Directory UMM :Data Elmu:jurnal:J-a:Journal of Computational And Applied Mathematics:Vol105.Issue1-2.1999:
Journal of Computational and Applied Mathematics 105 (1999) 425–436
Quasiconformality of harmonic extensions
Dariusz Partykaa; ∗; 1 , Ken-Ichi Sakanb;2; 3
a
The Catholic University of Lublin, Institute of Mathematics, Al. Raclawickie 14, P.O. Box 129,
20-950 Lublin, Poland
b
Department of Mathematics, Graduate School of Science, Osaka City University, Sugimoto, Sumiyoshi-ku,
Osaka, 558, Japan
Received 16 October 1997; received in revised form 14 April 1998
Dedicated to Professor Haakon Waadeland on the occasion of his 70th birthday
Abstract
Continued from Partyka and Sakan (Bull. Soc. Sci. Letters Lodz 47 (1997) 51–63) this paper aims at giving necessary
and sucient conditions on sense-preserving homeomorphisms of the unit circle for the quasiconformality of their harmonic
extensions to the unit disk. In particular, all such homeomorphisms with a bounded derivative are well characterized. In
c 1999 Elsevier Science B.V. All rights reserved.
consequence, a generalization of Martio’s result is obtained.
MSC: 30C55; 30C62
Keywords: Harmonic mapping in the plane; Quasiconformal mapping in the plane; Poisson integral
1. Introduction
Let Hom+ (T ) stand for the class of all sense-preserving homeomorphic self-mappings
of the
unit circle T :={z ∈ C: |z| = 1} and let L1 (T ) be the space of all complex-valued functions Lebesgue
integrable on T . According to the famous Rado–Kneser–Choquet theorem (cf. e.g. [1, p. 22])
the Poisson extension P[
] of
∈ Hom+ (T ) to the unit disk :={z ∈ C: |z|¡1} is a harmonic and
homeomorphic self-mapping of , where for every f ∈ L1 (T ),
1
P[f](z ):=
2
Z
T
f(u)Re
u+z
|d u|;
u−z
z ∈ :
∗
Corresponding author.
E-mail address: [email protected] (D. Partyka).
1
Supported by KBN (Scienti c Research Council) grant No. 2 PO3 A 002 08.
2
Partially supported by Grant-in-Aid for Scienti c Research No. 08304014, Ministry of Education, Japan.
3
E-mail: [email protected].
c 1999 Elsevier Science B.V. All rights reserved.
0377-0427/99/$ - see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 9 ) 0 0 0 0 9 - 6
(1.1)
426
D. Partyka, K.-I. Sakan / Journal of Computational and Applied Mathematics 105 (1999) 425–436
[
]|2 is positive on
Moreover, P[
] is sense-preserving, i.e. the Jacobian J[P[
]] = |@P[
]|2 − |@P
1
1
, where @:= 2 ((@=@x) − i(@=@y)) and @:= 2 ((@=@x) + i(@=@y)) are the so-called formal derivatives
operators; cf. e.g. [6, pp. 42– 43]. For K¿1 let QT (K ) be the class of Sall
∈ Hom+ (T ) which
admit a K -quasiconformal (K -qc. for short) extension to , and let QT := K¿1 QT (K ). Obviously,
by de nition any
∈ Hom+ (T ) belongs to QT provided P[
] is a qc. mapping. Thus the natural
question is whether the Poisson extension P[
] is qc. provided
∈ QT . The answer is negative. Let
Q∗T denote the class of all
∈ QT such that P[
] is a qc. mapping. Yang has shown in [10] that
Q∗T 6= QT . Next, Laugesen improved this by showing (cf. [4, Corollary 3]) that for each K¿1 there
exists
∈ QT (K )\Q∗T . Several very simple and explicit examples of
∈ QT (K )\Q∗T for each K¿1
were presented in [7, Examples 4:1–4:3]. The reader can nd in [7] general simple techniques of
constructing for each K¿1 an explicit homeomorphism
∈ QT (K )\Q∗T . This shows that the class
Q∗T is substantially smaller than QT . However, the question how big is the class Q∗T is still open.
In other words, the problem of characterizing homeomorphisms
∈ Q∗T arises. So far as the authors
know, Martio was the rst who studied this problem provided
∈ Hom+ (T ) is suciently smooth;
cf. [5]. In what follows, we proceed with the study of the problem for irregular
. In fact, our paper
is a continuation of [7]. In Section 2 we derive some lower estimate of the radial limiting values
of the Jacobian J[P[
]] on the boundary T . Then as an application we present several necessary
and sucient conditions for
∈ Hom+ (T ) to belong to Q∗T . Section 3 deals with the general case
of arbitrary
∈ Hom+ (T ). In Section 4 we restrict our attention to
∈ Hom+ (T) with a bounded
derivative, in particular to
∈ Hom+ (T) being a Lipschitz function. We show that a homeomorphism
with bounded derivative has a qc. harmonic extension if and only if its derivative is bounded away
from zero and the Cauchy–Stieltjes transform of its di erential is bounded; see Theorem 4.1. In
Section 5 we give a few examples and compare our results with Martio’s theorem; cf. [5, Theorem
1]. These results were presented by the rst named author on the conference “Continued Fractions
and Geometric Function Theory”, Trondheim (Norway), June 24 –28, 1997. In general, we adopt all
notations from our paper [7]. Thus for a function F : → C (resp. F : C\ → C) and z ∈ T we
de ne
−
@ˆr F (z ):= lim− F (tz )
t→1
+
(resp: @ˆr F (z ):= lim+ F (tz ));
t→1
+
−
whenever the limit exists, while @ˆr F (z ):=0 (resp. @ˆr F (z ):=0) otherwise.
−
2. A lower estimate of @ˆr J[P[
]]
Given a function f : T → C and z ∈ T we de ne
f(u) − f(z )
T∋u→z
u−z
f′ (z ):= lim
provided the limit exists, while f′ (z ):=0 otherwise. If the limit exists we say that f has the
derivative f′ (z ) at z . As in [7], for any
∈ Hom+ (T ) we can consider the Riemann–Stieltjes integral
C
(z ):=
1
2 i
Z
T
d
(u)
;
u−z
z ∈ ∪ (C\ )
and
C
(∞):=0:
D. Partyka, K.-I. Sakan / Journal of Computational and Applied Mathematics 105 (1999) 425–436
427
[
] have nonLemma 2.1. If
∈ Hom+ (T ) then for a.e. z ∈ T; both the functions @P[
] and @P
tangential limiting values at z and
z
d
(z ) − P[
](rz )
z
′
′
P[
](rz ) + z
(z ) = lim−
lim @P[
](rz ) = lim−
+ z
(z )
(2.1)
r→1−
2 r→1 d r
2 r→1
1−r
and
z
d
(z ) − P[
](rz )
z
′
′
P[
](rz ) − z
(z ) = lim−
− z
(z ) :
(2.2)
lim @P[
](rz ) = lim−
r→1−
2 r→1 d r
2 r→1
1−r
Proof. As is remarked just below (3.9), C
has nontangential limiting values a.e. on T . Hence by
[
] have nontangential limiting values a.e. on T , as well.
(3.1) both the functions @P[
] and @P
From (1.1) it follows that for every r ∈ (0; 1) and for every z ∈ T ,
1
@P[
](rz ) =
2
=−
u
1 1
(u)
|d u| = −
2
(u − rz )
2 2irz
T
Z
1 1
2 2irz
=
=
(eit )
0
"Z
1 1
=−
2 2irz
1 1
=
2 2irz
2
Z
2
0
"Z
2
0
1 1
2 2irz
Z
2
1 1
2 2irz
Z
2
eit + rz
eit − rz
Re
Re
2
(eit )
0
−2irz eit
dt
(eit − rz )2
dt
d
eit + rz
(eit ) Re it
dt + i
dt
e − rz
eit + rz
Re it
d
(eit ) + ir
e − rz
0
0
d
dt
!
Z
Z
2
0
d
eit + rz
(eit ) Im it
dt
d t e − rz
2z eit
(e )Re it
dt
(e − rz )2
2
it
0
eit + rz
1 1
d
(eit ) +
it
e − rz
2 2z
Z
Z
2
(eit )
0
#
#
@
eit + rz
Re it
dt
@r
e − rz
eit + rz
1 d
P[
](rz ):
d
(eit ) +
it
e − rz
2z d r
(2.3)
Assume
has the derivative
′ (z ) at z = ei ∈ T . Then (d = d t )
(eit )|t= = iz
′ (z ) and by the Fatou
theorem on Poisson integrals (cf. [9, Theorem 11.12]) we have
2
1 1
eit + rz
1
Re it
d
(eit ) =
′ (z ):
(2.4)
r→1 2 2irz 0
e − rz
2
By (2.3) and (2.4), the limit limr→1− @P[
](rz ) exists i the limit limr→1− (1= 2z )(d = d r )P[
](rz )
exists. As shown in the proof of [7, Theorem 1.1] both the radial limits
lim−
Z
−
lim− @P[
](rz ) = @ˆr C
(z ) and
r→1
+
[
](rz ) = z 2 @ˆr C
(z )
lim− @P
r→1
(2.5)
exist for a.e. z ∈ T . Since
has the derivative
′ (z ) for a.e. z ∈ T , we conclude from (2.3), (2.4)
and (2.5) that the rst equality in (2.1) holds. If for z ∈ T the limit limr→1− (d = d r )P[
](rz ) exists,
then
Z 1
d
1
d
(z ) − P[
](rz )
= lim−
P[
](tz ) d t = lim− P[
](rz ):
lim
r→1
r→1
r→1−
1−r
1 − r r dt
dr
428
D. Partyka, K.-I. Sakan / Journal of Computational and Applied Mathematics 105 (1999) 425–436
This yields the second equality in (2.1). By (2.5), we can rewrite [7, (1.8)] as
[
](rz )) =
′ (z ) for a:e: z ∈ T:
lim− (@P[
](rz ) − z2 @P
r→1
Combining this with (2.1) we obtain (2.2).
