants with and . If is -quasiconformal in with and if then has a continuous extension to and for , where the constant depends only on , , , and the uniformity constant for . In the case , (18) can be replaced by the stronger conclusion that

Example 6. The mapping , is -qc with bounded in the unit ball. Hence satisfies the hypothesis of Lemma 5 with . Since , we see that when , that is, , the conclusion (18) in Lemma 5 cannot be replaced by the stronger assertion .

Let and and . If there is a positive constant such that , , we write on . If there is a positive constant such that we write (or ) on .

Let be a domain and let , be a harmonic mapping. This means that is a map from into and both and are harmonic functions, that is, solutions of the two-dimensional Laplace equation The above definition of a harmonic mapping extends in a natural way to the case of vector-valued mappings , , defined on a domain , .

Let be a harmonic univalent orientation preserving mapping on a domain , and . If has the form, where and are analytic, we define , and .

2.1. Quasihyperbolic Metrics and the Average Jacobian

For harmonic qc mappings we refer the interested reader to [2528] and references cited therein.

Proposition 7. Suppose and are proper domains in . If is -qc and harmonic, then it is bi-Lipschitz with respect to quasihyperbolic metrics on and .

Results of this type have been known to the participants of Belgrade Complex Analysis seminar; see, for example, [29, 30] and Section 2.2 (Proposition 13, Remark 14 and Corollary 16). This version has been proved by Manojlović [31] as an application of Lemma 4. In [8], we refine her approach.

Proof. Let be local representation on .
Since is -qc, then on and since is harmonic,
Hence,
Hence, and . Using Astala-Gehring result, we get
This pointwise result, combined with integration along curves, easily gives
Note that we do not use that is a subharmonic function.

The following follows from the proof of Proposition 7:(I) and ; see below for the definition of average jacobian .

When underlining a symbol (we also use other latex-symbols) we want to emphasize that there is a special meaning of it; for example we denote by a constant and by , , some specific constants.

Our next result concerns the quantity associated with a quasiconformal mapping ; here is the Jacobian of , while stands for the ball and for its volume.

Define

Using the distortion property of qc (see [24, page 63]) we give short proof of a quasiconformal analogue of Koebe’s theorem (related to Astala and Gehring’s results from [4], cited as Lemma 4 here).

Theorem 8. Suppose that and are domains in : If is -quasiconformal, then where is a constant which depends only on and .

Proof. By the distortion property of qc (see [23, page 383], [24, page 63]), there are the constants and which depend on and only, such that where and . Hence and therefore we prove Theorem 8.

We only outline proofs in the rest of this subsection.

Suppose that and are domains in different from .

Theorem 9. Suppose is a and the following hold.(i1) is c-Lipschitz with respect to quasihyperbolic metrics on and ; then for every .(i2)If is a qc -quasihyperbolic-isometry, then is a -qc mapping,.

Proof. Since and are different from , there are quasihyperbolic metrics on and . Then for a fixed , we have
Hence, (i1) implies (I1). If is a -quasihyperbolic-isometry, then and . Hence, (I2) and (I3) follow.

Theorem 10. Suppose is a qc homeomorphism. The following conditions are equivalent:(a.1) is bi-Lipschitz with respect to quasihyperbolic metrics on and ,(b.1),(c.1),(d.1),where and .

Proof. It follows from Theorem 9 that (a) is equivalent to (b) (see, e.g., [32, 33]).
Theorem 8 states that . By Lemma 4, and therefore (b) is equivalent to (c). The rest of the proof is straightforward.

Lemma 11. If is a -quasiconformal mapping defined in a domain , then provided that . The constant is sharp.

If , then it is bi-Lipschitz with respect to Euclidean and quasihyperbolic metrics on and .

It is a natural question whether is there an analogy of Theorem 10 if we drop the hypothesis.

Suppose is onto qc mapping. We can consider the following conditions:(a.2) is bi-Lipschitz with respect to quasihyperbolic metrics on and ;(b.2) a.e. in ;(c.2) a.e. in ;(d.2) a.e. in .

It seems that the above conditions are equivalent, but we did not check details.

If is a planar domain and harmonic qc, then we proved that holds (see Proposition 18).

2.2. Quasi-Isometry in Planar Case

For a function , we use notations and ; we also use notations and instead of and , respectively, when it seems convenient. Now we give another proof of Proposition 7, using the following.

Proposition 12 (see [29]). Let be an euclidean harmonic orientation preserving univalent mapping of the unit disc into such that contains a disc and . Then
If, in addition, is -qc, then

For more details, in connection with material considered in this subsection, see also Appendix A, in particular, Proposition A.7.

Let be a harmonic univalent orientation preserving mapping on the unit disk , , , , and . By the harmonic analogue of the Koebe Theorem, then and therefore If, in addition, is -qc, then Using Proposition 12, we also prove

If we summarize the above considerations, we have proved the part of the following proposition.

