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 [1–5, 7, 13–22].

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 [25–28] 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 23–24 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 Denote a ball in with center and radius by In particular, denotes the unit ball and the sphere . Set , the open unit disk in , and let be the unit circle in .

A continuous complex-valued function defined in a domain is said to be pluriharmonic if, for fixed and , the function is harmonic in , where denotes the distance from to the boundary of . It is easy to verify that the real part of any holomorphic function is pluriharmonic; cf. [40].

Let with be a continuous function. We say that is a majorant if(1) is increasing,(2) is nonincreasing for .

If, in addition, there is a constant depending only on such that for some , then we say that is a regular majorant. A majorant is called fast (resp., slow) if condition (81) (resp., (82)) is fulfilled.

Given a majorant , we define (resp., ) to be the Lipschitz-type space consisting of all complex-valued functions for which there exists a constant such that, for all and (resp., and ),

Using Lemma 25, one can prove the following.

Proposition 26. Let be a regular majorant and let be harmonic mapping in a simply connected -extension domain . Then if and only if .

It is easy to verify that the real part of any holomorphic function is pluriharmonic. It is interesting that the converse is true in simply connected domains.

Lemma 27. (i) Let be pluriharmonic in . Then there is an analytic function in such that .
(ii) Let be simply connected and be pluriharmonic in . Then there is analytic function in such that .

Proof. (i) Let , , and ; define form
and . Then (i) holds for , which is analytic on .
(ii) If , there is a chain in such that is center of and, by the lemma, there is analytic chain , . We define . As in in the proof of monodromy theorem in one complex variable, one can show that this definition does not depend of chains and that in .

The following three theorems in [9] are a generalization of the corresponding one in [15].

Theorem 28. Let be a fast majorant, and let be a pluriharmonic mapping in a simply connected -extension domain . Then the following are equivalent:(1);(2) and ;(3) and ;(4) and ,where denotes the class of all continuous functions on which satisfy (83) with some positive constant , whenever and .

Define and , where , and , .

Theorem 29. Let be a domain and analytic in . Then
If , then .
If is a pluriharmonic mapping, where and are analytic in , then and . In particular, .

Proof. Using a version of Koebe theorem for analytic functions (we can also use Bloch theorem), we outline a proof. Let , , , , , and .
By the version of Koebe theorem for analytic functions, for every line which contains , there are points such that , , and . Hence and .
Define . Since , we find , where . Hence
Finally if , we have .

For the following result is proved in [9].

Theorem 30. Let be a regular majorant and let be a simply connected -extension domain. A function pluriharmonic in belongs to if and only if, for each , and for some constant depending only on , , , and .

We only outline a proof: let . Note that and .

We can also use Proposition 26.

3.2. Lipschitz-Type Spaces

Let be a function and . We denote by the class of functions which satisfy the following condition: for every .

It was observed in [8] that . In [7], we proved the following results.

Theorem 31. Suppose that(a1) is a -extension domain in , , and is continuous on which is a -quasiconformal mapping of onto ;(a2) is connected;(a3) is Hölder on with exponent ;(a4). Then is Hölder on with exponent

The proof in [7] is based on Lemmas 3 and 8 in the paper of Martio and Nakki [18]. In the setting of Lemma 8, . In the setting of Lemma 3, using the fact that is connected, we get similar estimate for small enough.

Theorem 32. Suppose that is a domain in , , and is harmonic (more generally ) in . Then one has the following:(i.1)  implies(ii.1), .(ii.1)implies(iii.1) .

4. Theorems of Koebe and Bloch Type for Quasiregular Mappings

We assume throughout that is an open connected set whose boundary, , is nonempty. Also is the open ball centered at with radius . If is a ball, then , , denotes the ball with the same center as and with radius equal to times that of .

The spherical (chordal) distance between two points is the number where is stereographic projection, defined by Explicitly, if ,

When is differentiable, we denote its Jacobi matrix by or and the norm of the Jacobi matrix as a linear transformation by . When exists a.e. we denote the local Dirichlet integral of at by , where . If there is no chance of confusion, we will omit the index . If , then and if is the unit ball, we write instead of .