−
Given
∈ Hom+ (T ) de ne d
:=ess inf z∈T |
′ (z )| and j
:=ess inf z∈T @ˆr J[P[
]](z ). We use Lemma
2.1 to nd a lower estimate of the functional j
. The following theorem is in fact a generalization
of Martio’s [5, Lemma 3].
Theorem 2.2. If
∈ Hom+ (T ) then for a.e. z ∈ T; the Jacobian J[P[
]] has a nontangential limiting value at z and
+
−
1 − |F
(0)|
1
;
lim J[P[
]](rz ) = |@ˆr C
(z )|2 − |@ˆr C
(z )|2 ¿ |
′ (z )|
2
1 + |F
(0)|
(2.6)
r→1−
where F
(0) is the point in which is uniquely determined by P[
](F
(0)) = 0. In particular;
1 1 − |F
(0)|
j
¿ d
:
2 1 + |F
(0)|
(2.7)
Proof. It is easy to check that for each r ∈ [0; 1) and u; z ∈ T ,
1 1
u + rz
1 1+r
1 1+r
1
¿
¿ :
Re
=
2
2
1 − r 2 u − rz 2 |u − rz|
2 (1 + r )
4
(2.8)
Given a ∈ write ha (u):=(u − a)= (1 − au
) for u ∈ C\{1= a}
and ha (1= a):=∞. Set a:= − F
(0) and
R
R 2 ˆ
:=
◦ ha . Then P[
◦ ha ](0) = P[
](ha (0)) = P[
](F
(0)) = 0, and so 0 = T (u)|d u| = 0 ei(t)
dt =
R 2
R 2
ˆ(t ) d t + i 0 sinˆ(t ) d t , where ˆ is the angular parametrization of ; cf. (3.3). Thus for
0 cos
R 2
every ∈ R, 0 cos(ˆ(t ) − ˆ()) d t = 0. From this, (2.8) and Lemma 2.1 we see, following the
proof of [5, Lemma 3], that for a.e. z = ei ∈ T ,
[](rz )|2 )
lim J[P[]](rz ) = lim− (|@P[](rz )|2 − |@P
r→1−
r→1
= lim− Re
r→1
(z ) − P[](rz ) ′
z (z )
1−r
= lim− |′ (z )|Re (z )
r→1
= lim− |′ (z )|
r→1
1 1
1 − r 2
(z ) − P[](rz )
1−r
Z
0
2
[1 − cos(ˆ(t ) − ˆ())]Re
eit + rz
dt
eit − rz
¿ 21 |′ (z )| = 12 |(
◦ ha )′ (z )| = 12 |
′ (ha (z ))||h′a (z )|:
(2.9)
Since P[] = P[
◦ ha ] = P[
] ◦ ha , (2.9) yields
lim J[P[
]](ha (rz ))|h′a (rz )|2 = lim− J[P[
] ◦ ha ](rz ) = 21 |
′ (ha (z ))||h′a (z )|:
r→1−
r→1
(2.10)
D. Partyka, K.-I. Sakan / Journal of Computational and Applied Mathematics 105 (1999) 425–436
429
From Lemma 2.1 it follows that for a.e. z ∈ T , the Jacobian J[P[
]] has a nontangential limiting
value at z , and (2.10) implies
1
1 − |F
(0)|
1
lim J[P[
]](rz )¿ |
′ (z )| inf |h′a (u)|−1 = |
′ (z )|
:
u∈T
2
2
1 + |F
(0)|
r→1−
Combining this with (2.5) we obtain (2.6). Moreover, by de nition
−
j
= ess inf @ˆr J[P[
]](z )¿ ess inf
z∈T
z∈T
1 − |F
(0)|
1 ′
|
(z )|
2
1 + |F
(0)|
!
1 1 − |F
(0)|
;
= d
2 1 + |F
(0)|
which proves (2.7).
Remark 2.3. It can be shown that j
¿d3
. However, the proof exceeds the scope of this paper and
will be published elsewhere.
3. The main results
Let
∈ Hom+ (T ). For the convenience of readers, we recall that for each z ∈ ,
1
@P[
](z ) =
2i
Z
T
d
(u)
= C
(z ) and
u−z
[
](z ) = 1 C
1 ;
@P
z
z2
(3.1)
cf. [7, (1.3) and (1.4)]. Moreover, for z ∈ \{0}
C
(z ) − C
1
1
=
z
2
Z
0
2
ei + z
Re i
e −z
!
d
˜()
(3.2)
(cf. [7, (1.7)], where
˜():=
Z
0
d
(eit )
;
ieit
∈ [0; 2]:
(3.3)
Following the proof of [7, Theorem 3.1] we rst show
Theorem 3.1. Suppose that
∈ Hom+ (T ) and that there exists a sequence n ∈ T; n ∈ N; such that
the derivative
′ (n ) = limu→n (
(u) −
(n ))= (u − n ) exists for each n ∈ N and
lim
′ (n ) = 0:
(3.4)
n→∞
Then P[
] is not a qc. mapping. In particular; if d
= 0 then P[
] is not a qc. mapping.
Proof. Fix n ∈ N. By the Fatou theorem on Poisson integrals (cf. [9, Theorem 11.12]), the Poisson–
Stieltjes integral in (3.2) has the nontangential limit
˜′ (’) =
′ (ei’ ) for each ’ ∈ [0; 2] such that
the derivative
˜′ (’) exists; cf. [2, pp. 4 –5]. Therefore, by (3.2) we obtain
C
(tn ) − C
(1=tn ) →
′ (n )
as (0; 1) ∋ t → 1:
(3.5)
430
D. Partyka, K.-I. Sakan / Journal of Computational and Applied Mathematics 105 (1999) 425–436
As shown in [7, (1.10)], 2|@P[
]()|2 ¿c(
)¿0 on , where c(
) is a positive constant from [7,
Theorem 1.1]. Hence by (3.1) we see that for 0¡t¡1
2
@P
[
](tn )
tn @P[
](tn )
061−
61 −
@P[
](tn )
@P[
](tn )
√
|C
(tn ) − C
(1=tn )|
2|C
(tn ) − C
(1=tn )|
√
6
6
:
|@P[
](tn )|
c(
)
Combining this with (3.5) and (3.4) we obtain
@P
[
](z )
sup
¿ lim inf
t→1−
z∈ @P[
](z )
s
@P
2 ′
[
](tn )
|
(n )| → 1
¿1 −
@P[
](tn )
c(
)
as n → ∞;
and consequently P[
] is not a qc. mapping.
If d
= 0, then obviously there exists a sequence n ∈ T , n ∈ N, such that
has the derivative at
each n and such that (3.4) is satis ed. Thus P[
] is not a qc. mapping.
We write L∞ (T ) for the class of all measurable functions f : T → C essentially bounded on T ,
i.e. ||f||∞ :=ess supz∈T |f(z )|¡∞. As a conclusion from Theorems 2.2 and 3.1 we derive a useful
sucient condition of quasiconformality of P[
].
−
Theorem 3.2. Suppose that
∈ Hom+ (T ). If d
¿0 and @ˆr C
∈ L∞ (T ); then
∈ Q∗T and
v
u
@P
j
[
](z ) u
:
sup
6t1 −
−
z∈ @P[
](z )
||@ˆr C
||2∞
(3.6)
Proof. From [7, Theorem 1.1] it follows that
ˆ+
@P
@
C
(
z
)
[
](
z
)
r
sup
= ess sup −
z∈T
z∈ @P[
](z )
@ˆ C (z )
r
and
(3.7)
−
|@ˆr C
(u)|2 ¿c(
)= 2¿0;
(3.8)
for a.e. u ∈ T , where c(
) is a constant. As shown in the proof of [7, Theorem 1.1] both the radial
−
+
limits @ˆr C
(u) and @ˆr C
(u) exist for a.e. u ∈ T . Hence Theorem 2.2 shows, by (3.1), that for a.e.
u∈T
+
−
[
](tu)|2 ) = lim J[P[
]](tu)¿j
:
|@ˆr C
(u)|2 − |@ˆr C
(u)|2 = lim− (|@P[
](tu)|2 − |@P
−
t→1
t→1
Applying now (3.8) we have
2
ˆ+
@r C
(u)
j
¿
1− −
−
@ˆr C
(z )
|@ˆr C
(u)|2
for a:e: u ∈ T:
D. Partyka, K.-I. Sakan / Journal of Computational and Applied Mathematics 105 (1999) 425–436
431
By the assumptions and (2.7),
2
ˆ+
@r C
(u)
j
61 −
¡1:
ess sup −
−
ˆ
ˆ
u∈T
@r C
(z )
||@r C
||2∞
From this and (3.7) it follows that
∈ Q∗T and (3.6) holds.
