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

Authors: Miodrag Mateljević
Journal: Abstract and Applied Analysis
 (2014)

---

## 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 inn-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.


---

## Body

## 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 a
f, defined in Section 2, plays analogous role as |
f
′
| when n
=
2 and f 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 Section2, we give a short proof of Lemma 4, using a version with the average Jacobian J
_
f instead of a
f, 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 D with the property that functions which satisfy a local Lipschitz condition in D for some α 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 t
α, 0
<
α
<
1.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 Section4, 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.ByR
n
=
{
x
=
(
x
1
,
…
,
x
n
)
:
x
1
,
…
,
x
n
∈
R
} denote the real vector space of dimension n. For a domain D in R
n with nonempty boundary, we define the distance function d
=
d
(
D
)
=
dist
⁡
(
D
) by d
(
x
)
=
d
(
x
;
∂
D
)
=
dist
⁡
(
D
)
(
x
)
=
inf
⁡
{
|
x
-
y
|
:
y
∈
∂
D
}, and if f maps D onto D
′
⊂
R
n, in some settings, it is convenient to use short notation d
*
=
d
f
(
x
) for d
(
f
(
x
)
;
∂
D
′
). It is clear that d
(
x
)
=
dist
⁡
(
x
,
D
c
), where D
c is the complement of D in R
n.LetG be an open set in R
n. A mapping f
:
G
→
R
m is differentiable at x
∈
G if there is a linear mapping f
′
(
x
)
:
R
n
→
R
n, called the derivative of f at x, such that
(1)
f
(
x
+
h
)
-
f
(
x
)
=
f
′
(
x
)
h
+
|
h
|
ɛ
(
x
,
h
)
,
where ɛ
(
x
,
h
)
→
0 as h
→
0. For a vector-valued function f
:
G
→
R
n, where G
⊂
R
n is a domain, we define
(2)
|
f
′
(
x
)
|
=
max
⁡
|
h
|
=
1
|
f
′
(
x
)
h
|
,
l
(
f
′
(
x
)
)
=
min
⁡
|
h
|
=
1
|
f
′
(
x
)
h
|
,
when f is differentiable at x
∈
G.In Section3, we review some results from [7, 8]. For example, in [7] under some conditions concerning a majorant ω, we showed the following.Letf
∈
C
1
(
D
,
R
n
) and let ω be a continuous majorant such that ω
*
(
t
)
=
ω
(
t
)
/
t is nonincreasing for t
>
0.Assumef satisfies the following property (which we call Hardy-Littlewood (
c
,
ω
)-property):
(3)
(
HL
(
c
,
ω
)
)
d
(
x
)
|
f
′
(
x
)
|
≤
c
ω
(
d
(
x
)
)
,
aaai
x
∈
D
,
where
d
(
x
)
=
dist
⁡
(
x
,
D
c
)
.Then(4)
(
loc
⁡
Λ
)
f
∈
loc
⁡
Λ
ω
(
G
)
.If, in addition,f is harmonic in D or, more generally, f
∈
O
C
2
(
D
), then (
HL
(
c
,
ω
)
) is equivalent to (
loc
⁡
Λ
). If D is a Λ
ω-extension domain, then (
HL
(
c
,
ω
)
) is equivalent to f
∈
Λ
ω
(
D
).In Section3, we also consider Lipschitz-type spaces of pluriharmonic mappings and extend some results from [9].In AppendicesA 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 byΩ, G, and D open subset of R
n, n
≥
1.LetB
n
(
x
,
r
)
=
{
z
∈
R
n
:
|
z
-
x
|
<
r
}, S
n
-
1
(
x
,
r
)
=
∂
B
n
(
x
,
r
) (abbreviated S
(
x
,
r
)) and let B
n, S
n
-
1 stand for the unit ball and the unit sphere in R
n, respectively. Sometimes we write D and T instead of B
2 and ∂
D, respectively. For a domain G
⊂
R
n let ρ
:
G
→
[
0
,
∞
) be a continuous function. We say that ρ is a weight function or a metric density if, for every locally rectifiable curve γ in G, the integral
(5)
l
ρ
(
γ
)
=
∫
γ
‍
ρ
(
x
)
d
s
exists. In this case we call l
ρ
(
γ
) the ρ-length of γ. A metric density defines a metric d
ρ
:
G
×
G
→
[
0
,
∞
) as follows. For a
,
b
∈
G, let
(6)
d
ρ
(
a
,
b
)
=
inf
⁡
γ
l
ρ
(
γ
)
,
where the infimum is taken over all locally rectifiable curves in G joining a and b.For the modern mapping theory, which also considers dimensionsn
≥
3, 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 κ
G and the distance ratio metric j
G of a domain G
⊂
R
n (see [11, 12]). The quasihyperbolic metric κ
=
κ
G of G is a particular case of the metric d
ρ when ρ
(
x
)
=
1
/
d
(
x
,
∂
G
) (see [11, 12]).Given a subsetE of C
n or R
n, a function f
:
E
→
C (or, more generally, a mapping f from E into C
m or R
m) is said to belong to the Lipschitz space Λ
ω
(
E
) if there is a constant L
=
L
(
f
)
=
L
(
f
;
E
)
=
L
(
f
,
ω
;
E
), which we call Lipschitz constant, such that
(7)
|
f
(
x
)
-
f
(
y
)
|
≤
L
ω
(
|
x
-
y
|
)
for all x
,
y
∈
E. The norm ∥
f
∥
Λ
ω
(
E
) is defined as the smallest L 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 majorantsω
α
(
t
)
:
=
t
α
(
0
<
α
≤
1
). The classes Λ
ω
(
E
) with ω
=
ω
α will be denoted by Λ
α (or by Lip
⁡
(
α
;
E
)
=
Lip
⁡
(
α
,
L
;
E
)). L
(
f
) and α are called, respectively, Lipschitz constant and exponent (of f on E). We say that a domain G
⊂
R
n is uniform if there are constants a and b such that each pair of points x
1
,
x
2
∈
G can be joined by rectifiable arc γ in G for which
(8)
l
(
γ
)
≤
a
|
x
1
-
x
2
|
,
min
⁡
l
(
γ
j
)
≤
b
d
(
x
,
∂
G
)
for each x
∈
γ; here l
(
γ
) denotes the length of γ and γ
1
,
γ
2 the components of γ
∖
{
x
}. We define C
(
G
)
=
max
⁡
{
a
,
b
}. The smallest C
(
G
) for which the previous inequalities hold is called the uniformity constant of G and we denote it by c
*
=
c
*
(
G
). Following [2, 17], we say that a function f belongs to the local Lipschitz space loc
⁡
Λ
ω
(
G
) if (7) holds, with a fixed C
>
0, whenever x
∈
G and |
y
-
x
|
<
(
1
/
2
)
d
(
x
,
∂
G
). We say that G is a Λ
ω-extension domain if Λ
ω
(
G
)
=
loc
⁡
Λ
ω
(
G
). In particular if ω
=
ω
α, we say that G is a Λ
α-extension domain; this class includes the uniform domains mentioned above.Suppose thatΓ is a curve family in R
¯
n. We denote by F
(
Γ
) the family whose elements are nonnegative Borel-measurable functions ρ which satisfy the condition ∫
γ
ρ
(
x
)
|
d
x
|
≥
1 for every locally rectifiable curve γ
∈
Γ, where d
s
=
|
d
x
| denotes the arc length element. For p
≥
1, with the notation
(9)
A
p
(
ρ
)
=
∫
R
n
|
ρ
(
x
)
|
p
d
V
(
x
)
,
where d
V
(
x
)
=
d
x denotes the Euclidean volume element d
x
1
d
x
2
⋯
d
x
n, we define the p-modulus of Γ by
(10)
M
p
(
Γ
)
=
inf
⁡
{
A
p
(
ρ
)
:
ρ
∈
F
(
Γ
)
}
.We will denoteM
n
(
Γ
) simply by M
(
Γ
) and call it the modulus of Γ.Suppose thatf
:
Ω
→
Ω
* is a homeomorphism. Consider a path family Γ in Ω and its image family Γ
*
=
{
f
∘
γ
:
γ
∈
Γ
}. We introduce the quantities
(11)
K
I
(
f
)
=
sup
⁡
M
(
Γ
*
)
M
(
Γ
)
,
K
O
(
f
)
=
sup
⁡
M
(
Γ
)
M
(
Γ
*
)
,
where the suprema are taken over all path families Γ in Ω such that M
(
Γ
) and M
(
Γ
*
) are not simultaneously 0 or ∞.Definition 1.
Suppose thatf
:
Ω
→
Ω
* is a homeomorphism; we call K
I
(
f
) the inner dilatation and K
O
(
f
) the outer dilatation of f. The maximal dilatation of f is K
(
f
)
=
max
⁡
{
K
O
(
f
)
,
K
I
(
f
)
}. If K
(
f
)
≤
K
<
∞, we say f is K-quasiconformal (abbreviated qc).Suppose thatf
:
Ω
→
Ω
* is a homeomorphism and x
∈
Ω, x
≠
∞, and f
(
x
)
≠
∞.For eachr
>
0 such that S
(
x
;
r
)
⊂
Ω we set L
(
x
,
f
,
r
)
=
max
⁡
|
y
-
x
|
=
r
|
f
(
y
)
-
f
(
x
)
|, l
(
x
,
f
,
r
)
=
min
⁡
|
y
-
x
|
=
r
|
f
(
y
)
-
f
(
x
)
|.Definition 2.
The linear dilatation off at x is
(12)
H
(
x
,
f
)
=
limsup
⁡
r
→
0
L
(
x
,
f
,
r
)
l
(
x
,
f
,
r
)
.Theorem 3 (the metric definition of quasiconformality).
A homeomorphismf
:
Ω
→
Ω
* is qc if and only if H
(
x
,
f
) is bounded on Ω.LetΩ be a domain in R
n and let f
:
Ω
→
R
n be continuous. We say that f is quasiregular (abbreviated qr) if(1)
fbelongs to Sobolev space W
1
,
loc
⁡
n
(
Ω
),
(2)
there existsK, 1
≤
K
<
∞, such that
(13)
|
f
′
(
x
)
|
n
≤
K
J
f
(
x
)
a
.
e
.The smallestK in (13) is called the outer dilatation K
O
(
f
). 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
K
≥
1
and
n
≥
2
, there exists a function
θ
K
n
:
(
0,1
)
→
R
with the following properties:
(1)
θ
K
n
is increasing,
(2)
lim
⁡
r
→
0
θ
K
n
(
r
)
=
0
,
(3)
lim
⁡
r
→
1
θ
K
n
(
r
)
=
∞
,
(4)
Let
Ω
and
Ω
′
be proper subdomains of
R
n
and let
f
:
Ω
→
Ω
′
be a
K
-qc. If
x
and
y
are points in
Ω
such that
0
<
|
y
-
x
|
<
d
(
x
,
∂
Ω
)
, then
(14)
|
f
(
y
)
-
f
(
x
)
|
d
(
f
(
x
)
,
∂
Ω
′
)
≤
θ
K
n
(
|
y
-
x
|
d
(
x
,
∂
Ω
)
)
.Introduce the quantity, mentioned in the introduction,(15)
a
f
(
x
)
=
a
f
,
G
(
x
)
:
=
exp
⁡
(
1
n
|
B
x
|
∫
B
x
‍
log
⁡
J
f
(
z
)
d
z
)
,
x
∈
G
,
associated with a quasiconformal mapping f
:
G
→
f
(
G
)
⊂
R
n; here J
f is the Jacobian of f, while B
x
=
B
x
,
G stands for the ball B
(
x
;
d
(
x
,
∂
G
)
) and |
B
x
| for its volume.Lemma 4 (see [4]).
Suppose thatG and G
′ are domains in R
n: If f
:
G
→
G
′ is K-quasiconformal, then
(16)
1
c
d
(
f
(
x
)
,
∂
G
′
)
d
(
x
,
∂
G
)
≤
a
f
,
G
(
x
)
≤
c
d
(
f
(
x
)
,
∂
G
′
)
d
(
x
,
∂
G
)
,
x
∈
G
,
where c is a constant which depends only on K and n.Setα
0
=
α
K
n
=
K
1
/
(
1
-
n
).Lemma 5 (see [4]).
Suppose thatD is a uniform domain in R
n and that α and m are constants with 0
<
α
≤
1 and m
≥
0. If f is K-quasiconformal in D with f
(
D
)
⊂
R
n and if
(17)
a
f
,
D
(
x
)
≤
m
d
(
x
,
∂
D
)
α
-
1
f
o
r
x
∈
D
,
then f has a continuous extension to D
¯
∖
{
∞
} and
(18)
|
f
(
x
1
)
-
f
(
x
2
)
|
≤
C
(
|
x
1
-
x
2
|
+
d
(
x
1
,
∂
D
)
)
α
for x
1
,
x
2
∈
D
¯
∖
{
∞
}, where the constant C
=
c
^
(
D
) depends only on K, n, a, m and the uniformity constant c
*
=
c
*
(
D
) for D. In the case α
≤
α
0, (18) can be replaced by the stronger conclusion that
(19)
|
f
(
x
1
)
-
f
(
x
2
)
|
≤
C
(
|
x
1
-
x
2
|
)
α
(
x
1
,
x
2
∈
D
¯
∖
{
∞
}
)
.Example 6.
The mappingf
(
x
)
=
|
x
|
a
-
1
x, a
=
K
1
/
(
1
-
n
) is K-qc with a
f bounded in the unit ball. Hence f satisfies the hypothesis of Lemma 5 with α
=
1. Since |
f
(
x
)
-
f
(
0
)
|
≤
|
x
-
0
|
a
=
|
x
|
a, we see that when K
>
1, that is, a
<
1
=
α, the conclusion (18) in Lemma 5 cannot be replaced by the stronger assertion |
f
(
x
1
)
-
f
(
x
2
)
|
≤
c
m
|
x
1
-
x
2
|.LetΩ
∈
R
n and R
+
=
[
0
,
∞
) and f
,
g
:
Ω
→
R
+. If there is a positive constant c such that f
(
x
)
≤
c
g
(
x
), x
∈
Ω, we write f
⪯
g on Ω. If there is a positive constant c such that
(20)
1
c
g
(
x
)
≤
f
(
x
)
≤
c
g
(
x
)
,
x
∈
Ω
,
we write f
≈
g (or f
≍
g) on Ω.LetG
⊂
R
2 be a domain and let f
:
G
→
R
2, f
=
(
f
1
,
f
2
) be a harmonic mapping. This means that f is a map from G into R
2 and both f
1 and f
2 are harmonic functions, that is, solutions of the two-dimensional Laplace equation
(21)
Δ
u
=
0
.
The above definition of a harmonic mapping extends in a natural way to the case of vector-valued mappings f
:
G
→
R
n, f
=
(
f
1
,
…
,
f
n
), defined on a domain G
⊂
R
n, n
≥
2.Leth be a harmonic univalent orientation preserving mapping on a domain D, D
′
=
h
(
D
) and d
h
(
z
)
=
d
(
h
(
z
)
,
∂
D
′
). If h
=
f
+
g
¯ has the form, where f and g are analytic, we define λ
h
(
z
)
=
D
-
(
z
)
=
|
f
′
(
z
)
|
-
|
g
′
(
z
)
|, and Λ
h
(
z
)
=
D
+
(
z
)
=
|
f
′
(
z
)
|
+
|
g
′
(
z
)
|.