When the measure is omitted from an integral, as here, integration with respect to -dimensional Lebesgue measure is assumed.

A continuous increasing function is a majorant if and if for all .

The main result of the paper [3] generalizes Lemma 5 to a quasiregular version involving a somewhat larger class of moduli of continuity than , .

Lemma 33 (see [3]). Suppose that is -extension domain in . If is -quasiregular in with and if then has a continuous extension to and for , where the constant depends only on , and .
If is uniform, the constant depends only on and the uniformity constant for .
Conversely if there exists a constant such that (92) holds for all , then (91) holds for all with depending only on , and .

Now suppose that and .

Also in [3], Nolder, using suitable modification of a theorem of Näkki and Palka [41], shows that (92) can be replaced by the stronger conclusion that

Remark 34. Simple examples show that the term cannot in general be omitted. For example, with is -quasiconformal in . is bounded over yet ; see Example 6.
If , then , where , and if , then and .
Note that we will show below that if , the conclusion (92) in Lemma 33 can be replaced by the stronger assertion .

Lemma 35 (see [3]). If is -quasiregular in with and if is a ball with , , then there exists a constant , depending only on , such that for all and all . Here and in some places we omit to write the volume and the surface element.

Lemma 36 (see[42], second version of Koebe theorem for analytic functions). Let ; let be holomorphic function on , , , and let the unbounded component of be not empty, and let . Then ; if, in addition, is simply connected, then contains the disk of radius , where .

The following result can be considered as a version of this lemma for quasiregular mappings in space.

Theorem 37. Suppose that is a -quasiregular mapping on the unit ball , and .
Then, there exists an absolute constant such that for every there exists a point on the half-line , which belongs to , such that .
If is a -quasiconformal mapping, then there exists an absolute constant such that contains .

For the proof of the theorem, we need also the following result, Theorem [10].

Lemma 38. For each there is with the following property. Let , let be a domain, and let denote the family of all qr mappings with and where are points in such that , . Then forms a normal family in .

Now we prove Theorem 37.

Proof. If we suppose that this result is not true then there is a sequence of positive numbers , which converges to zero, and a sequence of -quasiregular functions , such that does not intersect , . Next, the functions map into and hence, by Lemma 38, the sequence is equicontinuous and therefore forms normal family. Thus, there is a subsequence, which we denote again by , which converges uniformly on compact subsets of to a quasiregular function . Since converges to and converges to infinity, we have a contradiction by Lemma 35.

A path-connected topological space with a trivial fundamental group is said to be simply connected. We say that a domain in is spatially simply connected if the fundamental group is trivial.

As an application of Theorem 37, we immediately obtain the following result, which we call the Koebe theorem for quasiregular mappings.

Theorem 39 (second version of Koebe theorem for -quasiregular functions). Let ; let   be -quasiregular function on , , , and let the unbounded component of be not empty, and let . Then there exists an absolute constant :(a.1);(a.2)if, in addition, and is spatially simply connected, then contains the disk of radius , where .

(A.1) Now, using Theorem 39 and Lemma 20, we will establish the characterization of Lipschitz-type spaces for quasiregular mappings by the average Jacobian and in particular an extension of Dyakonov’s theorem for quasiregular mappings in space (without Dyakonov’s hypothesis that it is a quasiregular local homeomorphism). In particular, our approach is based on the estimate below.

Theorem 40 (Theorem-DyMa). Let , , and . Suppose is a -extension domain in and is a -quasiregular mapping of onto . The following are equivalent:(i.4),(ii.4),(iii.4) for every ball .

Proof. Suppose that is a -quasiregular mapping of onto . We first establish that for every ball , one has the following:(a.3), where .
Let be line throughout and and denote by the unbounded component of , where . Then, using a similar procedure as in the proof of Theorem 22, the part , one can show that there is such that , where .
Take a point . Then and . Since . Then using Theorem 39, we find (a.3).
Now we suppose , that is, .
Thus we have . Hence, we get (a.4), where .
It is clear that is denoted here as (a.4). The implication is thus established.
If we suppose , then an application of Lemma 20 shows that , and since is a -extension domain, we conclude that . Thus implies .
Finally, the implication is a clear corollary of the triangle inequality.