From [7, (1.6)] it follows that for z ∈ C\T , C
(z ) is represented by the Cauchy–Stieltjes type
integral of
˜ de ned by (3.3),
u d
(u)
1 2 ei
1
=
d
˜():
(3.9)
C
(z ) =
2 T u − z i u
2 0 ei − z
For every function f : [0; 2] → C of bounded variation write CT [d f] for the Cauchy–Stieltjes type
singular integral, i.e. for every z = eix ∈ T
Z
Z
ei
1 2 ei
1
CT [d f](z ):=PV
d
f
(
):=
lim
d f()
→0+ 2 ¡|−x|6 ei − z
2 0 ei − z
whenever the limit exists and CT [d f](z ):=0 otherwise.
By Smirnov theorem [8, p. 65] and (3.9) the function C
belongs to the Hardy class H p () for
arbitrary p ∈ (0; 1). Hence C
has nontangential limiting values a.e. on T ; cf. [8, p. 56]. Then the
classical Privalov’s Theorem (cf. [8, p. 135]) says that the singular integral exists a.e. on T and
the formulas
−
+
@ˆr C
(u) = 1
˜′ (x) + CT [d
˜](u) and @ˆr C
(u) = − 1
˜′ (x) + CT [d
˜](u)
Z
Z
2
2
′
ix
′
ix
hold for a.e. u = e ∈ T . Since
˜ (x) =
(e ) for a.e. x ∈ [0; 2] we have
−
@ˆr C
= 12
′ + CT [d
˜]
and
+
@ˆr C
= − 12
′ + CT [d
˜]
a:e: on T:
(3.10)
Applying the equalities (3.10) we can extend [7, Theorem 1.2 and Corollary 2.1] to an arbitrary
homeomorphism
∈ Hom+ (T ) with CT (
′ ) replaced by CT [d
˜] as follows.
Theorem 3.3. If
∈ Hom+ (T ); then for a.e. z ∈ T
and
where
|
′ (z ) + 2CT [d
˜](z )|2 ¿2c(
)¿0
′
s
@P
(z ) − 2CT [d
˜](z )
[
](z )
= 1 − m
;
sup
= ess sup ′
@P
[
](
z
)
(
z
)
+
2
C
[d
˜
](
z
)
1 + m
z∈T
z∈
T
(3.11)
(3.12)
4Re[
′ (z )CT [d
˜](z )]
(3.13)
z∈T
|
′ (z )|2 + 4|CT [d
˜](z )|2
and c(
) is a constant from [7; Theorem 1:1]. In particular; m
¿0; and the mapping P[
] is qc.
i m
¿0.
m
:= ess inf
Proof. The theorem follows from [7, Theorem 1.1] and the equalities (3.10). The second equality
in (3.12) is obtained by simple computations.
432
D. Partyka, K.-I. Sakan / Journal of Computational and Applied Mathematics 105 (1999) 425–436
Corollary 3.4. Suppose that
∈ Hom+ (T ). Then
∈ Q∗T i d
¿0 and
CT [d
˜](z )
CT [d
˜](z )
¡∞:
0¡ ess inf Re
6 ess sup
′
z∈T
(z )
′ (z )
z∈T
(3.14)
Proof. Assume P[
] is a qc. mapping. Theorem 3.1 then implies that d
¿0. From Theorem 3.3 it
follows that m
¿0. Hence (3.14) follows from (3.13). Conversely, if d
¿0 and (3.14) holds then
from (3.13) it follows that m
¿0. Applying Theorem 3.3 we see that
∈ Q∗T as claimed.
Remark 3.5. If
′ (z ) 6= 0 for a.e. z ∈ T and (3.14) holds, then from (3.11) it follows that d
¿0.
Thus in Corollary 3.4 the condition d
¿0 may be replaced by the condition that
′ (z ) 6= 0 for a.e.
z ∈ T.
4. The case of homeomorphisms with a bounded derivative
In this section, we discuss the case where
∈ Hom+ (T ) has a bounded derivative. Our most
essential result is
Theorem 4.1. Suppose that
∈ Hom+ (T ) satis es
′ ∈ L∞ (T ). Then
∈ Q∗T i d
¿0 and CT [d
˜] ∈
L∞ (T ).
Proof. Assume
∈ Q∗T . Corollary 3.4 shows that d
¿0 and by (3.14),
||CT [d
˜]||∞ 6||
′ ||∞ ||CT [d
˜]=
′ ||∞ ¡∞;
so that CT [d
˜] ∈ L∞ (T ). Conversely, if
′ ∈ L∞ (T ) and CT [d
˜] ∈ L∞ (T ); then by (3.10) we obtain
−
@ˆr C
=(1= 2)
′ +CT [d
˜] ∈ L∞ (T ). Since d
¿0 we conclude from Theorem 3.2 that
∈ Q∗T as claimed.
We now turn to the case where
∈ Hom+ (T ) is absolutely continuous. For every f ∈ L1 (T ) de ne
Z
1
f(u)|d u|
fT :=
2 T
and write CT (f) for the Cauchy singular integral, i.e. for every z ∈ T
Z
Z
f(u)
f (u )
1
1
CT (f)(z ):=PV
d u:= lim+
d u;
→0
2i T u − z
2i T\T(z;) u − z
whenever the limit exists and CT (f)(z ):=0 otherwise, where T (eix ; ):={eit ∈ T : |t − x|¡}. We
recall that the harmonic conjugation operator A is de ned by the singular integral
Z
x−t
1
lim+
f(eit )cot
d t; z = eix ∈ T;
A(f)(z ):=
2 →0 ¡|t−x|6
2
whenever the limit exists and AR(f)(z ):=0 otherwise; cf. e.g. [3, 11]. If
∈ Hom+ (T ) is absolutely
continuous then
′ ∈ L1 (T ) and T |
′ (u)||d u| = 2. Moreover, by [7, (1.14)] we have for a.e. z ∈ T
CT [d
˜](z ) = CT (
′ )(z ) = 21
′T + 2i A(
′ )(z ):
(4.1)
D. Partyka, K.-I. Sakan / Journal of Computational and Applied Mathematics 105 (1999) 425–436
433
Applying now Theorem 4.1 to Lipschitz functions we obtain
Corollary 4.2. Suppose that
∈ Hom+ (T ) is a Lipschitz function; i.e. there exists a constant L¿0
such that
|
(u) −
(w)|6L|u − w|;
u; w ∈ T:
(4.2)
Then
∈ Q∗T i d
¿0 and CT (
′ ) ∈ L∞ (T ).
The same is true with CT (
′ ) replaced by A(
′ ).
Proof. Since
∈ Hom+ (T ) satis es (4.2), it follows that
is absolutely continuous and
′ ∈ L∞ (T )
with ||
′ ||∞ 6L. Therefore, the corollary follows from Theorem 4.1 and (4.1).
Each
∈ Hom+ (T ) de nes a unique continuous function
ˆ satisfying 06
ˆ(0)¡2 and
ˆ
(eit ) = ei
(t)
;
t ∈ R:
(4.3)
Actually,
ˆ is an increasing homeomorphism of R onto itself satisfying
ˆ(t + 2) −
ˆ(t ) = 2 for
t ∈ R; called the angular parametrization of
. Sometimes it is more convenient to use the real-valued
function |
′ | instead of
′ . To this end we prove.
Corollary 4.3. Suppose that
∈ Hom+ (T ) is a Lipschitz function. Then
∈ Q∗T i d
¿0 and
A(|
′ |) ∈ L∞ (T ).
ˆ
Proof. From (4.3) it follows that for a.e. t ∈ R;
′ (eit ) =
ˆ′ (t )ei(
(t)−t)
. Hence for a.e. x ∈ R;
ˆ
|A(
′ )(eix ) − ei(
(x)−x)
A(|
′ |)(eix )|
Z
Z
1
= lim+
2 →0
1
= lim+
2 →0
¡|t−x|6
Z
||
′ ||∞
2
Z
x+
||
′ ||∞
2
Z
x+
′
Z
x+
6
6
6
||
||∞
2
x+
x−
x−
′
it
i(
(x)−x)
ˆ
|
(e )|(e
i(
(t)−t)
ˆ
x − t
|
(e )|)cot
dt
2
′
−e
it
i(
(x)−x)
ˆ
|x − t|
ei(
(t)−t)
ˆ
ˆ
− ei(
(x)−x)
dt
|sin x−t |
x−t
x − t
dt
) cot
2
2
x−
x−
it
(
(e ) − e
¡|t−x|6
||
′ ||∞
6
2
′
!
eit − eix ei
(t)
ˆ
ˆ
− ei
(x)
+
dt
t−x
t−x
ˆ(t ) −
ˆ(x)
dt
x−t
1 +
1+
1
|t − x|
6||
′ ||∞ (1 + ||
′ ||∞ )¡∞:
Z
x
t
ˆ′ (s)d s d t
Therefore, A(
′ ) ∈ L∞ (T ) i A(|
′ |) ∈ L∞ (T ); and the corollary follows from Corollary 4.2.