### 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.
SupposeD and D
′ are proper domains in R
2. If h
:
D
→
D
′ is K-qc and harmonic, then it is bi-Lipschitz with respect to quasihyperbolic metrics on D and D
′.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.
Leth
=
f
+
g
¯ be local representation on B
z.
Sinceh is K-qc, then
(22)
(
1
-
k
2
)
|
f
′
|
2
≤
J
h
≤
K
|
f
′
|
2
on B
z and since log
⁡
|
f
′
(
ζ
)
| is harmonic,
(23)
log
⁡
|
f
′
(
z
)
|
=
1
2
|
B
z
|
∫
B
z
‍
log
⁡
|
f
′
(
ζ
)
|
2
d
ξ
d
η
.
Hence,(24)
log
⁡
a
h
,
D
(
z
)
≤
1
2
log
⁡
K
+
1
2
|
B
z
|
∫
B
z
‍
log
⁡
|
f
′
(
ζ
)
|
2
d
ξ
d
η
=
log
⁡
K
|
f
′
(
z
)
|
.
Hence,a
h
,
D
(
z
)
≤
K
|
f
′
(
z
)
| and 1
-
k
|
f
′
(
z
)
|
≤
a
h
,
D
(
z
). Using Astala-Gehring result, we get
(25)
Λ
(
h
,
z
)
≍
d
(
h
z
,
∂
D
′
)
d
(
z
,
∂
D
)
≍
λ
(
h
,
z
)
.
This pointwise result, combined with integration along curves, easily gives(26)
k
D
′
(
h
(
z
1
)
,
h
(
z
2
)
)
≍
k
D
(
z
1
,
z
2
)
,
z
1
,
z
2
∈
D
.
Note that we do not use thatlog
⁡
(
1
/
J
h
) is a subharmonic function.The following follows from the proof of Proposition7:(I)
Λ
(
h
,
z
)
≈
a
h
,
D
≈
λ
(
h
,
z
) and J
f
2
≈
J
_
f; see below for the definition of average jacobian J
_
f.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 byc a constant and by c
_, c
^, c
~ some specific constants.Our next result concerns the quantity(27)
E
_
f
,
G
(
x
)
:
=
1
|
B
_
x
|
∫
B
_
x
‍
J
f
(
z
)
d
z
,
x
∈
G
,
associated with a quasiconformal mapping f
:
G
→
f
(
G
)
⊂
R
n; here J
f is the Jacobian of f, while B
_
x
=
B
_
x
,
G stands for the ball B
(
x
,
d
(
x
,
∂
G
)
/
2
) and |
B
x
| for its volume.Define(28)
J
_
f
=
J
_
f
,
G
=
E
_
f
,
G
n
.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 thatG and G
′ are domains in R
n: If f
:
G
→
G
′ is K-quasiconformal, then
(29)
1
c
d
(
f
(
x
)
,
∂
G
′
)
d
(
x
,
∂
G
)
≤
J
_
f
,
G
(
x
)
≤
c
d
(
f
(
x
)
,
∂
G
′
)
d
(
x
,
∂
G
)
,
x
∈
G
,
where c is a constant which depends only on K and n.Proof.
By the distortion property of qc (see [23, page 383], [24, page 63]), there are the constants C
* and c
* which depend on n and K only, such that
(30)
B
(
f
(
x
)
,
c
*
d
*
)
⊂
f
(
B
x
)
⊂
B
(
f
(
x
)
,
C
*
d
*
)
,
x
∈
G
,
where d
*
(
x
)
:
=
d
(
f
(
x
)
)
=
d
(
f
(
x
)
,
∂
G
′
) and d
(
x
)
:
=
d
(
x
,
∂
G
). Hence
(31)
|
B
(
f
(
x
)
,
c
*
d
*
)
|
≤
∫
B
x
‍
J
f
(
z
)
d
z
≤
|
B
(
f
(
x
)
,
C
*
d
*
)
|
and therefore we prove Theorem 8.We only outline proofs in the rest of this subsection.Suppose thatΩ and Ω
′ are domains in R
n different from R
n.Theorem 9.
Supposef
:
Ω
→
Ω
′ is a C
1 and the following hold.(i1)
f is c-Lipschitz with respect to quasihyperbolic metrics on Ω and Ω
′; then
(I1)
d
|
f
′
(
x
0
)
|
≤
c
d
*
(
x
0
) for every x
0
∈
Ω.
(i2)
Iff is a qc c-quasihyperbolic-isometry, then
(I2)
f is a c
2-qc mapping,
(
I
3
)
|
f
′
|
≈
d
*
/
d.Proof.
SinceΩ and Ω
′ are different from R
n, there are quasihyperbolic metrics on Ω and Ω
′. Then for a fixed x
0
∈
Ω, we have
(32)
k
(
x
,
x
0
)
=
d
(
x
0
)
-
1
|
x
-
x
0
|
(
1
+
o
(
1
)
)
,
x
⟶
x
0
,
k
(
f
(
x
)
,
f
(
x
0
)
)
=
|
f
(
x
)
-
f
(
x
0
)
|
d
*
(
f
(
x
0
)
)
(
1
+
o
(
1
)
)
,
yyyyyyyyyyyyyyyyyyyyyyyy
when
x
⟶
x
0
.
Hence, (i1) implies (I1). Iff is a c-quasihyperbolic-isometry, then d
|
f
′
(
x
0
)
|
≤
c
d
* and d
l
(
f
′
(
x
0
)
)
≥
d
*
/
c. Hence, (I2) and (I3) follow.Theorem 10.
Supposef
:
Ω
→
Ω
′ is a C
1 qc homeomorphism. The following conditions are equivalent:(a.1)
f is bi-Lipschitz with respect to quasihyperbolic metrics on Ω and Ω
′,
(b.1)
J
f
n
≈
d
*
/
d,
(c.1)
J
f
n
≈
a
f,
(d.1)
J
f
n
≈
J
_
f,where d
(
x
)
=
d
(
x
,
∂
Ω
) and d
*
(
x
)
=
d
(
f
(
x
)
,
∂
Ω
′
).Proof.
It follows from Theorem9 that (a) is equivalent to (b) (see, e.g., [32, 33]).
Theorem8 states that J
_
f
≈
d
*
/
d. By Lemma 4, a
f
≈
d
*
/
d and therefore (b) is equivalent to (c). The rest of the proof is straightforward.Lemma 11.
Iff
∈
C
1,1 is a K-quasiconformal mapping defined in a domain Ω
⊂
R
n
(
n
≥
3
), then
(33)
J
f
(
x
)
>
0
,
x
∈
Ω
provided that K
<
2
n
-
1. The constant 2
n
-
1 is sharp.IfG
¯
⊂
Ω, then it is bi-Lipschitz with respect to Euclidean and quasihyperbolic metrics on G and G
′
=
f
(
G
).It is a natural question whether is there an analogy of Theorem10 if we drop the C
1 hypothesis.Supposef
:
Ω
→
Ω
′ is onto qc mapping. We can consider the following conditions:(a.2)
f is bi-Lipschitz with respect to quasihyperbolic metrics on Ω and Ω
′;
(b.2)
J
f
n
≈
d
*
/
d a.e. in Ω;
(c.2)
J
f
n
≈
a
f a.e. in Ω;
(d.2)
J
f
n
≈
J
_
f a.e. in Ω.It seems that the above conditions are equivalent, but we did not check details.IfΩ is a planar domain and f harmonic qc, then we proved that (
d
.
2
) holds (see Proposition 18).


### 2.2. Quasi-Isometry in Planar Case

For a functionh, we use notations ∂
h
=
(
1
/
2
)
(
h
x
′
-
i
h
y
′
) and ∂
¯
h
=
(
1
/
2
)
(
h
x
′
+
i
h
y
′
); we also use notations D
h and D
¯
h instead of ∂
h and ∂
¯
h, respectively, when it seems convenient. Now we give another proof of Proposition 7, using the following.Proposition 12 (see [29]).
Leth be an euclidean harmonic orientation preserving univalent mapping of the unit disc D into C such that h
(
D
) contains a disc B
R
=
B
(
a
;
R
) and h
(
0
)
=
a. Then
(34)
|
∂
h
(
0
)
|
≥
R
4
.
If, in addition,h is K-qc, then
(35)
λ
h
(
0
)
≥
1
-
k
4
R
.For more details, in connection with material considered in this subsection, see also AppendixA, in particular, Proposition A.7.Leth
=
f
+
g
¯ be a harmonic univalent orientation preserving mapping on the unit disk D, Ω
=
h
(
D
), d
h
(
z
)
=
d
(
h
(
z
)
,
∂
Ω
), λ
h
(
z
)
=
D
-
(
z
)
=
|
f
′
(
z
)
|
-
|
g
′
(
z
)
|, and Λ
h
(
z
)
=
D
+
(
z
)
=
|
f
′
(
z
)
|
+
|
g
′
(
z
)
|. By the harmonic analogue of the Koebe Theorem, then
(36)
1
16
D
-
(
0
)
≤
d
h
(
0
)
and therefore
(37)
(
1
-
|
z
|
2
)
1
16
D
-
(
z
)
≤
d
h
(
z
)
.
If, in addition, h is K-qc, then
(38)
(
1
-
|
z
|
2
)
D
+
(
z
)
≤
16
K
d
h
(
z
)
.
Using Proposition 12, we also prove
(39)
(
1
-
|
z
|
2
)
λ
h
(
z
)
≥
1
-
k
4
d
h
(
z
)
.If we summarize the above considerations, we have proved the part(
a
.
3
) of the following proposition.Proposition 13 (e-qch, hyperbolic distance version).
(a.3). Leth
=
f
+
g
¯ be a harmonic univalent orientation preserving K-qc mapping on the unit disk D, Ω
=
h
(
D
). Then, for z
∈
D,
(40)
(
1
-
|
z
|
2
)
Λ
h
(
z
)
≤
16
K
d
h
(
z
)
,
(
1
-
|
z
|
2
)
λ
h
(
z
)
≥
1
-
k
4
d
h
(
z
)
.
(
b
.
3
) If h is a harmonic univalent orientation preserving K-qc mapping of domain D onto D
′, then
(41)
d
(
z
)
Λ
h
(
z
)
≤
16
K
d
h
(
z
)
,
d
(
z
)
λ
h
(
z
)
≥
1
-
k
4
d
h
(
z
)
,
and
(42)
1
-
k
4
κ
D
(
z
,
z
′
)
≤
κ
D
′
(
h
z
,
h
z
′
)
≤
16
K
κ
D
(
z
,
z
′
)
,
where z
,
z
′
∈
D.Remark 14.
In particular, we have Proposition7, 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 B
z, z
∈
D, we get (41). It is clear that (41) implies (42).For a planar hyperbolic domain inC, we denote by ρ
=
ρ
D and d
hyp
=
d
hyp
;
D the hyperbolic density and metric of D, respectively.We say that a domainD
⊂
C isstrongly 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 diskD
′
=
{
0
<
|
z
|
<
1
} is obtained by mapping its universal covering, an infinitely-sheeted disk, on the half plane Π
-
=
{
Re
w
<
0
} by means of w
=
ln
⁡
z (i.e., z
=
e
w). The metric is
(43)
|
d
w
|
|
Re
w
|
=
|
d
z
|
|
z
|
ln
⁡
(
1
/
|
z
|
)
.
Since a boundary component is the point0, the punctured disk is not a strongly hyperbolic domain. Note also that ρ
/
d and 1
/
ln
⁡
(
1
/
d
) tend to 0 if z tends to 0. Here d
=
d
(
z
)
=
dist
⁡
(
z
,
∂
D
′
) and ρ is the hyperbolic density of D
′. Therefore one has the following:(
A
.
1
)
forz
∈
D
′, κ
D
′
(
z
,
z
′
)
=
ln
⁡
(
1
/
d
(
z
)
)
d
hyp
,
D
′
(
z
,
z
′
) when z
′
→
z.
There is no constantc such that (
A
.
2
)
κ
D
′
(
z
,
z
′
)
≤
c
κ
Π
-
(
w
,
w
′
) for every w
,
w
′
∈
Π
-, where z
=
e
w and z
′
=
e
w
′.
Sinced
hyp
,
Π
-
≈
κ
Π
-, if (
A
.
2
) holds, we conclude that κ
D
′
≈
d
hyp
,
D
′, which is a contradiction by (
A
.
1
).LetD be a planar hyperbolic domain. Then if D is simply connected,
(44)
1
2
d
≤
ρ
≤
2
d
.For general domain asz
→
∂
D,
(45)
1
+
o
(
1
)
d
ln
⁡
(
1
/
d
)
≤
ρ
≤
2
d
.Henced
hyp
(
z
,
z
′
)
≤
2
κ
D
(
z
,
z
′
), z
,
z
′
∈
D.IfΩ is a strongly hyperbolic domain, then there is a hyperbolic density ρ on the domain Ω, such that ρ
≈
d
-
1, where d
(
w
)
=
d
(
w
,
∂
Ω
); see, for example, [34]. Thus d
hyp
;
D
≈
κ
D. Hence, we find the following.Corollary 16.
Everye-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.
LetD be a hyperbolic domain, let h  be e-harmonic quasiconformal mapping of D onto Ω, and let ϕ
:
D
→
D be a covering and h
*
=
h
∘
ϕ. (a.4)
Suppose thatD is simply connected. Thus ϕ and h
* are one-to-one. Then
(46)
(
1
-
|
z
|
2
)
λ
h
*
(
ζ
)
≥
1
-
k
4
d
h
(
z
)
,where z
=
ϕ
(
ζ
).
Hence, sinceλ
h
*
(
ζ
)
=
l
h
(
z
)
|
ϕ
′
(
ζ
)
| and ρ
hyp
;
D
(
z
)
|
ϕ
′
(
ζ
)
|
=
ρ
D
(
ζ
),
(47)
λ
h
(
z
)
≥
1
-
k
4
d
h
(
z
)
ρ
D
(
z
)
.
Using Hall’s sharp result, one can also improve the constant in the second inequality in Propositions13 and 12 (i.e., the constant 1
/
4 can be replaced by τ
0; see below for more details):
(48)
(
1
-
|
z
|
2
)
λ
h
(
z
)
≥
(
1
-
k
)
τ
0
d
h
(
z
)
,
where τ
0
=
3
3
/
2
π.(b.4)
Suppose thatD is not a simply connected. Then h
* is not one-to-one and we cannot apply the procedure as in (
a
.
4
).
(c.4)
It seems natural to consider whether there is an analogue in higher dimensions of Proposition13.Proposition 18.
For everye-harmonic quasiconformal mapping f of the unit disc (more generally of a hyperbolic domain D) the following holds: (e.2)
J
f
≈
d
*
/
d. In particular, it is a quasi-isometry with respect to quasihyperbolic distances.Proof.
Forz
∈
D, by the distortion property,
(49)
|
f
(
z
1
)
-
f
(
z
)
|
≤
c
d
(
f
(
z
)
)
for
every
z
1
∈
B
_
z
.
Hence, by Schwarz lemma for harmonic maps, d
|
f
′
(
z
)
|
≤
c
1
d
(
f
(
z
)
). Proposition 12 yields (e) and an application of Proposition 13 gives the proof.RecallB
_
x stands for the ball B
(
x
;
d
(
x
,
∂
G
)
/
2
) and |
B
_
x
| for its volume. If V is a subset of R
n and u
:
V
→
R
m, we define
(50)
os
c
V
u
:
=
sup
⁡
{
|
u
(
x
)
-
u
(
y
)
|
:
x
,
y
∈
V
}
,
ω
u
(
r
,
x
)
:
=
sup
⁡
{
|
u
(
x
)
-
u
(
y
)
|
:
|
x
-
y
|
=
r
}
.Suppose thatG
⊂
R
n and B
_
x
=
B
(
x
,
d
(
x
)
/
2
). Let O
C
1
(
G
)
=
O
C
1
(
G
;
c
1
) denote the class of f
∈
C
1
(
G
) such that
(51)
d
(
x
)
|
f
′
(
x
)
|
≤
c
1
os
c
B
_
x
f
for every x
∈
G.Proposition 19.
Supposef
:
Ω
→
Ω
′ is a C
1. Then f
∈
O
C
1
(
Ω
;
c
), if and only if f is c-Lipschitz with respect to quasihyperbolic metrics on B and f
(
B
) for every ball B
⊂
Ω.