Now we give another outline that (i.4) is equivalent to (ii.4).

Here, we use approach as in [15]. In particular, implies that the condition (91) holds.

We consider two cases:(1);(2).

Then we apply Lemma 33 on and in Case and Case , respectively.

In more detail, if , then for every ball , by (a.3), and the condition (91) holds. Then Lemma 33 tells us that (92) holds, with a fixed constant, for all balls and all pairs of points . Next, we pick two points with and apply (92) with , where , 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 being trivially true.

The consideration in shows that (i.4) and (ii.4) are equivalent with .

Appendices

A. Distortion of Harmonic Maps

Recall by and we denote the unit disc and the unit circle respectively, and we also use notation . For a function we denote by and (or sometimes by , , and ) partial derivatives with respect to , , and , respectively. Let be harmonic, where and are analytic, and every complex valued harmonic function on simply connected set is of this form. Then , , , and . If is univalent, then and therefore and .

After writing this paper and discussion with some colleagues (see Remark A.11 below), the author found out that it is useful to add this section. For origins of this section see also [29].

Theorem A.1. Suppose that (a) is an euclidean univalent harmonic mapping from an open set which contains into ;(b) is a convex set in ;(c) contains a disc ,  , and belongs to the boundary of .Then (d), .

A generalization of this result to several variables has been communicated at Analysis Belgrade Seminar, cf. [32, 33].

Proposition A.2. Suppose that (a′) is an euclidean harmonic orientation preserving univalent mapping from an open set which contains into ;(b′) is a convex set in ;(c′) contains a disc and .
Then (d′), .

By (d), we have(e) on .

Since is an euclidean univalent harmonic mapping, . Using (e) and applying Maximum Principle to the analytic function , we obtain Proposition A.2.

Proof of Theorem A.1. Without loss of generality we can suppose that . Let be arbitrary. Since is a bounded convex set in there exists such that harmonic function , defined by , where , has a maximum on at .
Define . By the mean value theorem, .
Since Poisson kernel for satisfies using Poisson integral representation of the function , , we obtain and hence .

Now we derive a slight generalization of Proposition A.2. More precisely, we show that we can drop the hypothesis and suppose weaker hypothesis .

Proposition A.3. Suppose that    is an euclidean harmonic orientation preserving univalent mapping of the unit disc onto convex domain . If contains a disc and , then

A proof of the proposition can be based on Proposition A.2 and the hereditary property of convex functions: (i) if an analytic function maps the unit disk univalently onto a convex domain, then it also maps each concentric subdisk onto a convex domain. It seems that we can also use the approach as in the proof of Proposition A.7, but an approximation argument for convex domain , which we outline here, is interesting in itself: (ii) approximation of convex domain with smooth convex domains.

Let be conformal mapping of onto , , , , is univalent mapping of the unit disc onto convex domain and .  (iii) Let be conformal mapping of onto , , , and . Since and , we can apply the Carathéodory theorem; tends to , uniformly on compacts, whence (). By the hereditary property is convex. (iv) Since the boundary of is an analytic Jordan curve, the mapping can be continued analytically across , which implies that has a harmonic extension across .

Thus we have the following. (v) are harmonic on , are smooth convex domains, and tends to , uniformly on compacts subset of .

Using (v), an application of Proposition A.2 to , gives the proof.

As a corollary of Proposition A.3 we obtain (A.4).

Proposition A.4. Let be an euclidean harmonic orientation preserving -qc mapping of the unit disc onto convex domain . If contains a disc and , then In particular, is Lipschitz on .

It is worthy to note that (A.4) holds (i.e., is Lipschitz) under assumption that is convex (without any smoothness hypothesis).

Example A.5. is univalent on . Since it follows that tends if tends to 1. This example shows that we cannot drop the hypothesis that is a convex domain in Proposition A.4.

Proof. Let . Since it follows that and therefore .

Hall, see [43, pages 66–68], proved the following.