434
D. Partyka, K.-I. Sakan / Journal of Computational and Applied Mathematics 105 (1999) 425–436
Following the proof of [7, Corollary 3.5] we easily obtain a version of Corollary 4.2 in terms of
a variant H of the Hilbert transform de ned for every locally integrable function f : R → C and
every x ∈ R by the singular integral
H(f)(x):=
1
lim
→0+
Z
¡|t−x|6
f (t )
d t;
x−t
whenever the limit exists and H(f)(x):=0 otherwise.
Corollary 4.4. Suppose that
∈ Hom+ (T ) and
ˆ is absolutely continuous on R satisfying
ess supt∈R
ˆ′ (t )¡∞. Then
∈ Q∗T i ess inf t∈R
ˆ′ (t )¿0 and ess supt∈R |H(
ˆ′ )(t )|¡∞.
We end this section with the following corollary, which points out a relation with the Hardy class
H ∞ () of bounded analytic functions on .
Corollary 4.5. Suppose
∈ Hom+ (T ) is a Lipschitz function. Then P[
] is a qc. mapping i there
exists F ∈ H ∞ () such that
−
@ˆr Re F (z ) = |
′ (z )|¿d
¿0
for a:e: z ∈ T:
(4.4)
Proof. Assume P[
] is a qc. mapping. Corollary 4.3 then shows that d
¿0 and ess supz∈T |A(|
′ |)(z )|
¡∞. By [3, p. 103], P[A(|
′ |)] is identical with the harmonic conjugate function v of P[|
′ |] satisfying v(0)=0. Thus F :=P[|
′ | +iA(|
′ |)] is an analytic function on . Since |
′ | +iA(|
′ |) ∈ L∞ (T );
we conclude from [2, Corollary 2 to Theorem 3.1] that F ∈ H ∞ (). By Fatou’s theorem on
−
Poisson integrals (cf. [9, Theorem 11.12]), the radial limit @ˆr Re F = |
′ | a.e. on T; which ends
the proof in the rst direction. Assume now that (4.4) holds for some F ∈ H ∞ (). A classi−
−
cal result (cf. e.g. [3, p. 103]) states that @ˆr Im F = Im F (0) + A(@ˆr Re F ) a.e. on T . Hence
ess supz∈T |A(|
′ |)(z )|6|F (0)| + supz∈ |F (z )|¡∞; and P[
] is qc. by Corollary 4.3.
5. Comparison between our results and Martio’s theorem
In this section, we compare our results with Martio’s theorem; cf. [5, Theorem 1]. We show that
Theorem 4.1 and Corollaries 4:2–4:5 essentially extend Martio’s theorem.
If
∈ Hom+ (T ) ∩ C 1 (T ) and |
′ | is Dini continuous on T; then a classical result (cf. e.g. [3,
p. 106]) shows that the function A(|
′ |) is continuous on T; and hence bounded. Thus Corollary
4.3 leads to
Corollary 5.1 (Martio [5]). Suppose
∈ Hom+ (T ) ∩ C 1 (T ). If |
′ | is Dini continuous on T; then
P[
] is qc. i d
¿0.
Let Di + (T ) denote the class of all sense-preserving di eomorphic self-mappings of T . For
0¡ 61 write C 1+ (T ) for the class of all complex-valued functions continuously di erentiable on
T; whose derivatives are -Holder continuous functions on T .
D. Partyka, K.-I. Sakan / Journal of Computational and Applied Mathematics 105 (1999) 425–436
Corollary 5.2. For each ∈ (0; 1];
Di + (T ) ∩ C 1+ (T ) ⊂ Q∗T :
435
(5.1)
Moreover;
Di + (T )\Q∗T 6= ∅ and
Q∗T \Di + (T ) 6= ∅:
(5.2)
Proof. Fix
∈ Di + (T ). By the de nition of a di eomorphism, d
¿0. For any ∈ (0; 1]; if
∈ C 1+
(T ) then |
′ | is Dini continuous on T; and the inclusion (5.1) follows from Corollary 5.1.
R Let F de∞
note the class of all real-valued functions f ∈ L (T ) such that ess inf z∈T f(z )¿0 and T f(u)|d u| =
2. Each function f ∈ F de nes a homeomorphism
f ∈ Hom+ (T ) whose angular parametrization
Rx
it
ˆf is determined by the equality
ˆf (x) = 0 f(e ) d t for x ∈ R. Obviously,
ˆf is a Lipschitz function
on R; and so is
f on T . Moreover, d
f ¿0. It is a simple matter to construct a continuous function
f ∈ F such that A(f) is unbounded; see Example 5.3. Then
f ∈ Di + (T ) and A(|
′f |) = A(f)
is unbounded. From Corollary 4.3 it follows that P[
f ] is not a qc. mapping, i.e.
f 6∈ Q∗T ; which
proves the rst assertion in (5.2).
−
It is easy to nd F ∈ H ∞ () such that f:=@ˆr Re F ∈ F but |
′f | is not continuous on T ; see
Example 5.4. Then
f 6∈ Di + (T ) and Corollary 4.5 shows that
f ∈ Q∗T ; which proves the second
assertion in (5.2).
Example 5.3. For every x ∈ R de ne
if x¿1=e;
1
p(x):=
−1= log x if 0¡x¡1=e;
0
if x60:
Since p ∈ C 1 (R\{0; 1=e}); H(p)(x) exists for every x ∈ R\{0; 1=e}. Moreover, p(t ) = 0 if t60 and
p(t )¿0 if t¿0. Then for each x ∈ (−1=e; 0) we have
Z
Z 1=e
Z
x+
p (t )
p(t )
p (t )
|H(p)(x)| = lim+
dt =
d t¿
dt
→0 ¡|t−x|6 x − t
−x=2 t − x
x− t − x
=
Z
1=e
t
1
−
t−x
t log t
−x=2
1
=−
3
Z
−1
log|x=2|
1
d t¿
3
Z
1=e
−x=2
−
dt
t log t
x
du 1
= log log → ∞;
u
3
2
(u = log t )
as x → 0− :
(5.3)
It is easily seen that there existsR a function q ∈ C 1 (R) such that (p + q)(−) = (p + q)()¿0 and
min−6t6 (p + q)(t )¿0. Set c:= − (p + q)(t ) d t¿0; and de ne f(eit ) = (2=c)(p(t ) + q(t )); −6t
6. Obviously, f ∈ F ∩ C (T ); and thus
f is a Lipschitz function on T . Since q ∈ C 1 (R); H(q)
is continuous on R; and thus bounded on [ − ; ]. Hence by (5.3) we see that
ess sup |H(
ˆ′f )(x)| = ∞:
(5.4)
x∈R
As in the proof of [7, Corollary 3.5], we can check that (5.4) implies
||A(
′f )||∞ = ∞:
(5.5)
436
D. Partyka, K.-I. Sakan / Journal of Computational and Applied Mathematics 105 (1999) 425–436
As shown in the proof of Corollary 4.3, we see that (5.5) implies ||A(f)||∞ = ||A(|
′f |)||∞ = ∞.
Example 5.4. For z ∈ de ne G (z ):=exp(−(1 + z )= (1 − z )). Clearly
|G (z )| = exp −Re
1+z
6e0 = 1;
1−z
so that G ∈ H ∞ (). Let c:=
R 2
0
−
z ∈ ;
(5.6)
2 + @ˆr Re G (eit ) d t¿0; and de ne F (z ):=(2=c)[2 + G (z )]; z ∈ .
−
By (5.6), F ∈ H () and f:=@ˆr Re F ∈ F. Moreover, if eix 6= 1; then
ˆ′f (x) = f(eix ) = (2=c)(2 +
cos(−cot (x= 2))). Hence
ˆ′f is discontinuous at x = 0; and so the function |
′f | is not continuous at
1 ∈ T.
∞
References
[1] D. Bshouty, W. Hengartner, Univalent harmonic mappings in the plane, Ann. Univ. Mariae Curie-Sklodowska, Sect.
A 48 (1994) 12–42.
[2] P. Duren, Theory of H p -Spaces, Academic Press, New York, 1970.
[3] J.B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981.
[4] R.S. Laugesen, Planar harmonic maps with inner and Blaschke dilatations, J. London Math. Soc. 56 (1997) 37–48.
[5] O. Martio, On harmonic quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A. I. Math. 425 (1968) 1–9.