### 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:SupposeG is a Λ
α-extension domain in R
n and f is a K-quasiconformal mapping of G onto f
(
G
)
⊂
R
n. Then f
∈
Λ
α
(
G
) if and only if |
f
|
∈
Λ
α
(
G
).This is TheoremB.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
(52)
E
_
f
,
G
(
x
)
:
=
1
|
B
_
x
|
∫
B
_
x
‍
J
f
(
z
)
d
z
,
x
∈
G
,
associated with a quasiconformal mapping f
:
G
→
f
(
G
)
⊂
R
n; here J
f is the Jacobian of f, while B
_
x
=
B
_
x
,
G stands for the ball B
(
x
;
d
(
x
,
∂
G
)
/
2
) and |
B
_
x
| for its volume.Defineα
0
=
α
K
n
=
K
1
/
(
1
-
n
), C
*
=
θ
K
n
(
1
/
2
), θ
K
n
(
c
*
)
=
1
/
2, and
(53)
J
_
f
,
G
=
E
_
f
,
G
n
.For a ballB
=
B
(
x
,
r
)
⊂
R
n and a mapping f
:
B
→
R
n, we define
(54)
E
f
(
B
;
x
)
:
=
1
|
B
|
∫
B
‍
J
f
(
z
)
d
z
,
x
∈
G
,
and J
f
a
v
(
B
;
x
)
=
E
f
(
B
;
x
)
n; we also use the notation J
f
(
B
;
x
) and J
f
(
x
;
r
) instead of J
f
a
v
(
B
;
x
).Forx
,
y
∈
R
n, we define the Euclidean inner product 〈
·
,
·
〉 by
(55)
〈
x
,
y
〉
=
x
1
y
1
+
⋯
+
x
n
y
n
.
By E
n we denote R
n with the Euclidean inner product and call it Euclidean space n-space (space of dimension n). In this paper, for simplicity, we will use also notation R
n for E
n. Then the Euclidean length of x
∈
R
n is defined by
(56)
|
x
|
=
〈
x
,
x
〉
1
/
2
=
(
|
x
1
|
2
+
⋯
+
|
x
n
|
2
)
1
/
2
.The minimal analytic assumptions necessary for a viable theory appear in the following definition.LetΩ be a domain in R
n and let f
:
Ω
→
R
n be continuous. We say that f has finite distortion if(1)
f belongs to Sobolev space W
1
,
loc
⁡
1
(
Ω
);
(2)
the Jacobian determinant off is locally integrable and does not change sign in Ω;
(3)
there is a measurable functionK
O
=
K
O
(
x
)
≥
1, finite a.e., such that
(57)
|
f
′
(
x
)
|
n
≤
K
O
(
x
)
|
J
f
(
x
)
|
a
.
e
.The assumptions(
1
), (
2
), and (
3
) do not imply that f
∈
W
1
,
loc
⁡
n
(
Ω
), unless of course K
O is a bounded function.IfK
O is a bounded function, then f is qr. In this setting, the smallest K in (57) is called the outer dilatation K
O
(
f
).Iff is qr, also
(58)
J
f
(
x
)
≤
K
′
l
(
f
′
(
x
)
)
n
a
.
e
.
for some K
′, 1
≤
K
′
<
∞, where l
(
f
′
(
x
)
)
=
inf
⁡
{
|
f
′
(
x
)
h
|
:
|
h
|
=
1
}. The smallest K
′ in (58) is called the inner dilatation K
I
(
f
) and K
(
f
)
=
max
⁡
(
K
O
(
f
)
,
K
I
(
f
)
) is called the maximal dilatation of f. If K
(
f
)
≤
K, f is called K-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, Lemma6.7
.
1, [10, page 170]).
Letf be a function of the Sobolev class W
1
,
p
(
B
,
E
n
) in the ball B
=
2
B
=
B
(
x
0
,
2
R
), B
_
=
B
(
x
0
,
R
), 1
≤
p
<
∞, such that
(59)
D
p
(
f
,
B
)
=
(
1
|
B
|
∫
B
‍
|
f
′
|
p
)
1
/
p
≤
M
r
γ
-
1
,
where 0
<
γ
≤
1, holds for every ball B
=
B
(
a
,
r
)
⊂
B. Then f is Hölder continuous in B
_ with exponent γ, and one has |
f
(
x
)
-
f
(
y
)
|
≤
C
|
x
-
y
|
γ, for all x
,
y
∈
B
_, where c
=
4
M
γ
-
1
2
-
γ. Here and in some places we omit to write the volume element d
x.We need a quasiregular version of this Lemma.Lemma 21.
Letf
:
B
→
R
n be a K-quasiregular mapping, such that
(60)
J
f
(
a
;
r
)
≤
M
r
γ
-
1
,
where 0
<
γ
≤
1, holds for every ball B
=
B
(
a
,
r
)
⊂
B. Then f is Hölder continuous in B
_ with exponent γ, and one has |
f
(
x
)
-
f
(
y
)
|
≤
C
|
x
-
y
|
γ, for all x
,
y
∈
B
_, where C
=
4
M
K
1
/
n
γ
-
1
2
-
γ.Proof.
By hypothesis,f satisfies (57) and therefore D
n
(
f
,
B
)
≤
K
1
/
n
J
f
(
a
;
r
). An application of Lemma 20 to p
=
n yields proof.(a.0)
ByB
^
(
G
) denote a family of ball B
=
B
(
x
)
=
B
(
x
,
r
) such that B
=
B
(
x
,
2
r
)
⊂
G. For B
∈
B
^
(
G
), define d
′
=
d
f
,
B
′
=
d
(
f
(
x
)
,
∂
B
′
) and R
(
f
(
x
)
,
B
′
)
:
=
c
*
d
′, where B
′
=
f
(
B
).Defineω
loc
⁡
(
f
,
r
)
=
ω
loc
⁡
,
G
(
f
,
r
)
=
sup
⁡
os
c
B
f and r
0
(
G
)
=
sup
⁡
r
(
B
), where supremum is taken over all balls B
=
B
(
a
)
=
B
(
a
,
r
) such that B
1
=
B
(
a
,
2
r
)
⊂
G and r
(
B
) denotes radius of B.Theorem 22.
Let0
<
α
≤
1, K
≥
1. Suppose that (a.5)  G is a domain in R
n and f
:
G
→
R
n is K-quasiconformal and G
′
=
f
(
G
). Then one has the following.(i.1)
For everyx
∈
G, there exists two points x
1
,
x
2
∈
B
_
x such that |
y
2
|
-
|
y
1
|
≥
R
(
x
), where y
k
=
f
(
x
k
), k
=
1
,
2, and R
(
x
)
:
=
c
*
d
*
(
x
).
(ii.1)
ω
loc
⁡
,
G
(
f
,
r
)
≤
2
c
0
ω
loc
⁡
,
G
(
|
f
|
,
r
), 0
≤
r
≤
r
0
(
G
), where c
0
=
2
C
*
/
c
*.
(iii.1)
If, in addition, one supposes that (b.5)|
f
|
∈
loc
⁡
Λ
α
(
G
), then for all balls B
=
B
(
x
)
=
B
(
x
,
r
) such that B
1
=
B
(
x
,
2
r
)
⊂
G, d
1
⪯
r
α and in particular R
(
x
)
:
=
c
*
d
*
(
x
)
≤
L
d
(
x
)
α, x
∈
G.
(iv.1)
There is a constantc
_ such that
(61)
J
f
(
x
;
r
)
≤
c
_
r
α
-
1
for every x
0
∈
G and B
=
B
(
x
,
r
)
⊂
B
_
x
0.
(v.1)
If, in addition, one supposes that (c.5)G is a Λ
α-extension domain in R
n, then f
∈  Lip
⁡
(
α
,
L
2
;
G
).Proof.
By the distortion property (30), we will prove the following. (vi.1)
For a ballB
=
B
(
x
)
=
B
(
x
,
r
) such that B
1
=
B
(
x
,
2
r
)
⊂
G there exist two points x
1
,
x
2
∈
B
(
x
) such that |
y
2
|
-
|
y
1
|
≥
R
(
x
), where y
k
=
f
(
x
k
), k
=
1
,
2.
Letl be line throughout 0 and f
(
x
) which intersects the S
(
f
(
x
)
,
(
x
)
) at points y
1 and y
2 and x
k
=
f
-
1
(
y
k
), k
=
1
,
2. By the left side of (30), x
1
,
x
2
∈
B
(
x
). We consider two cases:(a)
if0
∉
B
(
f
(
x
)
,
R
(
x
)
) and |
y
2
|
≥
|
y
1
|, then |
y
2
|
-
|
y
1
|
=
2
R
(
x
) and |
y
2
|
-
|
y
1
|
=
2
R
(
x
);
(b)
if0
∈
B
(
f
(
x
)
,
R
(
x
)
), then, for example, 0
∈
[
y
1
,
f
(
x
)
] and if we choose x
1
=
x, we find |
y
2
|
-
|
f
(
x
)
|
=
R
(
x
) and this yields (
vi
.
1
).
Letx
∈
G. An application of (
vi
.
1
) to B
=
B
_
x yields (
i
.
1
). If |
x
-
x
′
|
≤
r, then c
*
d
′
≤
|
y
2
|
-
|
y
1
| and therefore we have the following:(vi.1′)
|
f
(
x
)
-
f
(
x
′
)
|
≤
C
*
d
′
≤
c
0
(
|
y
2
|
-
|
y
1
|
), where c
0
=
C
*
/
c
*.
Ifz
1
,
z
2
∈
B
(
x
,
r
), then |
f
(
z
1
)
-
f
(
z
2
)
|
≤
|
f
(
z
1
)
-
f
(
x
)
|
+
|
f
(
z
2
)
-
f
(
x
)
|. Hence, using (
vi
.
1
′
), we find (
ii
.
1
). Proof of (
iii
.
1
). If the hypothesis (b) holds with multiplicative constant L, and |
x
-
x
′
|
=
r, then c
*
d
′
≤
|
y
2
|
-
|
y
1
|
≤
2
α
L
r
α and therefore |
f
(
x
)
-
f
(
x
′
)
|
≤
C
*
d
′
≤
c
1
r
α, where c
1
=
(
C
*
/
c
*
)
2
α
L. Hence d
1
⪯
r
α and therefore, in particular, R
(
x
)
:
=
c
*
d
*
(
x
)
≤
L
d
(
x
)
α, x
∈
G.
It is clear, by the hypothesis (b), that there is a fixed constantL
1 such that (
vii
.
1
)
|
f
| belongs to Lip
⁡
(
α
,
L
1
;
B
) for every ball B
=
B
(
x
,
r
) such that B
1
=
B
(
x
,
2
r
)
⊂
G and, by (
ii
.
1
), we have the following:(viii.1)
L
(
f
,
B
)
≤
L
2 for a fixed constant L
2 and f
∈
loc
⁡ Lip
⁡
(
α
,
L
2
;
G
).
Hence, since, by the hypothesis (c.5),G is a L
α-extension domain, we get (
v
.
1
).
Proof of(
iv
.
1
). Let B
=
B
(
x
)
=
B
(
x
,
r
) be a ball such that B
=
B
(
x
,
2
r
)
⊂
G. Then E
g
(
B
;
x
)
≈
d
1
n
/
r
n, J
f
(
B
;
x
)
≈
d
1
/
r, and, by (
iii
.
1
), d
1
⪯
r
α on G. Hence
(62)
J
f
(
B
;
x
)
⪯
r
α
-
1
,
on G.
Note that one can also combine(
iii
.
1
) and Lemma 33 (for details see proof of Theorem 40 below) to obtain (
v
.
1
).Note that as an immediate corollary of(
ii
.
1
) 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 Theorems23 and 24 mentioned in the introduction; in particular, these results give further extensions of Theorem-Dy.Theorem 23.
Let0
<
α
≤
1, K
≥
1, and n
≥
2. Suppose G is a Λ
α-extension domain in R
n and f is a K-quasiconformal mapping of G onto f
(
G
)
⊂
R
n. The following are equivalent:(i.2)
f
∈
Λ
α
(
G
),
(ii.2)
|
f
|
∈
Λ
α
(
G
),
(iii.2)
there is a constantc
3 such that
(63)
J
f
(
x
;
r
)
≤
c
3
r
α
-
1
for every x
0
∈
G and B
=
B
(
x
,
r
)
⊂
B
_
x
0. If, in addition, G is a uniform domain and if α
≤
α
0
=
K
1
/
(
1
-
n
), then (
i
.
2
) and (
i
i
.
2
) are equivalent to
(iv.2)
|
f
|
∈
Λ
α
(
G
,
∂
G
).Proof.
Suppose (ii.2) holds, so that|
f
| is L-Lipschitz in G. Then (
iv
.
1
) shows that (
iii
.
2
) holds. By Lemma 21, (
iii
.
2
) implies (
i
.
2
).We outline less direct proof that(
ii
.
2
) implies (
ii
.
1
).One can show first that(
ii
.
2
) implies (18) (or more generally (
iv
.
1
); see Theorem 22). Using A
(
z
)
=
a
+
R
z, we conclude that c
*
(
B
(
a
,
R
)
)
=
c
*
(
B
) and c
^
(
B
)
=
c
^ is a fixed constant (which depends only on K
,
n
,
c
1), for all balls B
⊂
G. Lemma 5 tells us that (18) holds, with a fixed constant, for all balls B
⊂
G and all pairs of points x
1
,
x
2
∈
B. Further, we pick two points x
,
y
∈
G with |
x
-
y
|
<
d
(
x
,
∂
G
) and apply (18) with D
=
B
(
x
;
|
x
-
y
|
), letting x
1
=
x and x
2
=
y. The resulting inequality
(64)
|
f
(
x
)
-
f
(
y
)
|
≤
const
|
x
-
y
|
α
shows that f
∈
loc
⁡
Λ
α
(
G
), and since G is a Λ
α-extension domain, we conclude that f
∈
Λ
α
(
G
).The implication (ii.4)⇒ (i.4) is thus established. The converse is clear. For the proof that (iii.4) implies (i.4) for α
≤
α
0, see [15].For a ballB
=
B
(
x
,
r
)
⊂
R
n and a mapping f
:
B
→
R
n, we define
(65)
E
f
(
B
;
x
)
:
=
1
|
B
|
∫
B
‍
J
f
(
z
)
d
z
,
x
∈
G
,
and J
f
(
B
;
x
)
=
E
f
(
B
;
x
)
n.We use the factorization of planar quasiregular mappings to prove the following.Theorem 24.
Let0
<
α
≤
1, K
≥
1. Suppose D is a Λ
α-extension domain in C and f is a K-quasiregular mapping of D onto f
(
D
)
⊂
C. The following are equivalent:(i.3)
f
∈
Λ
α
(
D
),
(ii.3)
|
f
|
∈
Λ
α
(
D
),
(iii.3)
there is a constantc
ˇ such that
(66)
J
f
(
B
;
z
)
≤
c
ˇ
r
α
-
1
(
z
)
,
for everyz
0
∈
D and B
=
B
(
z
,
r
)
⊂
B
_
z
0.Proof.
LetD and W be domains in C and f
:
D
→
W qr mapping. Then there is a domain G and analytic function ϕ on G such that f
=
ϕ
∘
g, where g is quasiconformal; see [36, page 247].
Our proof will rely on distortion property of quasiconformal mappings. By the triangle inequality,(
i
.
4
) implies (
ii
.
4
). Now, we prove that (
ii
.
4
) implies (
iii
.
4
).
Letz
0
∈
D, ζ
0
=
g
(
z
0
), G
0
=
g
(
B
z
0
), W
0
=
ϕ
(
G
0
), d
=
d
(
z
0
)
=
d
(
z
0
,
∂
D
), and d
1
=
d
1
(
ζ
0
)
=
d
(
ζ
0
,
∂
G
). Let B
=
B
(
z
,
r
)
⊂
B
z
0. As in analytic case there is ζ
1
∈
g
(
B
) such that |
|
ϕ
(
ζ
1
)
|
-
|
ϕ
(
ζ
0
)
|
|
≥
c
d
1
(
ζ
0
)
|
ϕ
′
(
ζ
0
)
|. If z
1
=
g
-
1
(
ζ
1
), then |
|
ϕ
(
ζ
1
)
|
-
|
ϕ
(
ζ
0
)
|
|
=
|
|
f
(
z
1
)
|
-
|
f
(
z
0
)
|
|. Hence, if |
f
| is α-Hölder, then d
1
(
ζ
)
|
ϕ
′
(
ζ
)
|
≤
c
1
r
α for ζ
∈
G
0. Hence, since
(67)
E
f
(
B
;
z
)
=
1
|
B
z
|
∫
B
z
‍
|
ϕ
′
(
ζ
)
|
J
g
(
z
)
d
x
d
y
,
we find
(68)
J
f
,
D
(
z
0
)
≤
c
2
r
α
d
1
(
ζ
0
)
J
g
,
G
(
z
0
)
and therefore, using E
g
(
B
;
z
)
≈
d
1
2
/
r
2, we get
(69)
J
f
(
B
;
z
)
≤
c
3
r
α
-
1
.
Thus we have (iii.3) withc
ˇ
=
c
3.
Proof of the implication(
iii
.
4
)
⇒
(
i
.
4
).
An application of Lemma21 (see also a version of Astala-Gehring lemma) shows that f is α-Hölder on B
_
z
0 with a Lipschitz (multiplicative) constant which depends only on α and it gives the result.