Proposition 13 (-qch, hyperbolic distance version). (a.3). Let be a harmonic univalent orientation preserving -qc mapping on the unit disk , . Then, for ,
If is a harmonic univalent orientation preserving -qc mapping of domain onto , then and where .

Remark 14. In particular, we have Proposition 7, but here the proof of Proposition 13 is very simple and it it is not based on Lemma 4.

Proof of (b.3). Applying (a.3) to the disk , , we get (41). It is clear that (41) implies (42).

For a planar hyperbolic domain in , we denote by and the hyperbolic density and metric of , respectively.

We say that a domain is strongly hyperbolic if it is hyperbolic and diameters of boundary components are uniformly bounded from below by a positive constant.

Example 15. The Poincaré metric of the punctured disk is obtained by mapping its universal covering, an infinitely-sheeted disk, on the half plane by means of (i.e., ). The metric is
Since a boundary component is the point , the punctured disk is not a strongly hyperbolic domain. Note also that and tend to if tends to . Here and is the hyperbolic density of . Therefore one has the following:for , when .
There is no constant such that for every , where and .
Since , if holds, we conclude that , which is a contradiction by .

Let be a planar hyperbolic domain. Then if is simply connected,

For general domain as ,

Hence , .

If is a strongly hyperbolic domain, then there is a hyperbolic density on the domain , such that , where ; see, for example, [34]. Thus . Hence, we find the following.

Corollary 16. Every -harmonic quasiconformal mapping of the unit disc (more generally of a strongly hyperbolic domain) is a quasi-isometry with respect to hyperbolic distances.

Remark 17. Let be a hyperbolic domain, let   be -harmonic quasiconformal mapping of onto , and let be a covering and . (a.4)Suppose that is simply connected. Thus and are one-to-one. Then where .
Hence, since and ,
Using Hall’s sharp result, one can also improve the constant in the second inequality in Propositions 13 and 12 (i.e., the constant can be replaced by ; see below for more details): where .(b.4)Suppose that is not a simply connected. Then is not one-to-one and we cannot apply the procedure as in .(c.4)It seems natural to consider whether there is an analogue in higher dimensions of Proposition 13.

Proposition 18. For every -harmonic quasiconformal mapping of the unit disc (more generally of a hyperbolic domain ) the following holds: (e.2). In particular, it is a quasi-isometry with respect to quasihyperbolic distances.

Proof. For , by the distortion property, Hence, by Schwarz lemma for harmonic maps, . Proposition 12 yields (e) and an application of Proposition 13 gives the proof.

Recall stands for the ball and for its volume. If is a subset of and , we define

Suppose that and . Let denote the class of such that for every .

Proposition 19. Suppose is a . Then , if and only if is -Lipschitz with respect to quasihyperbolic metrics on and for every ball .

2.3. Dyk-Type Results

The characterization of Lipschitz-type spaces for quasiconformal mappings in space and planar quasiregular mappings by the average Jacobian are the main results in this subsection. In particular, using the distortion property of qc mappings we give a short proof of a quasiconformal version of a Dyakonov theorem which states:Suppose is a -extension domain in and is a -quasiconformal mapping of onto . Then if and only if .

This is Theorem B.3, in Appendix B below. It is convenient to refer to this result as Theorem Dy; see also Theorems 2324 and Proposition A, Appendix B.

First we give some definitions and auxiliary results. Recall, Dyakonov [15] used the quantity associated with a quasiconformal mapping ; here is the Jacobian of , while stands for the ball and for its volume.

Define , , , and

For a ball and a mapping , we define and ; we also use the notation and instead of .

For , we define the Euclidean inner product by By we denote with the Euclidean inner product and call it Euclidean space -space (space of dimension ). In this paper, for simplicity, we will use also notation for . Then the Euclidean length of is defined by

The minimal analytic assumptions necessary for a viable theory appear in the following definition.

Let be a domain in and let be continuous. We say that has finite distortion if(1) belongs to Sobolev space ;(2)the Jacobian determinant of is locally integrable and does not change sign in ;(3)there is a measurable function , finite a.e., such that

The assumptions , , and do not imply that , unless of course is a bounded function.

If is a bounded function, then is qr. In this setting, the smallest in (57) is called the outer dilatation .

If is qr, also for some , , where . The smallest in (58) is called the inner dilatation and is called the maximal dilatation of . If , is called -quasiregular.

In a highly significant series of papers published in 1966–1969 Reshetnyak proved the fundamental properties of qr mappings and in particular the main theorem concerning topological properties of qr mappings: every nonconstant qr map is discrete and open; cf. [11, 35] and references cited there.

Lemma 20 (see Morrey’s Lemma, Lemma , [10, page 170]). Let be a function of the Sobolev class in the ball , , , such that where , holds for every ball . Then is Hölder continuous in with exponent , and one has , for all , where . Here and in some places we omit to write the volume element .