[6] D. Partyka, The generalized Neumann–Poincare operator and its spectrum, Dissertationes Math. No. 366, Institute
of Mathematics, Polish Academy of Sciences, Warszawa, 1997.
[7] D. Partyka, K. Sakan, A note on non-quasiconformal harmonic extensions, Bull. Soc. Sci. Lettres Lodz 47 (1997)
51– 63; Serie: Recherches sur les deformations 23.
[8] I.I. Priwalow, Randeigenschaften analytischer Funktionen, VEB Deutscher Verlag der Wissenschaften, Berlin, 1956.
[9] W. Rudin, Real and Complex Analysis, McGraw-Hill series in higher mathematics, McGraw-Hill, New York, 1966.
[10] S. Yang, Harmonic extensions and extremal quasiconformal extensions, Hunan Univ. J., in press.
[11] A. Zygmund, Trigonometric Series, vol. 1, Cambridge University Press, Cambridge, 1968.
Quasiconformality of harmonic extensions
Dariusz Partykaa; ∗; 1 , Ken-Ichi Sakanb;2; 3
a
The Catholic University of Lublin, Institute of Mathematics, Al. Raclawickie 14, P.O. Box 129,
20-950 Lublin, Poland
b
Department of Mathematics, Graduate School of Science, Osaka City University, Sugimoto, Sumiyoshi-ku,
Osaka, 558, Japan
Received 16 October 1997; received in revised form 14 April 1998
Dedicated to Professor Haakon Waadeland on the occasion of his 70th birthday
Abstract
Continued from Partyka and Sakan (Bull. Soc. Sci. Letters Lodz 47 (1997) 51–63) this paper aims at giving necessary
and sucient conditions on sense-preserving homeomorphisms of the unit circle for the quasiconformality of their harmonic
extensions to the unit disk. In particular, all such homeomorphisms with a bounded derivative are well characterized. In
c 1999 Elsevier Science B.V. All rights reserved.
consequence, a generalization of Martio’s result is obtained.
MSC: 30C55; 30C62
Keywords: Harmonic mapping in the plane; Quasiconformal mapping in the plane; Poisson integral
1. Introduction
Let Hom+ (T ) stand for the class of all sense-preserving homeomorphic self-mappings
of the
unit circle T :={z ∈ C: |z| = 1} and let L1 (T ) be the space of all complex-valued functions Lebesgue
integrable on T . According to the famous Rado–Kneser–Choquet theorem (cf. e.g. [1, p. 22])
the Poisson extension P[
] of
∈ Hom+ (T ) to the unit disk :={z ∈ C: |z|¡1} is a harmonic and
homeomorphic self-mapping of , where for every f ∈ L1 (T ),
1
P[f](z ):=
2
Z
T
f(u)Re
u+z
|d u|;
u−z
z ∈ :
∗
Corresponding author.
E-mail address: [email protected] (D. Partyka).
1
Supported by KBN (Scienti c Research Council) grant No. 2 PO3 A 002 08.
2
Partially supported by Grant-in-Aid for Scienti c Research No. 08304014, Ministry of Education, Japan.
3
E-mail: [email protected].
c 1999 Elsevier Science B.V. All rights reserved.
0377-0427/99/$ - see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 9 ) 0 0 0 0 9 - 6
(1.1)
426
D. Partyka, K.-I. Sakan / Journal of Computational and Applied Mathematics 105 (1999) 425–436
[
]|2 is positive on
Moreover, P[
] is sense-preserving, i.e. the Jacobian J[P[
]] = |@P[
]|2 − |@P
1
1
, where @:= 2 ((@=@x) − i(@=@y)) and @:= 2 ((@=@x) + i(@=@y)) are the so-called formal derivatives
operators; cf. e.g. [6, pp. 42– 43]. For K¿1 let QT (K ) be the class of Sall
∈ Hom+ (T ) which
admit a K -quasiconformal (K -qc. for short) extension to , and let QT := K¿1 QT (K ). Obviously,
by de nition any
∈ Hom+ (T ) belongs to QT provided P[
] is a qc. mapping. Thus the natural
question is whether the Poisson extension P[
] is qc. provided
∈ QT . The answer is negative. Let
Q∗T denote the class of all
∈ QT such that P[
] is a qc. mapping. Yang has shown in [10] that
Q∗T 6= QT . Next, Laugesen improved this by showing (cf. [4, Corollary 3]) that for each K¿1 there
exists
∈ QT (K )\Q∗T . Several very simple and explicit examples of
∈ QT (K )\Q∗T for each K¿1
were presented in [7, Examples 4:1–4:3]. The reader can nd in [7] general simple techniques of
constructing for each K¿1 an explicit homeomorphism
∈ QT (K )\Q∗T . This shows that the class
Q∗T is substantially smaller than QT . However, the question how big is the class Q∗T is still open.
In other words, the problem of characterizing homeomorphisms
∈ Q∗T arises. So far as the authors
know, Martio was the rst who studied this problem provided
∈ Hom+ (T ) is suciently smooth;
cf. [5]. In what follows, we proceed with the study of the problem for irregular
. In fact, our paper
is a continuation of [7]. In Section 2 we derive some lower estimate of the radial limiting values
of the Jacobian J[P[
]] on the boundary T . Then as an application we present several necessary
and sucient conditions for
∈ Hom+ (T ) to belong to Q∗T . Section 3 deals with the general case
of arbitrary
∈ Hom+ (T ). In Section 4 we restrict our attention to
∈ Hom+ (T) with a bounded
derivative, in particular to
∈ Hom+ (T) being a Lipschitz function. We show that a homeomorphism
with bounded derivative has a qc. harmonic extension if and only if its derivative is bounded away
from zero and the Cauchy–Stieltjes transform of its di erential is bounded; see Theorem 4.1. In
Section 5 we give a few examples and compare our results with Martio’s theorem; cf. [5, Theorem
1]. These results were presented by the rst named author on the conference “Continued Fractions
and Geometric Function Theory”, Trondheim (Norway), June 24 –28, 1997. In general, we adopt all
notations from our paper [7]. Thus for a function F : → C (resp. F : C\ → C) and z ∈ T we
de ne
−
@ˆr F (z ):= lim− F (tz )
t→1
+
(resp: @ˆr F (z ):= lim+ F (tz ));
t→1
+
−
whenever the limit exists, while @ˆr F (z ):=0 (resp. @ˆr F (z ):=0) otherwise.
−
2. A lower estimate of @ˆr J[P[
]]
Given a function f : T → C and z ∈ T we de ne
f(u) − f(z )
T∋u→z
u−z
f′ (z ):= lim
provided the limit exists, while f′ (z ):=0 otherwise. If the limit exists we say that f has the
derivative f′ (z ) at z . As in [7], for any
∈ Hom+ (T ) we can consider the Riemann–Stieltjes integral
C
(z ):=
1
2 i
Z
T
d
(u)
;
u−z
z ∈ ∪ (C\ )
and
C
(∞):=0:
D. Partyka, K.-I. Sakan / Journal of Computational and Applied Mathematics 105 (1999) 425–436
427
[
] have nonLemma 2.1. If
∈ Hom+ (T ) then for a.e. z ∈ T; both the functions @P[
] and @P
tangential limiting values at z and
z
d
(z ) − P[
](rz )
z
′
′
P[
](rz ) + z
(z ) = lim−
lim @P[
](rz ) = lim−
+ z
(z )
(2.1)
r→1−
2 r→1 d r
2 r→1
1−r
and
z
d
(z ) − P[
](rz )
z
′
′
P[
](rz ) − z
(z ) = lim−
− z
(z ) :
(2.2)
lim @P[
](rz ) = lim−
r→1−
2 r→1 d r
2 r→1
1−r
Proof. As is remarked just below (3.9), C
has nontangential limiting values a.e. on T . Hence by
[
] have nontangential limiting values a.e. on T , as well.
(3.1) both the functions @P[
] and @P
From (1.1) it follows that for every r ∈ (0; 1) and for every z ∈ T ,
1
@P[
](rz ) =
2
=−
u
1 1
(u)
|d u| = −
2
(u − rz )
2 2irz
T
Z
1 1
2 2irz
=
=
(eit )
0
"Z
1 1
=−
2 2irz
1 1
=
2 2irz
2
Z
2
0
"Z
2
0
1 1
2 2irz
Z
2
1 1
2 2irz
Z
2
eit + rz
eit − rz
Re
Re
2
(eit )
0
−2irz eit
dt
(eit − rz )2
dt
d
eit + rz
(eit ) Re it
dt + i
dt
e − rz
eit + rz
Re it
d
(eit ) + ir
e − rz
0
0
d
dt
!