## 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.
SupposeD and D
′ are proper domains in R
2. If h
:
D
→
D
′ is K-qc and harmonic, then it is bi-Lipschitz with respect to quasihyperbolic metrics on D and D
′.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.
Leth
=
f
+
g
¯ be local representation on B
z.
Sinceh is K-qc, then
(22)
(
1
-
k
2
)
|
f
′
|
2
≤
J
h
≤
K
|
f
′
|
2
on B
z and since log
⁡
|
f
′
(
ζ
)
| is harmonic,
(23)
log
⁡
|
f
′
(
z
)
|
=
1
2
|
B
z
|
∫
B
z
‍
log
⁡
|
f
′
(
ζ
)
|
2
d
ξ
d
η
.
Hence,(24)
log
⁡
a
h
,
D
(
z
)
≤
1
2
log
⁡
K
+
1
2
|
B
z
|
∫
B
z
‍
log
⁡
|
f
′
(
ζ
)
|
2
d
ξ
d
η
=
log
⁡
K
|
f
′
(
z
)
|
.
Hence,a
h
,
D
(
z
)
≤
K
|
f
′
(
z
)
| and 1
-
k
|
f
′
(
z
)
|
≤
a
h
,
D
(
z
). Using Astala-Gehring result, we get
(25)
Λ
(
h
,
z
)
≍
d
(
h
z
,
∂
D
′
)
d
(
z
,
∂
D
)
≍
λ
(
h
,
z
)
.
This pointwise result, combined with integration along curves, easily gives(26)
k
D
′
(
h
(
z
1
)
,
h
(
z
2
)
)
≍
k
D
(
z
1
,
z
2
)
,
z
1
,
z
2
∈
D
.
Note that we do not use thatlog
⁡
(
1
/
J
h
) is a subharmonic function.The following follows from the proof of Proposition7:(I)
Λ
(
h
,
z
)
≈
a
h
,
D
≈
λ
(
h
,
z
) and J
f
2
≈
J
_
f; see below for the definition of average jacobian J
_
f.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 byc a constant and by c
_, c
^, c
~ some specific constants.Our next result concerns the quantity(27)
E
_
f
,
G
(
x
)
:
=
1
|
B
_
x
|
∫
B
_
x
‍
J
f
(
z
)
d
z
,
x
∈
G
,
associated with a quasiconformal mapping f
:
G
→
f
(
G
)
⊂
R
n; here J
f is the Jacobian of f, while B
_
x
=
B
_
x
,
G stands for the ball B
(
x
,
d
(
x
,
∂
G
)
/
2
) and |
B
x
| for its volume.Define(28)
J
_
f
=
J
_
f
,
G
=
E
_
f
,
G
n
.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 thatG and G
′ are domains in R
n: If f
:
G
→
G
′ is K-quasiconformal, then
(29)
1
c
d
(
f
(
x
)
,
∂
G
′
)
d
(
x
,
∂
G
)
≤
J
_
f
,
G
(
x
)
≤
c
d
(
f
(
x
)
,
∂
G
′
)
d
(
x
,
∂
G
)
,
x
∈
G
,
where c is a constant which depends only on K and n.Proof.
By the distortion property of qc (see [23, page 383], [24, page 63]), there are the constants C
* and c
* which depend on n and K only, such that
(30)
B
(
f
(
x
)
,
c
*
d
*
)
⊂
f
(
B
x
)
⊂
B
(
f
(
x
)
,
C
*
d
*
)
,
x
∈
G
,
where d
*
(
x
)
:
=
d
(
f
(
x
)
)
=
d
(
f
(
x
)
,
∂
G
′
) and d
(
x
)
:
=
d
(
x
,
∂
G
). Hence
(31)
|
B
(
f
(
x
)
,
c
*
d
*
)
|
≤
∫
B
x
‍
J
f
(
z
)
d
z
≤
|
B
(
f
(
x
)
,
C
*
d
*
)
|
and therefore we prove Theorem 8.We only outline proofs in the rest of this subsection.Suppose thatΩ and Ω
′ are domains in R
n different from R
n.Theorem 9.
Supposef
:
Ω
→
Ω
′ is a C
1 and the following hold.(i1)
f is c-Lipschitz with respect to quasihyperbolic metrics on Ω and Ω
′; then
(I1)
d
|
f
′
(
x
0
)
|
≤
c
d
*
(
x
0
) for every x
0
∈
Ω.
(i2)
Iff is a qc c-quasihyperbolic-isometry, then
(I2)
f is a c
2-qc mapping,
(
I
3
)
|
f
′
|
≈
d
*
/
d.Proof.
SinceΩ and Ω
′ are different from R
n, there are quasihyperbolic metrics on Ω and Ω
′. Then for a fixed x
0
∈
Ω, we have
(32)
k
(
x
,
x
0
)
=
d
(
x
0
)
-
1
|
x
-
x
0
|
(
1
+
o
(
1
)
)
,
x
⟶
x
0
,
k
(
f
(
x
)
,
f
(
x
0
)
)
=
|
f
(
x
)
-
f
(
x
0
)
|
d
*
(
f
(
x
0
)
)
(
1
+
o
(
1
)
)
,
yyyyyyyyyyyyyyyyyyyyyyyy
when
x
⟶
x
0
.
Hence, (i1) implies (I1). Iff is a c-quasihyperbolic-isometry, then d
|
f
′
(
x
0
)
|
≤
c
d
* and d
l
(
f
′
(
x
0
)
)
≥
d
*
/
c. Hence, (I2) and (I3) follow.Theorem 10.
Supposef
:
Ω
→
Ω
′ is a C
1 qc homeomorphism. The following conditions are equivalent:(a.1)
f is bi-Lipschitz with respect to quasihyperbolic metrics on Ω and Ω
′,
(b.1)
J
f
n
≈
d
*
/
d,
(c.1)
J
f
n
≈
a
f,
(d.1)
J
f
n
≈
J
_
f,where d
(
x
)
=
d
(
x
,
∂
Ω
) and d
*
(
x
)
=
d
(
f
(
x
)
,
∂
Ω
′
).Proof.
It follows from Theorem9 that (a) is equivalent to (b) (see, e.g., [32, 33]).
Theorem8 states that J
_
f
≈
d
*
/
d. By Lemma 4, a
f
≈
d
*
/
d and therefore (b) is equivalent to (c). The rest of the proof is straightforward.Lemma 11.
Iff
∈
C
1,1 is a K-quasiconformal mapping defined in a domain Ω
⊂
R
n
(
n
≥
3
), then
(33)
J
f
(
x
)
>
0
,
x
∈
Ω
provided that K
<
2
n
-
1. The constant 2
n
-
1 is sharp.IfG
¯
⊂
Ω, then it is bi-Lipschitz with respect to Euclidean and quasihyperbolic metrics on G and G
′
=
f
(
G
).It is a natural question whether is there an analogy of Theorem10 if we drop the C
1 hypothesis.Supposef
:
Ω
→
Ω
′ is onto qc mapping. We can consider the following conditions:(a.2)
f is bi-Lipschitz with respect to quasihyperbolic metrics on Ω and Ω
′;
(b.2)
J
f
n
≈
d
*
/
d a.e. in Ω;
(c.2)
J
f
n
≈
a
f a.e. in Ω;
(d.2)
J
f
n
≈
J
_
f a.e. in Ω.It seems that the above conditions are equivalent, but we did not check details.IfΩ is a planar domain and f harmonic qc, then we proved that (
d
.
2
) holds (see Proposition 18).


## 2.2. Quasi-Isometry in Planar Case

For a functionh, we use notations ∂
h
=
(
1
/
2
)
(
h
x
′
-
i
h
y
′
) and ∂
¯
h
=
(
1
/
2
)
(
h
x
′
+
i
h
y
′
); we also use notations D
h and D
¯
h instead of ∂
h and ∂
¯
h, respectively, when it seems convenient. Now we give another proof of Proposition 7, using the following.Proposition 12 (see [29]).
Leth be an euclidean harmonic orientation preserving univalent mapping of the unit disc D into C such that h
(
D
) contains a disc B
R
=
B
(
a
;
R
) and h
(
0
)
=
a. Then
(34)
|
∂
h
(
0
)
|
≥
R
4
.
If, in addition,h is K-qc, then
(35)
λ
h
(
0
)
≥
1
-
k
4
R
.For more details, in connection with material considered in this subsection, see also AppendixA, in particular, Proposition A.7.Leth
=
f
+
g
¯ be a harmonic univalent orientation preserving mapping on the unit disk D, Ω
=
h
(
D
), d
h
(
z
)
=
d
(
h
(
z
)
,
∂
Ω
), λ
h
(
z
)
=
D
-
(
z
)
=
|
f
′
(
z
)
|
-
|
g
′
(
z
)
|, and Λ
h
(
z
)
=
D
+
(
z
)
=
|
f
′
(
z
)
|
+
|
g
′
(
z
)
|. By the harmonic analogue of the Koebe Theorem, then
(36)
1
16
D
-
(
0
)
≤
d
h
(
0
)
and therefore
(37)
(
1
-
|
z
|
2
)
1
16
D
-
(
z
)
≤
d
h
(
z
)
.
If, in addition, h is K-qc, then
(38)
(
1
-
|
z
|
2
)
D
+
(
z
)
≤
16
K
d
h
(
z
)
.
Using Proposition 12, we also prove
(39)
(
1
-
|
z
|
2
)
λ
h
(
z
)
≥
1
-
k
4
d
h
(
z
)
.If we summarize the above considerations, we have proved the part(
a
.
3
) of the following proposition.Proposition 13 (e-qch, hyperbolic distance version).
(a.3). Leth
=
f
+
g
¯ be a harmonic univalent orientation preserving K-qc mapping on the unit disk D, Ω
=
h
(
D
). Then, for z
∈
D,
(40)
(
1
-
|
z
|
2
)
Λ
h
(
z
)
≤
16
K
d
h
(
z
)
,
(
1
-
|
z
|
2
)
λ
h
(
z
)
≥
1
-
k
4
d
h
(
z
)
.
(
b
.
3
) If h is a harmonic univalent orientation preserving K-qc mapping of domain D onto D
′, then
(41)
d
(
z
)
Λ
h
(
z
)
≤
16
K
d
h
(
z
)
,
d
(
z
)
λ
h
(
z
)
≥
1
-
k
4
d
h
(
z
)
,
and
(42)
1
-
k
4
κ
D
(
z
,
z
′
)
≤
κ
D
′
(
h
z
,
h
z
′
)
≤
16
K
κ
D
(
z
,
z
′
)
,
where z
,
z
′
∈
D.Remark 14.
In particular, we have Proposition7, 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 B
z, z
∈
D, we get (41). It is clear that (41) implies (42).For a planar hyperbolic domain inC, we denote by ρ
=
ρ
D and d
hyp
=
d
hyp
;
D the hyperbolic density and metric of D, respectively.We say that a domainD
⊂
C isstrongly 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 diskD
′
=
{
0
<
|
z
|
<
1
} is obtained by mapping its universal covering, an infinitely-sheeted disk, on the half plane Π
-
=
{
Re
w
<
0
} by means of w
=
ln
⁡
z (i.e., z
=
e
w). The metric is
(43)
|
d
w
|
|
Re
w
|
=
|
d
z
|
|
z
|
ln
⁡
(
1
/
|
z
|
)
.
Since a boundary component is the point0, the punctured disk is not a strongly hyperbolic domain. Note also that ρ
/
d and 1
/
ln
⁡
(
1
/
d
) tend to 0 if z tends to 0. Here d
=
d
(
z
)
=
dist
⁡
(
z
,
∂
D
′
) and ρ is the hyperbolic density of D
′. Therefore one has the following:(
A
.
1
)
forz
∈
D
′, κ
D
′
(
z
,
z
′
)
=
ln
⁡
(
1
/
d
(
z
)
)
d
hyp
,
D
′
(
z
,
z
′
) when z
′
→
z.
There is no constantc such that (
A
.
2
)
κ
D
′
(
z
,
z
′
)
≤
c
κ
Π
-
(
w
,
w
′
) for every w
,
w
′
∈
Π
-, where z
=
e
w and z
′
=
e
w
′.
Sinced
hyp
,
Π
-
≈
κ
Π
-, if (
A
.
2
) holds, we conclude that κ
D
′
≈
d
hyp
,
D
′, which is a contradiction by (
A
.
1
).LetD be a planar hyperbolic domain. Then if D is simply connected,
(44)
1
2
d
≤
ρ
≤
2
d
.For general domain asz
→
∂
D,
(45)
1
+
o
(
1
)
d
ln
⁡
(
1
/
d
)
≤
ρ
≤
2
d
.Henced
hyp
(
z
,
z
′
)
≤
2
κ
D
(
z
,
z
′
), z
,
z
′
∈
D.IfΩ is a strongly hyperbolic domain, then there is a hyperbolic density ρ on the domain Ω, such that ρ
≈
d
-
1, where d
(
w
)
=
d
(
w
,
∂
Ω
); see, for example, [34]. Thus d
hyp
;
D
≈
κ
D. Hence, we find the following.Corollary 16.
Everye-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.
LetD be a hyperbolic domain, let h  be e-harmonic quasiconformal mapping of D onto Ω, and let ϕ
:
D
→
D be a covering and h
*
=
h
∘
ϕ. (a.4)
Suppose thatD is simply connected. Thus ϕ and h
* are one-to-one. Then
(46)
(
1
-
|
z
|
2
)
λ
h
*
(
ζ
)
≥
1
-
k
4
d
h
(
z
)
,where z
=
ϕ
(
ζ
).
Hence, sinceλ
h
*
(
ζ
)
=
l
h
(
z
)
|
ϕ
′
(
ζ
)
| and ρ
hyp
;
D
(
z
)
|
ϕ
′
(
ζ
)
|
=
ρ
D
(
ζ
),
(47)
λ
h
(
z
)
≥
1
-
k
4
d
h
(
z
)
ρ
D
(
z
)
.
Using Hall’s sharp result, one can also improve the constant in the second inequality in Propositions13 and 12 (i.e., the constant 1
/
4 can be replaced by τ
0; see below for more details):
(48)
(
1
-
|
z
|
2
)
λ
h
(
z
)
≥
(
1
-
k
)
τ
0
d
h
(
z
)
,
where τ
0
=
3
3
/
2
π.(b.4)
Suppose thatD is not a simply connected. Then h
* is not one-to-one and we cannot apply the procedure as in (
a
.
4
).
(c.4)
It seems natural to consider whether there is an analogue in higher dimensions of Proposition13.Proposition 18.
For everye-harmonic quasiconformal mapping f of the unit disc (more generally of a hyperbolic domain D) the following holds: (e.2)
J
f
≈
d
*
/
d. In particular, it is a quasi-isometry with respect to quasihyperbolic distances.Proof.
Forz
∈
D, by the distortion property,
(49)
|
f
(
z
1
)
-
f
(
z
)
|
≤
c
d
(
f
(
z
)
)
for
every
z
1
∈
B
_
z
.
Hence, by Schwarz lemma for harmonic maps, d
|
f
′
(
z
)
|
≤
c
1
d
(
f
(
z
)
). Proposition 12 yields (e) and an application of Proposition 13 gives the proof.RecallB
_
x stands for the ball B
(
x
;
d
(
x
,
∂
G
)
/
2
) and |
B
_
x
| for its volume. If V is a subset of R
n and u
:
V
→
R
m, we define
(50)
os
c
V
u
:
=
sup
⁡
{
|
u
(
x
)
-
u
(
y
)
|
:
x
,
y
∈
V
}
,
ω
u
(
r
,
x
)
:
=
sup
⁡
{
|
u
(
x
)
-
u
(
y
)
|
:
|
x
-
y
|
=
r
}
.Suppose thatG
⊂
R
n and B
_
x
=
B
(
x
,
d
(
x
)
/
2
). Let O
C
1
(
G
)
=
O
C
1
(
G
;
c
1
) denote the class of f
∈
C
1
(
G
) such that
(51)
d
(
x
)
|
f
′
(
x
)
|
≤
c
1
os
c
B
_
x
f
for every x
∈
G.Proposition 19.
Supposef
:
Ω
→
Ω
′ is a C
1. Then f
∈
O
C
1
(
Ω
;
c
), if and only if f is c-Lipschitz with respect to quasihyperbolic metrics on B and f
(
B
) for every ball B
⊂
Ω.