Lemma A.6 (Hall Lemma). For all harmonic univalent mappings of the unit disk onto itself with , where , , and .

Set . Now we derive a slight generalization of Proposition A.3. More precisely, we show that we can drop the hypothesis that the image of the unit disc is convex.

Proposition A.7. Let be an euclidean harmonic orientation preserving univalent mapping of the unit disc into such that contains a disc and .   Then   The constant in inequality (A.8) can be replaced with sharp constant .  If in addition is -qc mapping and , then

Proof. Let and and let be a conformal mapping of the unit disc onto such that and let . By Schwarz lemma The function has continuous partial derivatives on . Since , by Proposition A.3, we get . Hence, using (A.10) we find and if tends to , we get (A.8). (i2)If and , then(vi).
Hence, by the Hall lemma, and therefore  (vii), where . Combining (vi) and (vii), we prove (i2).(i3)If , where and are analytic, then and . Set and . By the Hall sharp form we get , then and therefore (A.9).

Also as a corollary of Proposition A.3 we obtain the following.

Proposition A.8 (see [27, 44]). Let be an euclidean harmonic diffeomorphism of the unit disc onto convex domain . If contains a disc and , then
where .

The following example shows that Theorem A.1 and Propositions A.2, A.3, A.4, A.7, and A.8 are not true if we omit the condition .

Example A.9. The mapping is a conformal automorphism of the unit disc onto itself and In particular .

Heinz proved (see [45]); that if is a harmonic diffeomorphism of the unit disc onto itself such that , then

Using Proposition A.3 we can prove another Heinz theorem.

Theorem A.10 (Heinz). There exists no euclidean harmonic diffeomorphism from the unit disc onto .

Note that this result was a key step in his proof of the Bernstein theorem for minimal surfaces in .

Remark A.11. Professor Kalaj turned my attention to the fact that in Proposition 12, the constant can be replaced with sharp constant which is approximately .
Thus Hall asserts the sharp form , where and therefore if , then , where .
If we combine Hall’s sharp form with the Schwarz lemma for harmonic mappings, we conclude that if is real, then .
Concerning general codomains, the author, using Hall’s sharp result, communicated around 1990 at Seminar University of Belgrade a proof of Corollary 16 and a version of Proposition 13; cf. [32, 33].

A.1. Characterization of Harmonic qc Mappings (See [25])

By we denote restriction of on . If , it is well-known that is a homeomorphism and . Now we give characterizations of in terms of its boundary value .

Suppose that is an orientation preserving diffeomorphism of onto itself, continuous on such that , and the is restriction of on . Recall if and only if there is analytic function such that is relatively compact subset of and a.e.

We give similar characterizations in the case of the unit disk and for smooth domains (see below).

Theorem A.12. Let be a continuous increasing function on such that , and . Then is qc if and only if the following hold:(1);(2)there is analytic function such that is relatively compact subset of and a.e.

In the setting of this theorem we write . The reader can use the above characterization and functions of the form , where is an inner function, to produce examples of HQC mappings of the unit disk onto itself so the partial derivatives of have no continuous extension to certain points on the unit circle. In particular we can take ; for the subject of this subsections cf. [25, 28] and references cited therein.

Remark A.13. Because of lack of space in this paper we could not consider some basic concepts related to the subject and in particular further distortion properties of qc maps as Gehring and Osgood inequality [12]. For an application of this inequality, see [25, 28].

B. Quasi-Regular Mappings

(A) The theory of holomorphic functions of one complex variable is the central object of study in complex analysis. It is one of the most beautiful and most useful parts of the whole mathematics.

Holomorphic functions are also sometimes referred to as analytic functions, regular functions, complex differentiable functions or conformal maps.

This theory deals only with maps between two-dimensional spaces (Riemann surfaces).

For a function which has a domain and range in the complex plane and which preserves angles, we call a conformal map. The theory of functions of several complex variables has a different character, mainly because analytic functions of several variables are not conformal.

Conformal maps can be defined between Euclidean spaces of arbitrary dimension, but when the dimension is greater than 2, this class of maps is very small. A theorem of J. Liouville states that it consists of Mobius transformations only; relaxing the smoothness assumptions does not help, as proved by Reshetnyak. This suggests the search of a generalization of the property of conformality which would give a rich and interesting class of maps in higher dimension.