We need a quasiregular version of this Lemma.

Lemma 21. Let be a -quasiregular mapping, such that where , holds for every ball . Then is Hölder continuous in with exponent , and one has , for all , where .

Proof. By hypothesis, satisfies (57) and therefore . An application of Lemma 20 to yields proof.

(a.0)By denote a family of ball such that . For , define and , where .

Define and , where supremum is taken over all balls such that and denotes radius of .

Theorem 22. Let , . Suppose that (a.5)   is a domain in and is -quasiconformal and . Then one has the following.(i.1)For every , there exists two points such that , where , , and .(ii.1), , where .(iii.1)If, in addition, one supposes that (b.5)  , then for all balls such that , and in particular , .(iv.1)There is a constant such that for every and .(v.1)If, in addition, one supposes that (c.5) is a -extension domain in , then   .

Proof. By the distortion property (30), we will prove the following. (vi.1)For a ball such that there exist two points such that , where , .
Let be line throughout and which intersects the at points and and , . By the left side of (30), . We consider two cases:(a)if and , then and ;(b)if , then, for example, and if we choose , we find and this yields .
Let . An application of to yields . If , then and therefore we have the following:(vi.1′), where .
If , then . Hence, using , we find . Proof of . If the hypothesis (b) holds with multiplicative constant , and , then and therefore , where . Hence and therefore, in particular, , .
It is clear, by the hypothesis (b), that there is a fixed constant such that belongs to for every ball such that and, by , we have the following:(viii.1) for a fixed constant and .
Hence, since, by the hypothesis (c.5), is a -extension domain, we get .
Proof of . Let be a ball such that . Then , , and, by , on . Hence on .
Note that one can also combine and Lemma 33 (for details see proof of Theorem 40 below) to obtain .

Note that as an immediate corollary of we get a simple proof of Dyakonov results for quasiconformal mappings (without appeal to Lemma 33 or Lemma 21, which is a version of Morrey’s Lemma).

We enclose this section by proving Theorems 23 and 24 mentioned in the introduction; in particular, these results give further extensions of Theorem-Dy.

Theorem 23. Let , , and . Suppose is a -extension domain in and is a -quasiconformal mapping of onto . The following are equivalent:(i.2),(ii.2),(iii.2) there is a constant such that for every and . If, in addition, is a uniform domain and if , then and are equivalent to(iv.2).

Proof. Suppose (ii.2) holds, so that is -Lipschitz in . Then shows that holds. By Lemma 21, implies .

We outline less direct proof that implies .

One can show first that implies (18) (or more generally ; see Theorem 22). Using , we conclude that and is a fixed constant (which depends only on ), for all balls . Lemma 5 tells us that (18) holds, with a fixed constant, for all balls and all pairs of points . Further, we pick two points with and apply (18) with , letting and . The resulting inequality shows that , and since is a -extension domain, we conclude that .

The implication (ii.4) (i.4) is thus established. The converse is clear. For the proof that (iii.4) implies (i.4) for , see [15].

For a ball and a mapping , we define and .

We use the factorization of planar quasiregular mappings to prove the following.

Theorem 24. Let , . Suppose is a -extension domain in and is a -quasiregular mapping of onto . The following are equivalent:(i.3),(ii.3),(iii.3)there is a constant such that for every and .

Proof. Let and be domains in and qr mapping. Then there is a domain and analytic function on such that , where is quasiconformal; see [36, page 247].
Our proof will rely on distortion property of quasiconformal mappings. By the triangle inequality, implies . Now, we prove that implies .
Let , , , , , and . Let . As in analytic case there is such that . If , then . Hence, if is -Hölder, then for . Hence, since we find and therefore, using , we get
Thus we have (iii.3) with .
Proof of the implication .
An application of Lemma 21 (see also a version of Astala-Gehring lemma) shows that is -Hölder on with a Lipschitz (multiplicative) constant which depends only on and it gives the result.

3. Lipschitz-Type Spaces of Harmonic and Pluriharmonic Mappings

3.1. Higher Dimensional Version of Schwarz Lemma

Before giving a proof of the higher dimensional version of the Schwarz lemma we first establish notation.

Suppose that is a continuous vector-valued function, harmonic on , and let

Let . A modification of the estimate in [37, Equation ] gives

We next extend this result to the case of vector-valued functions. See also [38] and [39, Theorem 6.16].

Lemma 25. Suppose that is a continuous mapping, harmonic in . Then

Proof. Without loss of generality, we may suppose that and . Let where is the volume of the unit ball in . Hence, as in [8], for , we have
Let be a unit vector and . For given , it is convenient to write and consider as function of .
Then
Since , we see that
This last inequality yields where is the surface element on the sphere, and the proof is complete.

Let denote the complex vector space of dimension . For , we define the Euclidean inner product by where denotes the complex conjugate of . Then the Euclidean length of is defined by