## 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:SupposeG is a Λ
α-extension domain in R
n and f is a K-quasiconformal mapping of G onto f
(
G
)
⊂
R
n. Then f
∈
Λ
α
(
G
) if and only if |
f
|
∈
Λ
α
(
G
).This is TheoremB.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
(52)
E
_
f
,
G
(
x
)
:
=
1
|
B
_
x
|
∫
B
_
x
‍
J
f
(
z
)
d
z
,
x
∈
G
,
associated with a quasiconformal mapping f
:
G
→
f
(
G
)
⊂
R
n; here J
f is the Jacobian of f, while B
_
x
=
B
_
x
,
G stands for the ball B
(
x
;
d
(
x
,
∂
G
)
/
2
) and |
B
_
x
| for its volume.Defineα
0
=
α
K
n
=
K
1
/
(
1
-
n
), C
*
=
θ
K
n
(
1
/
2
), θ
K
n
(
c
*
)
=
1
/
2, and
(53)
J
_
f
,
G
=
E
_
f
,
G
n
.For a ballB
=
B
(
x
,
r
)
⊂
R
n and a mapping f
:
B
→
R
n, we define
(54)
E
f
(
B
;
x
)
:
=
1
|
B
|
∫
B
‍
J
f
(
z
)
d
z
,
x
∈
G
,
and J
f
a
v
(
B
;
x
)
=
E
f
(
B
;
x
)
n; we also use the notation J
f
(
B
;
x
) and J
f
(
x
;
r
) instead of J
f
a
v
(
B
;
x
).Forx
,
y
∈
R
n, we define the Euclidean inner product 〈
·
,
·
〉 by
(55)
〈
x
,
y
〉
=
x
1
y
1
+
⋯
+
x
n
y
n
.
By E
n we denote R
n with the Euclidean inner product and call it Euclidean space n-space (space of dimension n). In this paper, for simplicity, we will use also notation R
n for E
n. Then the Euclidean length of x
∈
R
n is defined by
(56)
|
x
|
=
〈
x
,
x
〉
1
/
2
=
(
|
x
1
|
2
+
⋯
+
|
x
n
|
2
)
1
/
2
.The minimal analytic assumptions necessary for a viable theory appear in the following definition.LetΩ be a domain in R
n and let f
:
Ω
→
R
n be continuous. We say that f has finite distortion if(1)
f belongs to Sobolev space W
1
,
loc
⁡
1
(
Ω
);
(2)
the Jacobian determinant off is locally integrable and does not change sign in Ω;
(3)
there is a measurable functionK
O
=
K
O
(
x
)
≥
1, finite a.e., such that
(57)
|
f
′
(
x
)
|
n
≤
K
O
(
x
)
|
J
f
(
x
)
|
a
.
e
.The assumptions(
1
), (
2
), and (
3
) do not imply that f
∈
W
1
,
loc
⁡
n
(
Ω
), unless of course K
O is a bounded function.IfK
O is a bounded function, then f is qr. In this setting, the smallest K in (57) is called the outer dilatation K
O
(
f
).Iff is qr, also
(58)
J
f
(
x
)
≤
K
′
l
(
f
′
(
x
)
)
n
a
.
e
.
for some K
′, 1
≤
K
′
<
∞, where l
(
f
′
(
x
)
)
=
inf
⁡
{
|
f
′
(
x
)
h
|
:
|
h
|
=
1
}. The smallest K
′ in (58) is called the inner dilatation K
I
(
f
) and K
(
f
)
=
max
⁡
(
K
O
(
f
)
,
K
I
(
f
)
) is called the maximal dilatation of f. If K
(
f
)
≤
K, f is called K-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, Lemma6.7
.
1, [10, page 170]).
Letf be a function of the Sobolev class W
1
,
p
(
B
,
E
n
) in the ball B
=
2
B
=
B
(
x
0
,
2
R
), B
_
=
B
(
x
0
,
R
), 1
≤
p
<
∞, such that
(59)
D
p
(
f
,
B
)
=
(
1
|
B
|
∫
B
‍
|
f
′
|
p
)
1
/
p
≤
M
r
γ
-
1
,
where 0
<
γ
≤
1, holds for every ball B
=
B
(
a
,
r
)
⊂
B. Then f is Hölder continuous in B
_ with exponent γ, and one has |
f
(
x
)
-
f
(
y
)
|
≤
C
|
x
-
y
|
γ, for all x
,
y
∈
B
_, where c
=
4
M
γ
-
1
2
-
γ. Here and in some places we omit to write the volume element d
x.We need a quasiregular version of this Lemma.Lemma 21.
Letf
:
B
→
R
n be a K-quasiregular mapping, such that
(60)
J
f
(
a
;
r
)
≤
M
r
γ
-
1
,
where 0
<
γ
≤
1, holds for every ball B
=
B
(
a
,
r
)
⊂
B. Then f is Hölder continuous in B
_ with exponent γ, and one has |
f
(
x
)
-
f
(
y
)
|
≤
C
|
x
-
y
|
γ, for all x
,
y
∈
B
_, where C
=
4
M
K
1
/
n
γ
-
1
2
-
γ.Proof.
By hypothesis,f satisfies (57) and therefore D
n
(
f
,
B
)
≤
K
1
/
n
J
f
(
a
;
r
). An application of Lemma 20 to p
=
n yields proof.(a.0)
ByB
^
(
G
) denote a family of ball B
=
B
(
x
)
=
B
(
x
,
r
) such that B
=
B
(
x
,
2
r
)
⊂
G. For B
∈
B
^
(
G
), define d
′
=
d
f
,
B
′
=
d
(
f
(
x
)
,
∂
B
′
) and R
(
f
(
x
)
,
B
′
)
:
=
c
*
d
′, where B
′
=
f
(
B
).Defineω
loc
⁡
(
f
,
r
)
=
ω
loc
⁡
,
G
(
f
,
r
)
=
sup
⁡
os
c
B
f and r
0
(
G
)
=
sup
⁡
r
(
B
), where supremum is taken over all balls B
=
B
(
a
)
=
B
(
a
,
r
) such that B
1
=
B
(
a
,
2
r
)
⊂
G and r
(
B
) denotes radius of B.Theorem 22.
Let0
<
α
≤
1, K
≥
1. Suppose that (a.5)  G is a domain in R
n and f
:
G
→
R
n is K-quasiconformal and G
′
=
f
(
G
). Then one has the following.(i.1)
For everyx
∈
G, there exists two points x
1
,
x
2
∈
B
_
x such that |
y
2
|
-
|
y
1
|
≥
R
(
x
), where y
k
=
f
(
x
k
), k
=
1
,
2, and R
(
x
)
:
=
c
*
d
*
(
x
).
(ii.1)
ω
loc
⁡
,
G
(
f
,
r
)
≤
2
c
0
ω
loc
⁡
,
G
(
|
f
|
,
r
), 0
≤
r
≤
r
0
(
G
), where c
0
=
2
C
*
/
c
*.
(iii.1)
If, in addition, one supposes that (b.5)|
f
|
∈
loc
⁡
Λ
α
(
G
), then for all balls B
=
B
(
x
)
=
B
(
x
,
r
) such that B
1
=
B
(
x
,
2
r
)
⊂
G, d
1
⪯
r
α and in particular R
(
x
)
:
=
c
*
d
*
(
x
)
≤
L
d
(
x
)
α, x
∈
G.
(iv.1)
There is a constantc
_ such that
(61)
J
f
(
x
;
r
)
≤
c
_
r
α
-
1
for every x
0
∈
G and B
=
B
(
x
,
r
)
⊂
B
_
x
0.
(v.1)
If, in addition, one supposes that (c.5)G is a Λ
α-extension domain in R
n, then f
∈  Lip
⁡
(
α
,
L
2
;
G
).Proof.
By the distortion property (30), we will prove the following. (vi.1)
For a ballB
=
B
(
x
)
=
B
(
x
,
r
) such that B
1
=
B
(
x
,
2
r
)
⊂
G there exist two points x
1
,
x
2
∈
B
(
x
) such that |
y
2
|
-
|
y
1
|
≥
R
(
x
), where y
k
=
f
(
x
k
), k
=
1
,
2.
Letl be line throughout 0 and f
(
x
) which intersects the S
(
f
(
x
)
,
(
x
)
) at points y
1 and y
2 and x
k
=
f
-
1
(
y
k
), k
=
1
,
2. By the left side of (30), x
1
,
x
2
∈
B
(
x
). We consider two cases:(a)
if0
∉
B
(
f
(
x
)
,
R
(
x
)
) and |
y
2
|
≥
|
y
1
|, then |
y
2
|
-
|
y
1
|
=
2
R
(
x
) and |
y
2
|
-
|
y
1
|
=
2
R
(
x
);
(b)
if0
∈
B
(
f
(
x
)
,
R
(
x
)
), then, for example, 0
∈
[
y
1
,
f
(
x
)
] and if we choose x
1
=
x, we find |
y
2
|
-
|
f
(
x
)
|
=
R
(
x
) and this yields (
vi
.
1
).
Letx
∈
G. An application of (
vi
.
1
) to B
=
B
_
x yields (
i
.
1
). If |
x
-
x
′
|
≤
r, then c
*
d
′
≤
|
y
2
|
-
|
y
1
| and therefore we have the following:(vi.1′)
|
f
(
x
)
-
f
(
x
′
)
|
≤
C
*
d
′
≤
c
0
(
|
y
2
|
-
|
y
1
|
), where c
0
=
C
*
/
c
*.
Ifz
1
,
z
2
∈
B
(
x
,
r
), then |
f
(
z
1
)
-
f
(
z
2
)
|
≤
|
f
(
z
1
)
-
f
(
x
)
|
+
|
f
(
z
2
)
-
f
(
x
)
|. Hence, using (
vi
.
1
′
), we find (
ii
.
1
). Proof of (
iii
.
1
). If the hypothesis (b) holds with multiplicative constant L, and |
x
-
x
′
|
=
r, then c
*
d
′
≤
|
y
2
|
-
|
y
1
|
≤
2
α
L
r
α and therefore |
f
(
x
)
-
f
(
x
′
)
|
≤
C
*
d
′
≤
c
1
r
α, where c
1
=
(
C
*
/
c
*
)
2
α
L. Hence d
1
⪯
r
α and therefore, in particular, R
(
x
)
:
=
c
*
d
*
(
x
)
≤
L
d
(
x
)
α, x
∈
G.
It is clear, by the hypothesis (b), that there is a fixed constantL
1 such that (
vii
.
1
)
|
f
| belongs to Lip
⁡
(
α
,
L
1
;
B
) for every ball B
=
B
(
x
,
r
) such that B
1
=
B
(
x
,
2
r
)
⊂
G and, by (
ii
.
1
), we have the following:(viii.1)
L
(
f
,
B
)
≤
L
2 for a fixed constant L
2 and f
∈
loc
⁡ Lip
⁡
(
α
,
L
2
;
G
).
Hence, since, by the hypothesis (c.5),G is a L
α-extension domain, we get (
v
.
1
).
Proof of(
iv
.
1
). Let B
=
B
(
x
)
=
B
(
x
,
r
) be a ball such that B
=
B
(
x
,
2
r
)
⊂
G. Then E
g
(
B
;
x
)
≈
d
1
n
/
r
n, J
f
(
B
;
x
)
≈
d
1
/
r, and, by (
iii
.
1
), d
1
⪯
r
α on G. Hence
(62)
J
f
(
B
;
x
)
⪯
r
α
-
1
,
on G.
Note that one can also combine(
iii
.
1
) and Lemma 33 (for details see proof of Theorem 40 below) to obtain (
v
.
1
).Note that as an immediate corollary of(
ii
.
1
) 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 Theorems23 and 24 mentioned in the introduction; in particular, these results give further extensions of Theorem-Dy.Theorem 23.
Let0
<
α
≤
1, K
≥
1, and n
≥
2. Suppose G is a Λ
α-extension domain in R
n and f is a K-quasiconformal mapping of G onto f
(
G
)
⊂
R
n. The following are equivalent:(i.2)
f
∈
Λ
α
(
G
),
(ii.2)
|
f
|
∈
Λ
α
(
G
),
(iii.2)
there is a constantc
3 such that
(63)
J
f
(
x
;
r
)
≤
c
3
r
α
-
1
for every x
0
∈
G and B
=
B
(
x
,
r
)
⊂
B
_
x
0. If, in addition, G is a uniform domain and if α
≤
α
0
=
K
1
/
(
1
-
n
), then (
i
.
2
) and (
i
i
.
2
) are equivalent to
(iv.2)
|
f
|
∈
Λ
α
(
G
,
∂
G
).Proof.
Suppose (ii.2) holds, so that|
f
| is L-Lipschitz in G. Then (
iv
.
1
) shows that (
iii
.
2
) holds. By Lemma 21, (
iii
.
2
) implies (
i
.
2
).We outline less direct proof that(
ii
.
2
) implies (
ii
.
1
).One can show first that(
ii
.
2
) implies (18) (or more generally (
iv
.
1
); see Theorem 22). Using A
(
z
)
=
a
+
R
z, we conclude that c
*
(
B
(
a
,
R
)
)
=
c
*
(
B
) and c
^
(
B
)
=
c
^ is a fixed constant (which depends only on K
,
n
,
c
1), for all balls B
⊂
G. Lemma 5 tells us that (18) holds, with a fixed constant, for all balls B
⊂
G and all pairs of points x
1
,
x
2
∈
B. Further, we pick two points x
,
y
∈
G with |
x
-
y
|
<
d
(
x
,
∂
G
) and apply (18) with D
=
B
(
x
;
|
x
-
y
|
), letting x
1
=
x and x
2
=
y. The resulting inequality
(64)
|
f
(
x
)
-
f
(
y
)
|
≤
const
|
x
-
y
|
α
shows that f
∈
loc
⁡
Λ
α
(
G
), and since G is a Λ
α-extension domain, we conclude that f
∈
Λ
α
(
G
).The implication (ii.4)⇒ (i.4) is thus established. The converse is clear. For the proof that (iii.4) implies (i.4) for α
≤
α
0, see [15].For a ballB
=
B
(
x
,
r
)
⊂
R
n and a mapping f
:
B
→
R
n, we define