The general trend of the geometric function theory in is to generalize certain aspects of the analytic functions of one complex variable. The category of mappings that one usually considers in higher dimensions is the mappings with finite distortion, thus, in particular, quasiconformal and quasiregular mappings.

For the dimensions and , the class of -quasi-regular mappings agrees with that of the complex-analytic functions. Injective quasi-regular mappings in dimensions are called quasiconformal. If is a domain in , , we say that a mapping is discrete if the preimage of a point is discrete in the domain . Planar quasiregular mappings are discrete and open (a fact usually proved via Stoïlow’s Theorem).

Theorem A (Stoïlow’s Theorem). For and , a -quasi-regular mapping can be represented in the form , where is a -quasi-conformal homeomorphism and is an analytic function on .

There is no such representation in dimensions in general, but there is representation of Stoïlow’s type for quasiregular mappings of the Riemann -sphere , cf. [46].

Every quasiregular map has a factorization , where is quasiconformal and is uniformly quasiregular.

Gehring-Lehto Lemma: let be a complex, continuous, and open mapping of a plane domain which has finite partial derivatives a.e. in . Then is differentiable a.e. in .

Let be an open set in , and let be a mapping. Set If is differentiable at , then . The theorem of Rademacher-Stepanov states that if a.e., then is differentiable a.e.

Note that Gehring-Lehto Lemma is used in dimension and Rademacher-Stepanov theorem to show the following.

For all dimensions, , a quasiconformal mapping , where is a domain in , is differentiable a.e. in .

Therefore the set of those points where it is not differentiable has Lebesgue measure zero. Of course, may be nonempty in general and the behaviour of the mapping may be very interesting at the points of this set. Thus, there is a substantial difference between the two cases and . This indicates that the higher dimensions theory of quasi-regular mappings is essentially different from the theory in the complex plane. There are several reasons for this:(a)there are neither general representation theorems of Stoïlow’s type nor counterparts of power series expansions in higher dimensions;(b)the usual methods of function theory based on Cauchy’s Formula, Morera’s Theorem, Residue Theorem, The Residue Calculus and Consequences, Laurent Series, Schwarz’s Lemma, Automorphisms of the Unit Disc, Riemann Mapping Theorem, and so forth, are not applicable in the higher-dimensional theory;(c)in the plane case the class of conformal mappings is very rich, while in higher dimensions it is very small (J. Liouville proved that for and , sufficiently smooth quasiconformal mappings are restrictions of Möbius transformations);(d)for dimensions the branch set (i.e., the set of those points at which the mapping fails to be a local homeomorphism) is more complicated than in the two-dimensional case; for instance, it does not contain isolated points.

Injective quasiregular maps are called quasiconformal. Using the interaction between different coordinate systems, for example, spherical coordinates and cylindrical coordinates , one can construct certain qc maps.

Define by . Then maps the cone for onto the infinite cylinder . We leave it to the reader as an exercise to check that the linear distortion depends only on and that .

For we obtain a qc map of the half-space onto the cylinder with the linear distortion bounded by .

Since the half space and ball are conformally equivalent, we find that there is a qc map of the unit ball onto the infinite cylinder with the linear distortion bounded by .

A simple example of noninjective quasiregular map is given in cylindrical coordinates in 3-space by the formula . This map is two-to-one and it is quasiregular on any bounded domain in whose closure does not intersect the -axis. The Jacobian is different from except on the -axis, and it is smooth everywhere except on the -axis.

Set and . The Zorich map is a quasiregular analogue of the exponential function. It can be defined as follows.(1)Choose a bi-Lipschitz map .(2)Define by .(3)Extend to all of by repeatedly reflecting in planes.Quasiregular maps of which generalize the sine and cosine functions have been constructed by Drasin, by Mayer, and by Bergweiler and Eremenko, see in Fletcher and Nicks paper [47].