Z
Z
2
0
d
eit + rz
(eit ) Im it
dt
d t e − rz
2z eit
(e )Re it
dt
(e − rz )2
2
it
0
eit + rz
1 1
d
(eit ) +
it
e − rz
2 2z
Z
Z
2
(eit )
0
#
#
@
eit + rz
Re it
dt
@r
e − rz
eit + rz
1 d
P[
](rz ):
d
(eit ) +
it
e − rz
2z d r
(2.3)
Assume
has the derivative
′ (z ) at z = ei ∈ T . Then (d = d t )
(eit )|t= = iz
′ (z ) and by the Fatou
theorem on Poisson integrals (cf. [9, Theorem 11.12]) we have
2
1 1
eit + rz
1
Re it
d
(eit ) =
′ (z ):
(2.4)
r→1 2 2irz 0
e − rz
2
By (2.3) and (2.4), the limit limr→1− @P[
](rz ) exists i the limit limr→1− (1= 2z )(d = d r )P[
](rz )
exists. As shown in the proof of [7, Theorem 1.1] both the radial limits
lim−
Z
−
lim− @P[
](rz ) = @ˆr C
(z ) and
r→1
+
[
](rz ) = z 2 @ˆr C
(z )
lim− @P
r→1
(2.5)
exist for a.e. z ∈ T . Since
has the derivative
′ (z ) for a.e. z ∈ T , we conclude from (2.3), (2.4)
and (2.5) that the rst equality in (2.1) holds. If for z ∈ T the limit limr→1− (d = d r )P[
](rz ) exists,
then
Z 1
d
1
d
(z ) − P[
](rz )
= lim−
P[
](tz ) d t = lim− P[
](rz ):
lim
r→1
r→1
r→1−
1−r
1 − r r dt
dr
428
D. Partyka, K.-I. Sakan / Journal of Computational and Applied Mathematics 105 (1999) 425–436
This yields the second equality in (2.1). By (2.5), we can rewrite [7, (1.8)] as
[
](rz )) =
′ (z ) for a:e: z ∈ T:
lim− (@P[
](rz ) − z2 @P
r→1
Combining this with (2.1) we obtain (2.2).
−
Given
∈ Hom+ (T ) de ne d
:=ess inf z∈T |
′ (z )| and j
:=ess inf z∈T @ˆr J[P[
]](z ). We use Lemma
2.1 to nd a lower estimate of the functional j
. The following theorem is in fact a generalization
of Martio’s [5, Lemma 3].
Theorem 2.2. If
∈ Hom+ (T ) then for a.e. z ∈ T; the Jacobian J[P[
]] has a nontangential limiting value at z and
+
−
1 − |F
(0)|
1
;
lim J[P[
]](rz ) = |@ˆr C
(z )|2 − |@ˆr C
(z )|2 ¿ |
′ (z )|
2
1 + |F
(0)|
(2.6)
r→1−
where F
(0) is the point in which is uniquely determined by P[
](F
(0)) = 0. In particular;
1 1 − |F
(0)|
j
¿ d
:
2 1 + |F
(0)|
(2.7)
Proof. It is easy to check that for each r ∈ [0; 1) and u; z ∈ T ,
1 1
u + rz
1 1+r
1 1+r
1
¿
¿ :
Re
=
2
2
1 − r 2 u − rz 2 |u − rz|
2 (1 + r )
4
(2.8)
Given a ∈ write ha (u):=(u − a)= (1 − au
) for u ∈ C\{1= a}
and ha (1= a):=∞. Set a:= − F
(0) and
R
R 2 ˆ
:=
◦ ha . Then P[
◦ ha ](0) = P[
](ha (0)) = P[
](F
(0)) = 0, and so 0 = T (u)|d u| = 0 ei(t)
dt =
R 2
R 2
ˆ(t ) d t + i 0 sinˆ(t ) d t , where ˆ is the angular parametrization of ; cf. (3.3). Thus for
0 cos
R 2
every ∈ R, 0 cos(ˆ(t ) − ˆ()) d t = 0. From this, (2.8) and Lemma 2.1 we see, following the
proof of [5, Lemma 3], that for a.e. z = ei ∈ T ,
[](rz )|2 )
lim J[P[]](rz ) = lim− (|@P[](rz )|2 − |@P
r→1−
r→1
= lim− Re
r→1
(z ) − P[](rz ) ′
z (z )
1−r
= lim− |′ (z )|Re (z )
r→1
= lim− |′ (z )|
r→1
1 1
1 − r 2
(z ) − P[](rz )
1−r
Z
0
2
[1 − cos(ˆ(t ) − ˆ())]Re
eit + rz
dt
eit − rz
¿ 21 |′ (z )| = 12 |(
◦ ha )′ (z )| = 12 |
′ (ha (z ))||h′a (z )|:
(2.9)
Since P[] = P[
◦ ha ] = P[
] ◦ ha , (2.9) yields
lim J[P[
]](ha (rz ))|h′a (rz )|2 = lim− J[P[
] ◦ ha ](rz ) = 21 |
′ (ha (z ))||h′a (z )|:
r→1−
r→1
(2.10)
D. Partyka, K.-I. Sakan / Journal of Computational and Applied Mathematics 105 (1999) 425–436
429
From Lemma 2.1 it follows that for a.e. z ∈ T , the Jacobian J[P[
]] has a nontangential limiting
value at z , and (2.10) implies
1
1 − |F
(0)|
1
lim J[P[
]](rz )¿ |
′ (z )| inf |h′a (u)|−1 = |
′ (z )|
:
u∈T
2
2
1 + |F
(0)|
r→1−
Combining this with (2.5) we obtain (2.6). Moreover, by de nition
−
j
= ess inf @ˆr J[P[
]](z )¿ ess inf
z∈T
z∈T
1 − |F
(0)|
1 ′
|
(z )|
2
1 + |F
(0)|
!
1 1 − |F
(0)|
;
= d
2 1 + |F
(0)|
which proves (2.7).
Remark 2.3. It can be shown that j
¿d3
. However, the proof exceeds the scope of this paper and
will be published elsewhere.
3. The main results
Let
∈ Hom+ (T ). For the convenience of readers, we recall that for each z ∈ ,
1
@P[
](z ) =
2i
Z
T
d
(u)
= C
(z ) and
u−z
[
](z ) = 1 C
1 ;
@P
z
z2
(3.1)
cf. [7, (1.3) and (1.4)]. Moreover, for z ∈ \{0}
C
(z ) − C
1
1
=
z
2
Z
0
2
ei + z
Re i
e −z
!
d
˜()
(3.2)
(cf. [7, (1.7)], where
˜():=
Z
0
d
(eit )
;
ieit
∈ [0; 2]:
(3.3)
Following the proof of [7, Theorem 3.1] we rst show
Theorem 3.1. Suppose that
∈ Hom+ (T ) and that there exists a sequence n ∈ T; n ∈ N; such that
the derivative
′ (n ) = limu→n (
(u) −
(n ))= (u − n ) exists for each n ∈ N and
lim
′ (n ) = 0:
(3.4)
n→∞
Then P[
] is not a qc. mapping. In particular; if d
= 0 then P[
] is not a qc. mapping.
Proof. Fix n ∈ N. By the Fatou theorem on Poisson integrals (cf. [9, Theorem 11.12]), the Poisson–
Stieltjes integral in (3.2) has the nontangential limit
˜′ (’) =
′ (ei’ ) for each ’ ∈ [0; 2] such that
the derivative
˜′ (’) exists; cf. [2, pp. 4 –5]. Therefore, by (3.2) we obtain
C
(tn ) − C
(1=tn ) →
′ (n )
as (0; 1) ∋ t → 1:
(3.5)
430
D. Partyka, K.-I. Sakan / Journal of Computational and Applied Mathematics 105 (1999) 425–436
As shown in [7, (1.10)], 2|@P[
]()|2 ¿c(
)¿0 on , where c(
) is a positive constant from [7,
Theorem 1.1]. Hence by (3.1) we see that for 0¡t¡1
2
@P
[
](tn )
tn @P[
](tn )
061−
61 −
@P[
](tn )
@P[
](tn )
√
|C
(tn ) − C
(1=tn )|
2|C
(tn ) − C
(1=tn )|
√
6
6
:
|@P[
](tn )|
c(
)
Combining this with (3.5) and (3.4) we obtain
@P
[
](z )
sup
¿ lim inf
t→1−
z∈ @P[
](z )
s
@P
2 ′
[
](tn )
|
(n )| → 1
¿1 −
@P[
](tn )
c(
)
as n → ∞;
and consequently P[
] is not a qc. mapping.
If d
= 0, then obviously there exists a sequence n ∈ T , n ∈ N, such that
has the derivative at
each n and such that (3.4) is satis ed. Thus P[
] is not a qc. mapping.
We write L∞ (T ) for the class of all measurable functions f : T → C essentially bounded on T ,
i.e. ||f||∞ :=ess supz∈T |f(z )|¡∞. As a conclusion from Theorems 2.2 and 3.1 we derive a useful
sucient condition of quasiconformality of P[
].