(65)
E
f
(
B
;
x
)
:
=
1
|
B
|
∫
B
‍
J
f
(
z
)
d
z
,
x
∈
G
,
and J
f
(
B
;
x
)
=
E
f
(
B
;
x
)
n.We use the factorization of planar quasiregular mappings to prove the following.Theorem 24.
Let0
<
α
≤
1, K
≥
1. Suppose D is a Λ
α-extension domain in C and f is a K-quasiregular mapping of D onto f
(
D
)
⊂
C. The following are equivalent:(i.3)
f
∈
Λ
α
(
D
),
(ii.3)
|
f
|
∈
Λ
α
(
D
),
(iii.3)
there is a constantc
ˇ such that
(66)
J
f
(
B
;
z
)
≤
c
ˇ
r
α
-
1
(
z
)
,
for everyz
0
∈
D and B
=
B
(
z
,
r
)
⊂
B
_
z
0.Proof.
LetD and W be domains in C and f
:
D
→
W qr mapping. Then there is a domain G and analytic function ϕ on G such that f
=
ϕ
∘
g, where g is quasiconformal; see [36, page 247].
Our proof will rely on distortion property of quasiconformal mappings. By the triangle inequality,(
i
.
4
) implies (
ii
.
4
). Now, we prove that (
ii
.
4
) implies (
iii
.
4
).
Letz
0
∈
D, ζ
0
=
g
(
z
0
), G
0
=
g
(
B
z
0
), W
0
=
ϕ
(
G
0
), d
=
d
(
z
0
)
=
d
(
z
0
,
∂
D
), and d
1
=
d
1
(
ζ
0
)
=
d
(
ζ
0
,
∂
G
). Let B
=
B
(
z
,
r
)
⊂
B
z
0. As in analytic case there is ζ
1
∈
g
(
B
) such that |
|
ϕ
(
ζ
1
)
|
-
|
ϕ
(
ζ
0
)
|
|
≥
c
d
1
(
ζ
0
)
|
ϕ
′
(
ζ
0
)
|. If z
1
=
g
-
1
(
ζ
1
), then |
|
ϕ
(
ζ
1
)
|
-
|
ϕ
(
ζ
0
)
|
|
=
|
|
f
(
z
1
)
|
-
|
f
(
z
0
)
|
|. Hence, if |
f
| is α-Hölder, then d
1
(
ζ
)
|
ϕ
′
(
ζ
)
|
≤
c
1
r
α for ζ
∈
G
0. Hence, since
(67)
E
f
(
B
;
z
)
=
1
|
B
z
|
∫
B
z
‍
|
ϕ
′
(
ζ
)
|
J
g
(
z
)
d
x
d
y
,
we find
(68)
J
f
,
D
(
z
0
)
≤
c
2
r
α
d
1
(
ζ
0
)
J
g
,
G
(
z
0
)
and therefore, using E
g
(
B
;
z
)
≈
d
1
2
/
r
2, we get
(69)
J
f
(
B
;
z
)
≤
c
3
r
α
-
1
.
Thus we have (iii.3) withc
ˇ
=
c
3.
Proof of the implication(
iii
.
4
)
⇒
(
i
.
4
).
An application of Lemma21 (see also a version of Astala-Gehring lemma) shows that f is α-Hölder on B
_
z
0 with a Lipschitz (multiplicative) constant which depends only on α 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 thath
:
B
¯
n
(
a
,
r
)
→
R
m is a continuous vector-valued function, harmonic on B
n
(
a
,
r
), and let
(70)
M
a
*
=
sup
⁡
{
|
h
(
y
)
-
h
(
a
)
|
:
y
∈
S
n
-
1
(
a
,
r
)
}
.Leth
=
(
h
1
,
h
2
,
…
,
h
m
). A modification of the estimate in [37, Equation (
2.31
)] gives
(71)
r
|
∇
h
k
(
a
)
|
≤
n
M
a
*
,
k
=
1
,
…
,
m
.We next extend this result to the case of vector-valued functions. See also [38] and [39, Theorem 6.16].Lemma 25.
Suppose thath
:
B
¯
n
(
a
,
r
)
→
R
m is a continuous mapping, harmonic in B
n
(
a
,
r
). Then
(72)
r
|
h
′
(
a
)
|
≤
n
M
a
*
.Proof.
Without loss of generality, we may suppose thata
=
0 and h
(
0
)
=
0. Let
(73)
K
(
x
,
y
)
=
K
y
(
x
)
=
r
2
-
|
x
|
2
n
ω
n
r
|
x
-
y
|
n
,
where ω
n is the volume of the unit ball in R
n. Hence, as in [8], for 1
≤
j
≤
n, we have
(74)
∂
∂
x
j
K
(
0
,
ξ
)
=
ξ
j
ω
n
r
n
+
1
.
Letη
∈
S
n
-
1 be a unit vector and |
ξ
|
=
r. For given ξ, it is convenient to write K
ξ
(
x
)
=
K
(
x
,
ξ
) and consider K
ξ as function of x.
Then(75)
K
ξ
′
(
0
)
η
=
1
ω
n
r
n
+
1
(
ξ
,
η
)
.
Since|
(
ξ
,
η
)
|
≤
|
ξ
|
|
η
|
=
r, we see that
(76)
|
K
ξ
′
(
0
)
η
|
≤
1
ω
n
r
n
,
and
therefore
|
∇
K
ξ
(
0
)
|
≤
1
ω
n
r
n
.
This last inequality yields(77)
|
h
′
(
0
)
(
η
)
|
≤
∫
S
n
-
1
(
a
,
r
)
‍
|
∇
K
y
(
0
)
|
|
h
(
y
)
|
d
σ
(
y
)
≤
M
0
*
n
ω
n
r
n
-
1
ω
n
r
n
=
M
0
*
n
r
,
where d
σ
(
y
) is the surface element on the sphere, and the proof is complete.LetC
n
=
{
z
=
(
z
1
,
…
,
z
n
)
:
z
1
,
…
,
z
n
∈
C
} denote the complex vector space of dimension n. For a
=
(
a
1
,
…
,
a
n
)
∈
C
n, we define the Euclidean inner product 〈
·
,
·
〉 by
(78)
〈
z
,
a
〉
=
z
1
a
¯
1
+
⋯
+
z
n
a
¯
n
,
where a
¯
k
(
k
∈
{
1
,
…
,
n
}
) denotes the complex conjugate of a
k. Then the Euclidean length of z is defined by
(79)
|
z
|
=
〈
z
,
z
〉
1
/
2
=
(
|
z
1
|
2
+
⋯
+
|
z
n
|
2
)
1
/
2
.
Denote a ball in C
n with center z
′ and radius r
>
0 by
(80)
B
n
(
z
′
,
r
)
=
{
z
∈
C
n
:
|
z
-
z
′
|
<
r
}
.
In particular, B
n denotes the unit ball B
n
(
0
,
1
) and S
n
-
1 the sphere {
z
∈
C
n
:
|
z
|
=
1
}. Set D
=
B
1, the open unit disk in C, and let T
=
S
0 be the unit circle in C.A continuous complex-valued functionf defined in a domain Ω
⊂
C
n is said to bepluriharmonic if, for fixed z
∈
Ω and θ
∈
S
n
-
1, the function f
(
z
+
θ
ζ
) is harmonic in {
ζ
∈
C
:
|
θ
ζ
-
z
|
<
d
Ω
(
z
)
}, where d
Ω
(
z
) denotes the distance from z to the boundary ∂
Ω of Ω. It is easy to verify that the real part of any holomorphic function is pluriharmonic; cf. [40].Letω
:
[
0
,
+
∞
)
→
[
0
,
+
∞
) with ω
(
0
)
=
0 be a continuous function. We say that ω is amajorant if(1)
ω
(
t
) is increasing,
(2)
ω
(
t
)
/
t is nonincreasing for t
>
0.If, in addition, there is a constantC
>
0 depending only on ω such that
(81)
∫
0
δ
‍
ω
(
t
)
t
d
t
≤
C
ω
(
δ
)
,
0
<
δ
<
δ
0
,
(82)
δ
∫
δ
∞
‍
ω
(
t
)
t
2
d
t
≤
C
ω
(
δ
)
,
0
<
δ
<
δ
0
for some δ
0, then we say that ω is aregular majorant. A majorant is calledfast (resp.,slow) if condition (81) (resp., (82)) is fulfilled.Given a majorantω, we define Λ
ω
(
Ω
) (resp., Λ
ω
(
∂
Ω
)) to be theLipschitz-type space consisting of all complex-valued functions f for which there exists a constant C such that, for all z and w
∈
Ω (resp., z and w
∈
∂
Ω),
(83)
|
f
(
z
)
-
f
(
w
)
|
≤
C
ω
(
|
z
-
w
|
)
.Using Lemma25, one can prove the following.Proposition 26.
Letω be aregular majorant and let f be harmonic mapping in a simply connected Λ
ω-extension domain D
⊂
R
n. Then h
∈
Λ
ω
(
D
) if and only if |
∇
f
|
≤
C
(
ω
(
1
-
|
z
|
)
/
(
1
-
|
z
|
)
).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) Letu be pluriharmonic in B
0
=
B
(
a
;
r
). Then there is an analytic function f in B such that f
=
u
+
i
v.
(ii) LetΩ be simply connected and u be pluriharmonic in Ω. Then there is analytic function f in Ω such that f
=
u
+
i
v.Proof.
(i) LetB
0
=
B
(
a
;
r
)
⊂
Ω, P
k
=
-
u
y
k, and Q
=
u
x
k; define form
(84)
ω
k
=
P
k
d
x
k
+
Q
k
d
y
k
=
-
u
y
k
d
x
k
+
u
x
k
d
y
k
,
ω
=
∑
k
=
1
n
‍
ω
k and v
(
x
,
y
)
=
∫
(
x
0
,
y
0
)
(
x
,
y
)
‍
ω. Then (i) holds for f
0
=
u
+
i
v, which is analytic on B
0.
(ii) Ifz
∈
Ω, there is a chain C
=
(
B
0
,
B
1
,
…
,
B
n
) in Ω such that z is center of B
n and, by the lemma, there is analytic chain (
B
k
,
f
k
), k
=
0
,
…
,
n. We define f
(
z
)
=
f
n
(
z
). As in in the proof of monodromy theorem in one complex variable, one can show that this definition does not depend of chains C and that f
=
u
+
i
v in Ω.The following three theorems in [9] are a generalization of the corresponding one in [15].Theorem 28.
Letω be a fast majorant, and let f
=
h
+
g
¯ be a pluriharmonic mapping in a simply connected Λ
ω-extension domain Ω. Then the following are equivalent:(1)
f
∈
Λ
ω
(
Ω
);
(2)
h
∈
Λ
ω
(
Ω
) and g
∈
Λ
ω
(
Ω
);
(3)
|
h
|
∈
Λ
ω
(
Ω
) and |
g
|
∈
Λ
ω
(
Ω
);
(4)
|
h
|
∈
Λ
ω
(
Ω
,
∂
Ω
) and |
g
|
∈
Λ
ω
(
Ω
,
∂
Ω
),where Λ
ω
(
Ω
,
∂
Ω
) denotes the class of all continuous functions f on Ω
∪
∂
Ω which satisfy (83) with some positive constant C, whenever z
∈
Ω and w
∈
∂
Ω.DefineD
f
=
(
D
1
f
,
…
,
D
n
f
) and D
¯
f
=
(
D
¯
1
f
,
…
,
D
¯
n
f
), where z
k
=
x
k
+
i
y
k, D
k
f
=
(
1
/
2
)
(
∂
x
k
f
-
i
∂
y
k
f
) and D
¯
k
f
=
(
1
/
2
)
(
∂
x
k
f
+
i
∂
y
k
f
), k
=
1
,
2
,
…
,
n.Theorem 29.
LetΩ
⊂
C
n be a domain and f analytic in Ω. Then
(85)
d
Ω
(
z
)
|
∇
f
(
z
)
|
≤
4
ω
|
f
|
(
d
Ω
(
z
)
)
.
If|
f
|
∈
Λ
ω
(
Ω
), then d
Ω
(
z
)
|
∇
f
(
z
)
|
≤
4
C
ω
(
d
Ω
(
z
)
).
Iff
=
h
+
g
¯ is a pluriharmonic mapping, where g and h are analytic in Ω, then |
g
′
|
≤
|
f
′
| and |
h
′
|
≤
|
f
′
|. In particular, |
D
i
g
|
,
|
D
i
h
|
≤
|
f
′
|
≤
|
D
g
|
+
|
D
h
|.Proof.
Using a version of Koebe theorem for analytic functions (we can also use Bloch theorem), we outline a proof. Letz
∈
Ω, a
=
(
a
1
,
…
,
a
n
)
∈
C
n, |
a
|
=
1, F
(
λ
)
=
F
(
λ
,
a
)
=
f
(
z
+
λ
a
), λ
∈
B
(
0
,
d
Ω
(
z
)
/
2
), and u
=
|
F
|.
By the version of Koebe theorem for analytic functions, for every lineL which contains w
=
f
(
z
), there are points w
1
,
w
2
∈
L such that d
Ω
(
z
)
|
F
′
(
0
)
|
≤
4
|
w
k
-
w
|, k
=
1
,
2, and w
∈
[
w
1
,
w
2
]. Hence d
Ω
(
z
)
|
F
′
(
0
)
|
≤
4
ω
u
(
d
Ω
(
z
)
) and d
Ω
(
z
)
|
F
′
(
0
)
|
≤
4
ω
|
f
|
(
d
Ω
(
z
)
).
Definef
′
(
z
)
=
(
f
z
1
′
(
z
)
,
…
f
z
n
′
(
z
)
). Since F
′
(
0
)
=
∑
k
=
1
n
‍
D
k
f
(
z
)
a
k
=
(
f
′
(
z
)
,
a
¯
), we find |
∇
f
(
z
)
|
=
|
f
′
(
z
)
|, where a
¯
=
(
a
1
¯
,
…
,
a
n
¯
). Hence
(86)
d
Ω
(
z
)
|
∇
f
(
z
)
|
≤
4
ω
u
(
d
Ω
(
z
)
)
≤
4
ω
|
f
|
(
d
Ω
(
z
)
)
.
Finally if|
f
|
∈
Λ
ω
(
Ω
), we have d
Ω
(
z
)
|
∇
f
(
z
)
|
≤
4
C
ω
(
d
Ω
(
z
)
).ForB
n the following result is proved in [9].Theorem 30.
Letω be a regular majorant and let Ω be a simply connected Λ
ω-extension domain. A function f pluriharmonic in belongs to Λ
ω
(
Ω
) if and only if, for each i
∈
{
1
,
2
,
…
,
n
}, d
Ω
(
z
)
|
D
i
f
(
z
)
|
≤
ω
(
d
Ω
(
z
)
) and d
Ω
(
z
)
|
D
¯
i
f
(
z
)
|
≤
ω
(
d
Ω
(
z
)
) for some constant C depending only on f, ω, Ω, and n.We only outline a proof: letf
=
h
+
g
¯. Note that D
i
f
=
D
i
h and D
¯
i
f
=
D
i
g
¯.We can also use Proposition26.