(B) There are some new phenomena concerning quasiregular maps which are local homeomorphisms in dimensions . A remarkable fact is that all smooth quasiregular maps are local homeomorphisms. Even more remarkable is the following result of Zorich [48].

Theorem B.1. Every quasiregular local homeomorphism , where , is a homeomorphism.

The result was conjectured by M. A. Lavrentev in 1938. The exponential function exp shows that there is no such result for . Zorich’s theorem was generalized by Martio et al., cf. [49] (for a proof see also, e.g., [35, Chapter III, Section 3]).

Theorem B.2. There is a number (which one calls the injectivity radius) such that every -quasiregular local homeomorphism of the unit ball is actually homeomorphic on .

An immediate corollary of this is Zorich’s result.

This explains why in the definition of quasiregular maps it is not reasonable to restrict oneself to smooth maps: all smooth quasiregular maps of to itself are quasiconformal.

In each dimension there is a positive number such that every nonconstant quasiregular mapping whose distortion function satisfies for some , is locally injective.

(C) Despite the differences between the two theories described in parts (A) and (B), many theorems about geometric properties of holomorphic functions of one complex variable have been extended to quasiregular maps. These extensions are usually highly nontrivial.(e)In a pioneering series of papers, Reshetnyak proved in 1966–1969 that these mappings share the fundamental topological properties of complex-analytic functions: nonconstant quasi-regular mappings are discrete, open, and sense-preserving, cf. [50]. Here we state only the following.

Open Mapping Theorem. If is open in and is a nonconstant qr function from to , we have that is open set. (Note that this does not hold for real analytic functions).

An immediate consequence of the open mapping theorem is the maximum modulus principle. It states that if is qr in a domain and achieves its maximum on , then is constant. This is clear. Namely, if then the open mapping theorem says that is an interior point of and hence there is a point in with larger modulus.(f)Reshetnyak also proved important convergence theorems for these mappings and several analytic properties: they preserve sets of zero Lebesgue-measure, are differentiable almost everywhere, and are Hölder continuous. The Reshetnyak theory (which uses the phrase mapping with bounded distortion for “quasi-regular mapping”) makes use of Sobolev spaces, potential theory, partial differential equations, calculus of variations, and differential geometry. Those mappings solve important first-order systems of PDEs analogous in many respects to the Cauchy-Riemann equation. The solutions of these systems can be viewed as “absolute” minimizers of certain energy functionals.(g)We have mentioned that all pure topological results about analytic functions (such as the Maximum Modulus Principle and Rouché’s theorem) extend to quasiregular maps. Perhaps the most famous result of this sort is the extension of Picard’s theorem which is due to Rickman, cf. [35]:

A -quasiregular map can omit at most a finite set.

When , this omitted set can contain at most two points (this is a simple extension of Picard’s theorem). But when , the omitted set can contain more than two points, and its cardinality can be estimated from above in terms of and . There is an integer such that every -qr mapping , where are disjoint, is constant. It was conjectured for a while that . Rickman gave a highly nontrivial example to show that it is not the case: for every positive integer there exists a nonconstant -qr mapping omitting points.(h)It turns out that these mappings have many properties similar to those of plane quasiconformal mappings. On the other hand, there are also striking differences. Probably the most important of these is that there exists no analogue of the Riemann mapping theorem when . This fact gives rise to the following two problems. Given a domain in Euclidean -space, does there exist a quasiconformal homeomorphism of onto the -dimensional unit ball ? Next, if such a homeomorphism exists, how small can the dilatation of be?

Complete answers to these questions are known when . For a plane domain can be mapped quasiconformally onto the unit disk if and only if is simply connected and has at least two boundary points. The Riemann mapping theorem then shows that if satisfies these conditions, there exists a conformal homeomorphism of onto . The situation is very much more complicated in higher dimensions, and the Gehring-Väisälä paper [51] is devoted to the study of these two questions in the case where .

(D) We close this subsection with short review of Dyakonov’s approach [15].

The main result of Dyakonov’s papers [6] (published in Acta Math.) is as follows.

Theorem B. Lipschitz-type properties are inherited from its modulus by analytic functions.