−
Theorem 3.2. Suppose that
∈ Hom+ (T ). If d
¿0 and @ˆr C
∈ L∞ (T ); then
∈ Q∗T and
v
u
@P
j
[
](z ) u
:
sup
6t1 −
−
z∈ @P[
](z )
||@ˆr C
||2∞
(3.6)
Proof. From [7, Theorem 1.1] it follows that
ˆ+
@P
@
C
(
z
)
[
](
z
)
r
sup
= ess sup −
z∈T
z∈ @P[
](z )
@ˆ C (z )
r
and
(3.7)
−
|@ˆr C
(u)|2 ¿c(
)= 2¿0;
(3.8)
for a.e. u ∈ T , where c(
) is a constant. As shown in the proof of [7, Theorem 1.1] both the radial
−
+
limits @ˆr C
(u) and @ˆr C
(u) exist for a.e. u ∈ T . Hence Theorem 2.2 shows, by (3.1), that for a.e.
u∈T
+
−
[
](tu)|2 ) = lim J[P[
]](tu)¿j
:
|@ˆr C
(u)|2 − |@ˆr C
(u)|2 = lim− (|@P[
](tu)|2 − |@P
−
t→1
t→1
Applying now (3.8) we have
2
ˆ+
@r C
(u)
j
¿
1− −
−
@ˆr C
(z )
|@ˆr C
(u)|2
for a:e: u ∈ T:
D. Partyka, K.-I. Sakan / Journal of Computational and Applied Mathematics 105 (1999) 425–436
431
By the assumptions and (2.7),
2
ˆ+
@r C
(u)
j
61 −
¡1:
ess sup −
−
ˆ
ˆ
u∈T
@r C
(z )
||@r C
||2∞
From this and (3.7) it follows that
∈ Q∗T and (3.6) holds.
From [7, (1.6)] it follows that for z ∈ C\T , C
(z ) is represented by the Cauchy–Stieltjes type
integral of
˜ de ned by (3.3),
u d
(u)
1 2 ei
1
=
d
˜():
(3.9)
C
(z ) =
2 T u − z i u
2 0 ei − z
For every function f : [0; 2] → C of bounded variation write CT [d f] for the Cauchy–Stieltjes type
singular integral, i.e. for every z = eix ∈ T
Z
Z
ei
1 2 ei
1
CT [d f](z ):=PV
d
f
(
):=
lim
d f()
→0+ 2 ¡|−x|6 ei − z
2 0 ei − z
whenever the limit exists and CT [d f](z ):=0 otherwise.
By Smirnov theorem [8, p. 65] and (3.9) the function C
belongs to the Hardy class H p () for
arbitrary p ∈ (0; 1). Hence C
has nontangential limiting values a.e. on T ; cf. [8, p. 56]. Then the
classical Privalov’s Theorem (cf. [8, p. 135]) says that the singular integral exists a.e. on T and
the formulas
−
+
@ˆr C
(u) = 1
˜′ (x) + CT [d
˜](u) and @ˆr C
(u) = − 1
˜′ (x) + CT [d
˜](u)
Z
Z
2
2
′
ix
′
ix
hold for a.e. u = e ∈ T . Since
˜ (x) =
(e ) for a.e. x ∈ [0; 2] we have
−
@ˆr C
= 12
′ + CT [d
˜]
and
+
@ˆr C
= − 12
′ + CT [d
˜]
a:e: on T:
(3.10)
Applying the equalities (3.10) we can extend [7, Theorem 1.2 and Corollary 2.1] to an arbitrary
homeomorphism
∈ Hom+ (T ) with CT (
′ ) replaced by CT [d
˜] as follows.
Theorem 3.3. If
∈ Hom+ (T ); then for a.e. z ∈ T
and
where
|
′ (z ) + 2CT [d
˜](z )|2 ¿2c(
)¿0
′
s
@P
(z ) − 2CT [d
˜](z )
[
](z )
= 1 − m
;
sup
= ess sup ′
@P
[
](
z
)
(
z
)
+
2
C
[d
˜
](
z
)
1 + m
z∈T
z∈
T
(3.11)
(3.12)
4Re[
′ (z )CT [d
˜](z )]
(3.13)
z∈T
|
′ (z )|2 + 4|CT [d
˜](z )|2
and c(
) is a constant from [7; Theorem 1:1]. In particular; m
¿0; and the mapping P[
] is qc.
i m
¿0.
m
:= ess inf
Proof. The theorem follows from [7, Theorem 1.1] and the equalities (3.10). The second equality
in (3.12) is obtained by simple computations.
432
D. Partyka, K.-I. Sakan / Journal of Computational and Applied Mathematics 105 (1999) 425–436
Corollary 3.4. Suppose that
∈ Hom+ (T ). Then
∈ Q∗T i d
¿0 and
CT [d
˜](z )
CT [d
˜](z )
¡∞:
0¡ ess inf Re
6 ess sup
′
z∈T
(z )
′ (z )
z∈T
(3.14)
Proof. Assume P[
] is a qc. mapping. Theorem 3.1 then implies that d
¿0. From Theorem 3.3 it
follows that m
¿0. Hence (3.14) follows from (3.13). Conversely, if d
¿0 and (3.14) holds then
from (3.13) it follows that m
¿0. Applying Theorem 3.3 we see that
∈ Q∗T as claimed.
Remark 3.5. If
′ (z ) 6= 0 for a.e. z ∈ T and (3.14) holds, then from (3.11) it follows that d
¿0.
Thus in Corollary 3.4 the condition d
¿0 may be replaced by the condition that
′ (z ) 6= 0 for a.e.
z ∈ T.
4. The case of homeomorphisms with a bounded derivative
In this section, we discuss the case where
∈ Hom+ (T ) has a bounded derivative. Our most
essential result is
Theorem 4.1. Suppose that
∈ Hom+ (T ) satis es
′ ∈ L∞ (T ). Then
∈ Q∗T i d
¿0 and CT [d
˜] ∈
L∞ (T ).
Proof. Assume
∈ Q∗T . Corollary 3.4 shows that d
¿0 and by (3.14),
||CT [d
˜]||∞ 6||
′ ||∞ ||CT [d
˜]=
′ ||∞ ¡∞;
so that CT [d
˜] ∈ L∞ (T ). Conversely, if
′ ∈ L∞ (T ) and CT [d
˜] ∈ L∞ (T ); then by (3.10) we obtain
−
@ˆr C
=(1= 2)
′ +CT [d
˜] ∈ L∞ (T ). Since d
¿0 we conclude from Theorem 3.2 that
∈ Q∗T as claimed.
We now turn to the case where
∈ Hom+ (T ) is absolutely continuous. For every f ∈ L1 (T ) de ne
Z
1
f(u)|d u|
fT :=
2 T
and write CT (f) for the Cauchy singular integral, i.e. for every z ∈ T
Z
Z
f(u)
f (u )
1
1
CT (f)(z ):=PV
d u:= lim+
d u;
→0
2i T u − z
2i T\T(z;) u − z
whenever the limit exists and CT (f)(z ):=0 otherwise, where T (eix ; ):={eit ∈ T : |t − x|¡}. We
recall that the harmonic conjugation operator A is de ned by the singular integral
Z
x−t
1
lim+
f(eit )cot
d t; z = eix ∈ T;
A(f)(z ):=
2 →0 ¡|t−x|6
2
whenever the limit exists and AR(f)(z ):=0 otherwise; cf. e.g. [3, 11]. If
∈ Hom+ (T ) is absolutely
continuous then
′ ∈ L1 (T ) and T |
′ (u)||d u| = 2. Moreover, by [7, (1.14)] we have for a.e. z ∈ T
CT [d
˜](z ) = CT (
′ )(z ) = 21
′T + 2i A(
′ )(z ):
(4.1)
D. Partyka, K.-I. Sakan / Journal of Computational and Applied Mathematics 105 (1999) 425–436
433
Applying now Theorem 4.1 to Lipschitz functions we obtain
Corollary 4.2. Suppose that
∈ Hom+ (T ) is a Lipschitz function; i.e. there exists a constant L¿0
such that
|
(u) −
(w)|6L|u − w|;
u; w ∈ T:
(4.2)
Then
∈ Q∗T i d
¿0 and CT (
′ ) ∈ L∞ (T ).
The same is true with CT (
′ ) replaced by A(
′ ).
Proof. Since
∈ Hom+ (T ) satis es (4.2), it follows that
is absolutely continuous and
′ ∈ L∞ (T )
with ||
′ ||∞ 6L. Therefore, the corollary follows from Theorem 4.1 and (4.1).
Each
∈ Hom+ (T ) de nes a unique continuous function
ˆ satisfying 06
ˆ(0)¡2 and
ˆ
(eit ) = ei
(t)
;
t ∈ R:
(4.3)
Actually,
ˆ is an increasing homeomorphism of R onto itself satisfying
ˆ(t + 2) −
ˆ(t ) = 2 for
t ∈ R; called the angular parametrization of
. Sometimes it is more convenient to use the real-valued
function |
′ | instead of
′ . To this end we prove.
Corollary 4.3. Suppose that
∈ Hom+ (T ) is a Lipschitz function. Then
∈ Q∗T i d
¿0 and
A(|
′ |) ∈ L∞ (T ).