### 3.2. Lipschitz-Type Spaces

Letf
:
G
→
G
′ be a C
2 function and B
_
x
=
B
(
x
,
d
(
x
)
/
2
). We denote by O
C
2
(
G
) the class of functions which satisfy the following condition:
(87)
sup
⁡
B
x
d
2
(
x
)
|
Δ
f
(
x
)
|
≤
c
os
c
B
x
f
,
for every x
∈
G.It was observed in [8] that O
C
2
(
G
)
⊂
O
C
1
(
G
). In [7], we proved the following results.Theorem 31.
Suppose that(a1)
D is a Λ
α-extension domain in R
n, 0
<
α
≤
1, and f is continuous on D
¯ which is a K-quasiconformal mapping of D onto G
=
f
(
D
)
⊂
R
n;
(a2)
∂
D is connected;
(a3)
f is Hölder on ∂
D with exponent α;
(a4)
f
∈
O
C
2
(
D
). Then f is Hölder on D
¯ with exponent αThe proof in [7] is based on Lemmas 3 and 8 in the paper of Martio and Nakki [18]. In the setting of Lemma 8, d
|
f
′
(
y
)
|
≤
M
^
d
α. In the setting of Lemma 3, using the fact that ∂
D is connected, we get similar estimate for d small enough.Theorem 32.
Suppose thatD is a domain in R
n, 0
<
α
≤
1, and f is harmonic (more generally O
C
2
(
D
)) in D. Then one has the following:(i.1)
f
∈
Lip
⁡
(
ω
,
c
,
D
)  implies
(ii.1)
|
f
′
(
x
)
|
≤
c
d
(
x
)
α
-
1, x
∈
D.
(ii.1)
implies
(iii.1)
f
∈
loc
⁡ Lip
⁡
(
α
,
L
1
;
D
).


## 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 thath
:
B
¯
n
(
a
,
r
)
→
R
m is a continuous vector-valued function, harmonic on B
n
(
a
,
r
), and let
(70)
M
a
*
=
sup
⁡
{
|
h
(
y
)
-
h
(
a
)
|
:
y
∈
S
n
-
1
(
a
,
r
)
}
.Leth
=
(
h
1
,
h
2
,
…
,
h
m
). A modification of the estimate in [37, Equation (
2.31
)] gives
(71)
r
|
∇
h
k
(
a
)
|
≤
n
M
a
*
,
k
=
1
,
…
,
m
.We next extend this result to the case of vector-valued functions. See also [38] and [39, Theorem 6.16].Lemma 25.
Suppose thath
:
B
¯
n
(
a
,
r
)
→
R
m is a continuous mapping, harmonic in B
n
(
a
,
r
). Then
(72)
r
|
h
′
(
a
)
|
≤
n
M
a
*
.Proof.
Without loss of generality, we may suppose thata
=
0 and h
(
0
)
=
0. Let
(73)
K
(
x
,
y
)
=
K
y
(
x
)
=
r
2
-
|
x
|
2
n
ω
n
r
|
x
-
y
|
n
,
where ω
n is the volume of the unit ball in R
n. Hence, as in [8], for 1
≤
j
≤
n, we have
(74)
∂
∂
x
j
K
(
0
,
ξ
)
=
ξ
j
ω
n
r
n
+
1
.
Letη
∈
S
n
-
1 be a unit vector and |
ξ
|
=
r. For given ξ, it is convenient to write K
ξ
(
x
)
=
K
(
x
,
ξ
) and consider K
ξ as function of x.
Then(75)
K
ξ
′
(
0
)
η
=
1
ω
n
r
n
+
1
(
ξ
,
η
)
.
Since|
(
ξ
,
η
)
|
≤
|
ξ
|
|
η
|
=
r, we see that
(76)
|
K
ξ
′
(
0
)
η
|
≤
1
ω
n
r
n
,
and
therefore
|
∇
K
ξ
(
0
)
|
≤
1
ω
n
r
n
.
This last inequality yields(77)
|
h
′
(
0
)
(
η
)
|
≤
∫
S
n
-
1
(
a
,
r
)
‍
|
∇
K
y
(
0
)
|
|
h
(
y
)
|
d
σ
(
y
)
≤
M
0
*
n
ω
n
r
n
-
1
ω
n
r
n
=
M
0
*
n
r
,
where d
σ
(
y
) is the surface element on the sphere, and the proof is complete.LetC
n
=
{
z
=
(
z
1
,
…
,
z
n
)
:
z
1
,
…
,
z
n
∈
C
} denote the complex vector space of dimension n. For a
=
(
a
1
,
…
,
a
n
)
∈
C
n, we define the Euclidean inner product 〈
·
,
·
〉 by
(78)
〈
z
,
a
〉
=
z
1
a
¯
1
+
⋯
+
z
n
a
¯
n
,
where a
¯
k
(
k
∈
{
1
,
…
,
n
}
) denotes the complex conjugate of a
k. Then the Euclidean length of z is defined by
(79)
|
z
|
=
〈
z
,
z
〉
1
/
2
=
(
|
z
1
|
2
+
⋯
+
|
z
n
|
2
)
1
/
2
.
Denote a ball in C
n with center z
′ and radius r
>
0 by
(80)
B
n
(
z
′
,
r
)
=
{
z
∈
C
n
:
|
z
-
z
′
|
<
r
}
.
In particular, B
n denotes the unit ball B
n
(
0
,
1
) and S
n
-
1 the sphere {
z
∈
C
n
:
|
z
|
=
1
}. Set D
=
B
1, the open unit disk in C, and let T
=
S
0 be the unit circle in C.A continuous complex-valued functionf defined in a domain Ω
⊂
C
n is said to bepluriharmonic if, for fixed z
∈
Ω and θ
∈
S
n
-
1, the function f
(
z
+
θ
ζ
) is harmonic in {
ζ
∈
C
:
|
θ
ζ
-
z
|
<
d
Ω
(
z
)
}, where d
Ω
(
z
) denotes the distance from z to the boundary ∂
Ω of Ω. It is easy to verify that the real part of any holomorphic function is pluriharmonic; cf. [40].Letω
:
[
0
,
+
∞
)
→
[
0
,
+
∞
) with ω
(
0
)
=
0 be a continuous function. We say that ω is amajorant if(1)
ω
(
t
) is increasing,
(2)
ω
(
t
)
/
t is nonincreasing for t
>
0.If, in addition, there is a constantC
>
0 depending only on ω such that
(81)
∫
0
δ
‍
ω
(
t
)
t
d
t
≤
C
ω
(
δ
)
,
0
<
δ
<
δ
0
,
(82)
δ
∫
δ
∞
‍
ω
(
t
)
t
2
d
t
≤
C
ω
(
δ
)
,
0
<
δ
<
δ
0
for some δ
0, then we say that ω is aregular majorant. A majorant is calledfast (resp.,slow) if condition (81) (resp., (82)) is fulfilled.Given a majorantω, we define Λ
ω
(
Ω
) (resp., Λ
ω
(
∂
Ω
)) to be theLipschitz-type space consisting of all complex-valued functions f for which there exists a constant C such that, for all z and w
∈
Ω (resp., z and w
∈
∂
Ω),
(83)
|
f
(
z
)
-
f
(
w
)
|
≤
C
ω
(
|
z
-
w
|
)
.Using Lemma25, one can prove the following.Proposition 26.
Letω be aregular majorant and let f be harmonic mapping in a simply connected Λ
ω-extension domain D
⊂
R
n. Then h
∈
Λ
ω
(
D
) if and only if |
∇
f
|
≤
C
(
ω
(
1
-
|
z
|
)
/
(
1
-
|
z
|
)
).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) Letu be pluriharmonic in B
0
=
B
(
a
;
r
). Then there is an analytic function f in B such that f
=
u
+
i
v.
(ii) LetΩ be simply connected and u be pluriharmonic in Ω. Then there is analytic function f in Ω such that f
=
u
+
i
v.Proof.
(i) LetB
0
=
B
(
a
;
r
)
⊂
Ω, P
k
=
-
u
y
k, and Q
=
u
x
k; define form
(84)
ω
k
=
P
k
d
x
k
+
Q
k
d
y
k
=
-
u
y
k
d
x
k
+
u
x
k
d
y
k
,
ω
=
∑
k
=
1
n
‍
ω
k and v
(
x
,
y
)
=
∫
(
x
0
,
y
0
)
(
x
,
y
)
‍
ω. Then (i) holds for f
0
=
u
+
i
v, which is analytic on B
0.
(ii) Ifz
∈
Ω, there is a chain C
=
(
B
0
,
B
1
,
…
,
B
n
) in Ω such that z is center of B
n and, by the lemma, there is analytic chain (
B
k
,
f
k
), k
=
0
,
…
,
n. We define f
(
z
)
=
f
n
(
z
). As in in the proof of monodromy theorem in one complex variable, one can show that this definition does not depend of chains C and that f
=
u
+
i
v in Ω.The following three theorems in [9] are a generalization of the corresponding one in [15].Theorem 28.
Letω be a fast majorant, and let f
=
h
+
g
¯ be a pluriharmonic mapping in a simply connected Λ
ω-extension domain Ω. Then the following are equivalent:(1)
f
∈
Λ
ω
(
Ω
);
(2)
h
∈
Λ
ω
(
Ω
) and g
∈
Λ
ω
(
Ω
);
(3)
|
h
|
∈
Λ
ω
(
Ω
) and |
g
|
∈
Λ
ω
(
Ω
);
(4)
|
h
|
∈
Λ
ω
(
Ω
,
∂
Ω
) and |
g
|
∈
Λ
ω
(
Ω
,
∂
Ω
),where Λ
ω
(
Ω
,
∂
Ω
) denotes the class of all continuous functions f on Ω
∪
∂
Ω which satisfy (83) with some positive constant C, whenever z
∈
Ω and w
∈
∂
Ω.DefineD
f
=
(
D
1
f
,
…
,
D
n
f
) and D
¯
f
=
(
D
¯
1
f
,
…
,
D
¯
n
f
), where z
k
=
x
k
+
i
y
k, D
k
f
=
(
1
/
2
)
(
∂
x
k
f
-
i
∂
y
k
f
) and D
¯
k
f
=
(
1
/
2
)
(
∂
x
k
f
+
i
∂
y
k
f
), k
=
1
,
2
,
…
,
n.Theorem 29.
LetΩ
⊂
C
n be a domain and f analytic in Ω. Then
(85)
d
Ω
(
z
)
|
∇
f
(
z
)
|
≤
4
ω
|
f
|
(
d
Ω
(
z
)
)
.
If|
f
|
∈
Λ
ω
(
Ω
), then d
Ω
(
z
)
|
∇
f
(
z
)
|
≤
4
C
ω
(
d
Ω
(
z
)
).
Iff
=
h
+
g
¯ is a pluriharmonic mapping, where g and h are analytic in Ω, then |
g
′
|
≤
|
f
′
| and |
h
′
|
≤
|
f
′
|. In particular, |
D
i
g
|
,
|
D
i
h
|
≤
|
f
′
|
≤
|
D
g
|
+
|
D
h
|.Proof.
Using a version of Koebe theorem for analytic functions (we can also use Bloch theorem), we outline a proof. Letz
∈
Ω, a
=
(
a
1
,
…
,
a
n
)
∈
C
n, |
a
|
=
1, F
(
λ
)
=
F
(
λ
,
a
)
=
f
(
z
+
λ
a
), λ
∈
B
(
0
,
d
Ω
(
z
)
/
2
), and u
=
|
F
|.
By the version of Koebe theorem for analytic functions, for every lineL which contains w
=
f
(
z
), there are points w
1
,
w
2
∈
L such that d
Ω
(
z
)
|
F
′
(
0
)
|
≤
4
|
w
k
-
w
|, k
=
1
,
2, and w
∈
[
w
1
,
w
2
]. Hence d
Ω
(
z
)
|
F
′
(
0
)
|
≤
4
ω
u
(
d
Ω
(
z
)
) and d
Ω
(
z
)
|
F
′
(
0
)
|
≤
4
ω
|
f
|
(
d
Ω
(
z
)
).
Definef
′
(
z
)
=
(
f
z
1
′
(
z
)
,
…
f
z
n
′
(
z
)
). Since F
′
(
0
)
=
∑
k
=
1
n
‍
D
k
f
(
z
)
a
k
=
(
f
′
(
z
)
,
a
¯
), we find |
∇
f
(
z
)
|
=
|
f
′
(
z
)
|, where a
¯
=
(
a
1
¯
,
…
,
a
n
¯
). Hence
(86)
d
Ω
(
z
)
|
∇
f
(
z
)
|
≤
4
ω
u
(
d
Ω
(
z
)
)
≤
4
ω
|
f
|
(
d
Ω
(
z
)
)
.
Finally if|
f
|
∈
Λ
ω
(
Ω
), we have d
Ω
(
z
)
|
∇
f
(
z
)
|
≤
4
C
ω
(
d
Ω
(
z
)
).ForB
n the following result is proved in [9].Theorem 30.
Letω be a regular majorant and let Ω be a simply connected Λ
ω-extension domain. A function f pluriharmonic in belongs to Λ
ω
(
Ω
) if and only if, for each i
∈
{
1
,
2
,
…
,
n
}, d
Ω
(
z
)
|
D
i
f
(
z
)
|
≤
ω
(
d
Ω
(
z
)
) and d
Ω
(
z
)
|
D
¯
i
f
(
z
)
|
≤
ω
(
d
Ω
(
z
)
) for some constant C depending only on f, ω, Ω, and n.We only outline a proof: letf
=
h
+
g
¯. Note that D
i
f
=
D
i
h and D
¯
i
f
=
D
i
g
¯.We can also use Proposition26.