A simple proof of this result was also published in Acta Math. by Pavlović, cf. [52].(D1)In this item we shortly discuss the Lipschitz-type properties for harmonic functions. The set of harmonic functions on a given open set can be seen as the kernel of the Laplace operator and is therefore a vector space over : sums, differences, and scalar multiples of harmonic functions are again harmonic. In several ways, the harmonic functions are real analogues to holomorphic functions. All harmonic functions are real analytic; that is, they can be locally expressed as power series, they satisfy the mean value theorem, there is Liouville’s type theorem for them, and so forth.

Harmonic quasiregular (briefly, qr) mappings in the plane were studied first by Martio in [53]; for a review of this subject and further results, see [25, 54] and the references cited there. The subject has grown to include study of qr maps in higher dimensions, which can be considered as a natural generalization of analytic function in plane and good candidate for a generalization of Theorem B.

For example, Chen et al. [19] and the author [55] have shown that Lipschitz-type properties are inherited from its modulo for -quasiregular and harmonic mappings in planar case. These classes include analytic functions in planar case, so this result is a generalization of Theorem A.(D2)Quasiregular mappings. Dyakonov [15] made further important step and roughly speaking showed that Lipschitz-type properties are inherited from its modulus by qc mappings (see two next results).

Theorem B.3 (Theorem  Dy, see [15, Theorem 4]). Let , , and . Suppose is a -extension domain in and is a -quasiconformal mapping of onto . Then the following conditions are equivalent: (i.5);(ii.5).If, in addition, is a uniform domain and if , then (i.5) and (ii.5) are equivalent to(iii.5).

An example in Section 2 [15] shows that the assumption cannot be dropped. For , we have the following generalization—but also a consequence of Theorem B.3 dealing with quasiregular mappings that are local homeomorphisms (i.e., have no branching points).

Theorem B.4 (see [15, Proposition 3]). Let , , and . Suppose is a -extension domain in and is a -quasiregular locally injective mapping of onto . Then if and only if .

We remark that a quasiregular mapping in dimension will be locally injective (or, equivalently, locally homeomorphic) if the dilatation is sufficiently close to 1. For more sophisticated local injectivity criteria, see the literature cited in [10, 15].

Here we only outline how to reduce the proof of Theorem B.4 to qc case.

It is known that, by Theorem B.2, for given and , there is a number such that every -quasiregular local homeomorphism of the unit ball is actually homeomorphic on . Applying this to the mappings , , where and , we see that is homeomorphic (and hence quasiconformal) on each ball . The constant , coming from the preceding statement, depends on and , but not on .

Note also the following.

(i0) Define and . We can express the distortion property of qc mappings in the following useful form.

Proposition A. Suppose that is a domain in and is -quasiconformal and . There is a constant such that for and . This form shows in an explicit way that the maximal dilatation of a qc mapping is controlled by minimal and it is convenient for some applications. For example, Proposition A also yields a simple proof of Theorem B.3.
Recall the following. (i1) Roughly speaking, quasiconformal and quasiregular mappings in , , are natural generalizations of conformal and analytic functions of one complex variable.
(i2) Dyakonov’s proof of Theorem 6.4 in [15] (as we have indicated above) is reduced to quasiconformal case, but it seems likely that he wanted to consider whether the theorem holds more generally for quasiregular mappings.
(i3) Quasiregular mappings are much more general than quasiconformal mappings and in particular analytic functions.

(E) Taking into account the above discussion it is natural to explore the following research problem.

Question  A. Is it possible to drop local homeomorphism hypothesis in Theorem B.4?

It seems that using the approach from [15], we cannot solve this problem and that we need new techniques.

(i4) However, we establish the second version of Koebe theorem for -quasiregular functions, Theorem 39, and the characterization of Lipschitz-type spaces for quasiregular mappings by the average Jacobian, Theorem 40. Using these theorems, we give a positive solution to Question A.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author is grateful to the anonymous referees for a number of helpful suggestions and in particular to the Editor in Chief Professor Cirić for patiently reading through the whole text at several stages of its revision. This research was partially supported by MNTRS, Serbia, Grant no. 174 032.