ˆ
Proof. From (4.3) it follows that for a.e. t ∈ R;
′ (eit ) =
ˆ′ (t )ei(
(t)−t)
. Hence for a.e. x ∈ R;
ˆ
|A(
′ )(eix ) − ei(
(x)−x)
A(|
′ |)(eix )|
Z
Z
1
= lim+
2 →0
1
= lim+
2 →0
¡|t−x|6
Z
||
′ ||∞
2
Z
x+
||
′ ||∞
2
Z
x+
′
Z
x+
6
6
6
||
||∞
2
x+
x−
x−
′
it
i(
(x)−x)
ˆ
|
(e )|(e
i(
(t)−t)
ˆ
x − t
|
(e )|)cot
dt
2
′
−e
it
i(
(x)−x)
ˆ
|x − t|
ei(
(t)−t)
ˆ
ˆ
− ei(
(x)−x)
dt
|sin x−t |
x−t
x − t
dt
) cot
2
2
x−
x−
it
(
(e ) − e
¡|t−x|6
||
′ ||∞
6
2
′
!
eit − eix ei
(t)
ˆ
ˆ
− ei
(x)
+
dt
t−x
t−x
ˆ(t ) −
ˆ(x)
dt
x−t
1 +
1+
1
|t − x|
6||
′ ||∞ (1 + ||
′ ||∞ )¡∞:
Z
x
t
ˆ′ (s)d s d t
Therefore, A(
′ ) ∈ L∞ (T ) i A(|
′ |) ∈ L∞ (T ); and the corollary follows from Corollary 4.2.
434
D. Partyka, K.-I. Sakan / Journal of Computational and Applied Mathematics 105 (1999) 425–436
Following the proof of [7, Corollary 3.5] we easily obtain a version of Corollary 4.2 in terms of
a variant H of the Hilbert transform de ned for every locally integrable function f : R → C and
every x ∈ R by the singular integral
H(f)(x):=
1
lim
→0+
Z
¡|t−x|6
f (t )
d t;
x−t
whenever the limit exists and H(f)(x):=0 otherwise.
Corollary 4.4. Suppose that
∈ Hom+ (T ) and
ˆ is absolutely continuous on R satisfying
ess supt∈R
ˆ′ (t )¡∞. Then
∈ Q∗T i ess inf t∈R
ˆ′ (t )¿0 and ess supt∈R |H(
ˆ′ )(t )|¡∞.
We end this section with the following corollary, which points out a relation with the Hardy class
H ∞ () of bounded analytic functions on .
Corollary 4.5. Suppose
∈ Hom+ (T ) is a Lipschitz function. Then P[
] is a qc. mapping i there
exists F ∈ H ∞ () such that
−
@ˆr Re F (z ) = |
′ (z )|¿d
¿0
for a:e: z ∈ T:
(4.4)
Proof. Assume P[
] is a qc. mapping. Corollary 4.3 then shows that d
¿0 and ess supz∈T |A(|
′ |)(z )|
¡∞. By [3, p. 103], P[A(|
′ |)] is identical with the harmonic conjugate function v of P[|
′ |] satisfying v(0)=0. Thus F :=P[|
′ | +iA(|
′ |)] is an analytic function on . Since |
′ | +iA(|
′ |) ∈ L∞ (T );
we conclude from [2, Corollary 2 to Theorem 3.1] that F ∈ H ∞ (). By Fatou’s theorem on
−
Poisson integrals (cf. [9, Theorem 11.12]), the radial limit @ˆr Re F = |
′ | a.e. on T; which ends
the proof in the rst direction. Assume now that (4.4) holds for some F ∈ H ∞ (). A classi−
−
cal result (cf. e.g. [3, p. 103]) states that @ˆr Im F = Im F (0) + A(@ˆr Re F ) a.e. on T . Hence
ess supz∈T |A(|
′ |)(z )|6|F (0)| + supz∈ |F (z )|¡∞; and P[
] is qc. by Corollary 4.3.
5. Comparison between our results and Martio’s theorem
In this section, we compare our results with Martio’s theorem; cf. [5, Theorem 1]. We show that
Theorem 4.1 and Corollaries 4:2–4:5 essentially extend Martio’s theorem.
If
∈ Hom+ (T ) ∩ C 1 (T ) and |
′ | is Dini continuous on T; then a classical result (cf. e.g. [3,
p. 106]) shows that the function A(|
′ |) is continuous on T; and hence bounded. Thus Corollary
4.3 leads to
Corollary 5.1 (Martio [5]). Suppose
∈ Hom+ (T ) ∩ C 1 (T ). If |
′ | is Dini continuous on T; then
P[
] is qc. i d
¿0.
Let Di + (T ) denote the class of all sense-preserving di eomorphic self-mappings of T . For
0¡ 61 write C 1+ (T ) for the class of all complex-valued functions continuously di erentiable on
T; whose derivatives are -Holder continuous functions on T .
D. Partyka, K.-I. Sakan / Journal of Computational and Applied Mathematics 105 (1999) 425–436
Corollary 5.2. For each ∈ (0; 1];
Di + (T ) ∩ C 1+ (T ) ⊂ Q∗T :
435
(5.1)
Moreover;
Di + (T )\Q∗T 6= ∅ and
Q∗T \Di + (T ) 6= ∅:
(5.2)
Proof. Fix
∈ Di + (T ). By the de nition of a di eomorphism, d
¿0. For any ∈ (0; 1]; if
∈ C 1+
(T ) then |
′ | is Dini continuous on T; and the inclusion (5.1) follows from Corollary 5.1.
R Let F de∞
note the class of all real-valued functions f ∈ L (T ) such that ess inf z∈T f(z )¿0 and T f(u)|d u| =
2. Each function f ∈ F de nes a homeomorphism
f ∈ Hom+ (T ) whose angular parametrization
Rx
it
ˆf is determined by the equality
ˆf (x) = 0 f(e ) d t for x ∈ R. Obviously,
ˆf is a Lipschitz function
on R; and so is
f on T . Moreover, d
f ¿0. It is a simple matter to construct a continuous function
f ∈ F such that A(f) is unbounded; see Example 5.3. Then
f ∈ Di + (T ) and A(|
′f |) = A(f)
is unbounded. From Corollary 4.3 it follows that P[
f ] is not a qc. mapping, i.e.
f 6∈ Q∗T ; which
proves the rst assertion in (5.2).
−
It is easy to nd F ∈ H ∞ () such that f:=@ˆr Re F ∈ F but |
′f | is not continuous on T ; see
Example 5.4. Then
f 6∈ Di + (T ) and Corollary 4.5 shows that
f ∈ Q∗T ; which proves the second
assertion in (5.2).
Example 5.3. For every x ∈ R de ne
if x¿1=e;
1
p(x):=
−1= log x if 0¡x¡1=e;
0
if x60:
Since p ∈ C 1 (R\{0; 1=e}); H(p)(x) exists for every x ∈ R\{0; 1=e}. Moreover, p(t ) = 0 if t60 and
p(t )¿0 if t¿0. Then for each x ∈ (−1=e; 0) we have
Z
Z 1=e
Z
x+
p (t )
p(t )
p (t )
|H(p)(x)| = lim+
dt =
d t¿
dt
→0 ¡|t−x|6 x − t
−x=2 t − x
x− t − x
=
Z
1=e
t
1
−
t−x
t log t
−x=2
1
=−
3
Z
−1
log|x=2|
1
d t¿
3
Z
1=e
−x=2
−
dt
t log t
x
du 1
= log log → ∞;
u
3
2
(u = log t )
as x → 0− :
(5.3)
It is easily seen that there existsR a function q ∈ C 1 (R) such that (p + q)(−) = (p + q)()¿0 and
min−6t6 (p + q)(t )¿0. Set c:= − (p + q)(t ) d t¿0; and de ne f(eit ) = (2=c)(p(t ) + q(t )); −6t
6. Obviously, f ∈ F ∩ C (T ); and thus
f is a Lipschitz function on T . Since q ∈ C 1 (R); H(q)
is continuous on R; and thus bounded on [ − ; ]. Hence by (5.3) we see that
ess sup |H(
ˆ′f )(x)| = ∞:
(5.4)
x∈R
As in the proof of [7, Corollary 3.5], we can check that (5.4) implies
||A(
′f )||∞ = ∞:
(5.5)
436
D. Partyka, K.-I. Sakan / Journal of Computational and Applied Mathematics 105 (1999) 425–436
As shown in the proof of Corollary 4.3, we see that (5.5) implies ||A(f)||∞ = ||A(|
′f |)||∞ = ∞.
Example 5.4. For z ∈ de ne G (z ):=exp(−(1 + z )= (1 − z )). Clearly
|G (z )| = exp −Re
1+z
6e0 = 1;
1−z
so that G ∈ H ∞ (). Let c:=
R 2
0
−
z ∈ ;
(5.6)
2 + @ˆr Re G (eit ) d t¿0; and de ne F (z ):=(2=c)[2 + G (z )]; z ∈ .
−
By (5.6), F ∈ H () and f:=@ˆr Re F ∈ F. Moreover, if eix 6= 1; then
ˆ′f (x) = f(eix ) = (2=c)(2 +
cos(−cot (x= 2))). Hence
ˆ′f is discontinuous at x = 0; and so the function |
′f | is not continuous at
1 ∈ T.
∞
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