## 3.2. Lipschitz-Type Spaces

Letf
:
G
→
G
′ be a C
2 function and B
_
x
=
B
(
x
,
d
(
x
)
/
2
). We denote by O
C
2
(
G
) the class of functions which satisfy the following condition:
(87)
sup
⁡
B
x
d
2
(
x
)
|
Δ
f
(
x
)
|
≤
c
os
c
B
x
f
,
for every x
∈
G.It was observed in [8] that O
C
2
(
G
)
⊂
O
C
1
(
G
). In [7], we proved the following results.Theorem 31.
Suppose that(a1)
D is a Λ
α-extension domain in R
n, 0
<
α
≤
1, and f is continuous on D
¯ which is a K-quasiconformal mapping of D onto G
=
f
(
D
)
⊂
R
n;
(a2)
∂
D is connected;
(a3)
f is Hölder on ∂
D with exponent α;
(a4)
f
∈
O
C
2
(
D
). Then f is Hölder on D
¯ with exponent αThe proof in [7] is based on Lemmas 3 and 8 in the paper of Martio and Nakki [18]. In the setting of Lemma 8, d
|
f
′
(
y
)
|
≤
M
^
d
α. In the setting of Lemma 3, using the fact that ∂
D is connected, we get similar estimate for d small enough.Theorem 32.
Suppose thatD is a domain in R
n, 0
<
α
≤
1, and f is harmonic (more generally O
C
2
(
D
)) in D. Then one has the following:(i.1)
f
∈
Lip
⁡
(
ω
,
c
,
D
)  implies
(ii.1)
|
f
′
(
x
)
|
≤
c
d
(
x
)
α
-
1, x
∈
D.
(ii.1)
implies
(iii.1)
f
∈
loc
⁡ Lip
⁡
(
α
,
L
1
;
D
).


## 4. Theorems of Koebe and Bloch Type for Quasiregular Mappings

We assume throughout thatG
⊂
R
n is an open connected set whose boundary, ∂
G, is nonempty. Also B
(
x
,
R
) is the open ball centered at x
∈
G with radius R. If B
⊂
R
n is a ball, then σ
B, σ
>
0, denotes the ball with the same center as B and with radius equal to σ times that of B.The spherical (chordal) distance between two pointsa
,
b
∈
R
¯
n is the number
(88)
q
_
(
a
,
b
)
=
|
p
(
a
)
-
p
(
b
)
|
,
where p
:
R
¯
n
→
S
(
e
n
+
1
/
2
,
1
/
2
) is stereographic projection, defined by
(89)
p
(
x
)
=
e
n
+
1
+
x
-
e
n
+
1
|
x
-
e
n
+
1
|
2
.
Explicitly, if a
≠
∞
≠
b,
(90)
q
_
(
a
,
b
)
=
|
a
-
b
|
(
1
+
|
a
|
2
)
-
1
/
2
(
1
+
|
b
|
2
)
-
1
/
2
.Whenf
:
G
→
R
n is differentiable, we denote its Jacobi matrix by D
f or f
′ and the norm of the Jacobi matrix as a linear transformation by |
f
′
|. When D
f exists a.e. we denote the local Dirichlet integral of f at x
∈
G by D
_
f
(
x
)
=
D
_
f
,
G
(
x
)
=
(
1
/
|
B
|
∫
B
_
‍
|
f
′
|
n
)
1
/
n, where B
_
=
B
_
x
=
B
_
x
,
G. If there is no chance of confusion, we will omit the index G. If B
=
B
x, then D
_
f
,
G
(
x
)
=
D
_
f
,
B
(
x
) and if B is the unit ball, we write D
_
(
f
) instead of D
_
f
,
B
(
0
).When the measure is omitted from an integral, as here, integration with respect ton-dimensional Lebesgue measure is assumed.A continuous increasing functionω
(
t
)
:
[
0
,
∞
)
→
[
0
,
∞
) is a majorant if ω
(
0
)
=
0 and if ω
(
t
1
+
t
2
)
≤
ω
(
t
1
)
+
ω
(
t
2
) for all t
1
,
t
2
≥
0.The main result of the paper [3] generalizes Lemma 5 to a quasiregular version involving a somewhat larger class of moduli of continuity than t
α, 0
<
α
<
1.Lemma 33 (see [3]).
Suppose thatG is Λ
ω-extension domain in R
n. If f is K-quasiregular in G with f
(
G
)
⊂
R
n and if
(91)
D
_
f
(
x
)
≤
C
1
ω
(
d
(
x
)
)
d
(
x
,
∂
D
)
-
1
f
o
r
x
∈
G
,
then f has a continuous extension to D
¯
∖
{
∞
} and
(92)
|
f
(
x
1
)
-
f
(
x
2
)
|
≤
C
2
ω
(
|
x
1
-
x
2
|
+
d
(
x
1
,
∂
G
)
)
,
for x
1
,
x
2
∈
D
¯
∖
{
∞
}, where the constant C
2 depends only on K
,
n
,
ω
,
C
1, and G.
IfG is uniform, the constant C
2
=
c
^
(
G
) depends only on K
,
n
,
C
1 and the uniformity constant c
*
=
c
*
(
G
) for G.
Conversely if there exists a constantC
2 such that (92) holds for all x
1
,
x
2
∈
G, then (91) holds for all x
∈
G with C
1 depending only on C
2
,
K
,
n
,
ω, and G.Now suppose thatω
=
ω
α and α
≤
α
0.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
(93)
|
f
(
x
1
)
-
f
(
x
2
)
|
≤
C
(
|
x
1
-
x
2
|
)
α
(
x
1
,
x
2
∈
D
¯
∖
{
∞
}
)
.Remark 34.
Simple examples show that the termd
(
x
1
,
∂
G
) cannot in general be omitted. For example, f
(
x
)
=
x
|
x
|
a
-
1 with a
=
K
1
/
(
1
-
n
) is K-quasiconformal in B
=
B
(
0
;
1
). D
_
f
(
x
) is bounded over x
∈
B yet f
∈
Li
p
a
(
B
); see Example 6.
Ifn
=
2, then |
f
′
(
z
)
|
≈
ρ
a
-
1, where z
=
ρ
e
i
θ, and if B
r
=
B
(
0
,
r
), then ∫
B
r
‍
|
f
′
(
z
)
|
d
x
d
y
=
∫
0
r
‍
∫
0
2
π
‍
ρ
2
(
a
-
1
)
ρ
d
ρ
d
θ
≈
r
2
a and D
_
f
,
B
r
(
0
)
≈
r
a
-
1.
Note that we will show below that if|
f
|
∈
L
α
(
G
), the conclusion (92) in Lemma 33 can be replaced by the stronger assertion |
f
(
x
1
)
-
f
(
x
2
)
|
≤
c
m
|
x
1
-
x
2
|
α.Lemma 35 (see [3]).
Iff
=
(
f
1
,
f
2
,
…
,
f
n
) is K-quasiregular in G with f
(
G
)
⊂
R
n and if B is a ball with σ
B
⊂
G, σ
>
1, then there exists a constant C, depending only on n, such that
(94)
(
1
|
B
|
∫
B
‍
|
f
′
|
n
)
1
/
n
≤
C
K
σ
σ
-
1
(
1
|
σ
B
|
∫
σ
B
‍
|
f
j
-
a
|
n
)
1
/
n
for all a
∈
R and all j
=
1
,
2
,
…
,
n. 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).
LetB
=
B
(
a
;
r
); let f be holomorphic function on B, D
=
f
(
B
), f
(
a
)
=
b, and let the unbounded component D
∞ of D be not empty, and let d
∞
=
d
∞
(
b
)
=
dist
⁡
(
b
,
D
∞
). Then (
a
.
1
)
r
|
f
′
(
a
)
|
≤
4
d
∞; (
b
.
1
) if, in addition, D is simply connected, then D contains the disk B
(
b
,
ρ
) of radius ρ, where ρ
=
ρ
f
(
a
)
=
r
|
f
′
(
a
)
|
/
4.The following result can be considered as a version of this lemma for quasiregular mappings in space.Theorem 37.
Suppose thatf is a K-quasiregular mapping on the unit ball B, f
(
0
)
=
0 and |
D
_
(
f
)
|
≥
1.
Then, there exists an absolute constantα such that for every x
∈
S there exists a point y on the half-line Λ
x
=
Λ
(
0
,
x
)
=
{
ρ
x
:
ρ
≥
0
}, which belongs to f
(
B
), such that |
y
|
≥
2
α.
Iff is a K-quasiconformal mapping, then there exists an absolute constant ρ
1 such that f
(
B
) contains B
(
0
;
ρ
1
).For the proof of the theorem, we need also the following result, Theorem18.8
.
1 [10].Lemma 38.
For eachK
≥
1 there is m
=
m
(
n
,
K
) with the following property. Let ϵ
>
0, let G
⊂
R
¯
n be a domain, and let F
^ denote the family of all qr mappings with max
⁡
{
K
O
(
x
,
f
)
,
K
I
(
x
,
f
)
}
≤
K and f
:
G
→
R
n
∖
{
a
1
,
f
,
a
2
,
f
,
…
,
a
m
,
f
} where a
i
,
f are points in R
¯
n such that q
_
(
a
i
,
f
,
a
j
,
f
)
>
ϵ, i
≠
j. Then F
^ forms a normal family in G.Now we prove Theorem37.Proof.
If we suppose that this result is not true then there is a sequence of positive numbersa
n, which converges to zero, and a sequence of K-quasiregular functions f
n, such that f
n
(
B
) does not intersect [
a
n
,
+
∞
), n
≥
1. Next, the functions g
n
=
f
n
/
a
n map B into G
=
R
n
∖
[
1
,
+
∞
) and hence, by Lemma 38, the sequence g
n is equicontinuous and therefore forms normal family. Thus, there is a subsequence, which we denote again by g
n, which converges uniformly on compact subsets of B to a quasiregular function g. Since D
_
(
g
n
) converges to D
_
(
g
) and |
D
_
(
g
n
)
|
=
|
D
_
(
f
n
)
|
/
a
n converges to infinity, we have a contradiction by Lemma 35.A path-connected topological spaceX with a trivial fundamental group π
1
(
X
) is said to be simply connected. We say that a domain V in R
3 is spatially simply connected if the fundamental group π
2
(
V
) is trivial.As an application of Theorem37, we immediately obtain the following result, which we call the Koebe theorem for quasiregular mappings.Theorem 39 (second version of Koebe theorem forK-quasiregular functions).
LetB
=
B
(
a
;
r
); let f  be K-quasiregular function on B
⊂
R
n, D
=
f
(
B
), f
(
a
)
=
b, and let the unbounded component D
∞ of D
c be not empty, and let d
∞
=
d
∞
(
b
)
=
dist
⁡
(
b
,
D
∞
). Then there exists an absolute constant c:(a.1)
r
|
D
_
f
(
a
)
|
≤
c
~
d
∞;
(a.2)
if, in addition,D
⊂
R
3 and D is spatially simply connected, then D contains the disk B
(
b
,
ρ
) of radius ρ, where ρ
=
ρ
f
(
a
)
=
r
|
D
_
f
(
a
)
|
/
c
~.(A.1) Now, using Theorem39 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 (
a
.
3
) below.Theorem 40 (Theorem-DyMa).
Let0
<
α
≤
1, K
≥
1, and n
≥
2. Suppose G is a L
α-extension domain in R
n and f is a K-quasiregular mapping of G onto f
(
G
)
⊂
R
n. The following are equivalent:(i.4)
f
∈
Λ
α
(
G
),
(ii.4)
|
f
|
∈
Λ
α
(
G
),
(iii.4)
|
D
_
f
(
r
,
x
)
|
≤
c
r
α
-
1 for every ball B
=
B
(
x
,
r
)
⊂
G.Proof.
Suppose thatf is a K-quasiregular mapping of G onto f
(
G
)
⊂
R
n. We first establish that for every ball B
=
B
(
x
,
r
)
⊂
G, one has the following:(a.3)
r
|
D
_
f
(
r
,
x
)
|
≤
c
~
ω
|
f
|
(
r
,
x
), where D
_
f
(
r
,
x
)
=
D
_
f
,
B
(
x
).
Letl be line throughout 0 and f
(
x
) and denote by D
∞
,
r the unbounded component of D
c, where D
=
f
(
B
(
x
,
r
)
). Then, using a similar procedure as in the proof of Theorem 22, the part (
iii
.
1
), one can show that there is z
′
∈
∂
D
∞
,
r
∩
l such that d
∞
(
x
′
)
≤
|
z
′
-
f
(
x
)
|
=
|
|
z
′
|
-
|
f
(
x
)
|
|, where x
′
=
f
(
x
).
Take a pointz
∈
f
-
1
(
z
′
). Then z
′
=
f
(
z
) and |
z
-
x
|
=
r. Since |
|
f
(
z
)
|
-
|
f
(
x
)
|
|
≤
ω
|
f
|
(
x
,
r
). Then using Theorem 39, we find (a.3).
Now we suppose(
ii
.
4
), that is, |
f
|
∈
Lip
⁡
(
α
,
L
;
G
).
Thus we haveω
|
f
|
(
x
,
r
)
≤
L
r
α. Hence, we get (a.4)
r
|
D
_
f
(
r
,
x
)
|
≤
c
r
α, where c
=
L
c
~.
It is clear that(
iii
.
4
) is denoted here as (a.4). The implication (
ii
.
4
)
⇒
(
iii
.
4
) is thus established.
If we suppose(
iii
.
4
), then an application of Lemma 20 shows that f
∈
loc
⁡
Λ
α
(
G
), and since G is a L
α-extension domain, we conclude that f
∈
Λ
α
(
G
). Thus (
iii
.
4
) implies (
i
.
4
).
Finally, the implication(
i
.
4
)
⇒
(
ii
.
4
) is a clear corollary of the triangle inequality.(
A
.
2
)
Now we give another outline that (i.4) is equivalent to (ii.4).Here, we use approach as in [15]. In particular, (
a
.
4
) implies that the condition (91) holds.We consider two cases:(1)
d
(
x
)
≤
2
|
x
-
y
|;
(2)
s
=
|
x
-
y
|
≤
d
(
x
)
/
2.Then we apply Lemma33 on G and A
=
B
(
x
,
2
s
) in Case (
1
) and Case (
2
), respectively.In more detail, if|
f
|
∈
L
α
(
G
), then for every ball B
(
x
,
r
)
⊂
G, by (a.3), r
|
D
_
f
(
x
)
|
≤
c
r
α
-
1 and the condition (91) holds. Then Lemma 33 tells us that (92) holds, with a fixed constant, for all balls B
⊂
G and all pairs of points x
1
,
x
2
∈
B. Next, we pick two points x
,
∈
G with |
x
-
y
|
<
d
(
x
,
∂
G
) and apply (92) with G
=
A, where A
=
B
(
x
;
|
x
-
y
|
), letting x
1
=
x and x
2
=
y. The resulting inequality
(95)
|
f
(
x
)
-
f
(
y
)
|
≤
const
|
x
-
y
|
α
shows that f
∈
loc
⁡
Λ
α
(
G
), and since G is a L
α-extension domain, we conclude that f
∈
Λ
α
(
G
).The implication (ii.4)⇒ (i.4) is thus established. The converse being trivially true.The consideration in(
A
.
1
) shows that (i.4) and (ii.4) are equivalent with (
a
.
4
).



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*Source: 895074-2014-10-21.xml*