on regularity of quasiconformal curves

Computational Methods and Function TheoryVolume 2 (2002), No. 2, 367–384

On Regularity of Quasiconformal Curves

Vladimir Gutlyanskiı and Olli Martio

(Communicated by Frederick W. Gehring)

Abstract. Regularity problems of a plane quasiconformal mapping f wherecomplex dilatation is close to zero are studied. Our approach is based on anew conformality result [11] which extends the classical Teichmuller-Wittich-Belinski Regularity Theorem.

Keywords. Quasiconformal curves, regularity of quasiconformal mappings.

2000 MSC. 30C55, 30C60.

1. Introduction and main theorems

We study regularity problems of a plane quasiconformal mapping f on a set Mwhere the complex dilatation µ of f is close to zero in a neighborhood of M.Our approach is based on the new conformality result [11] which extends theclassical Teichmuller-Wittich-Belinski Regularity Theorem [16], [18], [4], see also[13, p. 240]. The regularity problems have been extensively studied and theyhave found many applications, see, e.g. [2], [3], [8] – [10].

The first problem concerns the smoothness of the boundary correspondence ofa quasiconformal homeomorphism f of the open upper half-plane H onto itself.A. Beurling and L. Ahlfors [5] have shown that f admits an extension to ahomeomorphism of the closed half-plane and generates a quasisymmetric self-mapping f : R → R which needs not be absolutely continuous on the real axis.In [7] L. Carleson proved that the condition

(1)

∫ 1

0

η(t)

tdt <∞, η(t) = ess sup

0<Im z<t|µ(z)|,

implies that the boundary correspondence x 7→ f(x) has a continuous derivativef ′(x). The following theorem is an extension of the Carleson result, see also [2],[3] [6], [12] and references therein.

Received April 28, 2002.

ISSN 1617-9447/$ 2.50 c© 2002 Heldermann Verlag

http://www.cmft.de

368 V. Gutlyanskiı and O. Martio CMFT

Theorem 1.1. If for some R > 0 and t ∈ R

(2)

∫ ∫|z|<RIm z>0

|µ(z + t)|2

|z|2dxdy <∞

and if

(3) limr→0

∫ ∫r<|z|<RIm z>0

Reµ(z + t)

z2dxdy

exists, then the boundary correspondence x 7→ f(x) has nonvanishing derivativef ′(t). If the above integrals converge locally uniformly for every t ∈ R, then f ′(x)is continuous.

Corollary 1.2. If the Teichmuller-Wittich-Belinski integral

(4)

∫ ∫|z|<1

Im z>0

|µ(z + t)||z|2

dxdy

converges locally uniformly for every t ∈ R, then the boundary correspondencex 7→ f(x) is continuously differentiable.

Remark 1.3. 1) Assume that the monotonic majorant η(t) of µ given by (1)satisfies the Dini condition ∫ 1

0

η(t)

tdt ≤M.

Then the elementary estimate∫ ∫|z|<1

Im z>0

|µ(z + t)||z|2

dxdy ≤ πM

implies that condition (2) holds locally uniformly with respect to t ∈ R and theimproper integral ∫ ∫

|z|<1Im z>0

Reµ(z + t)

z2dxdy

converges absolutely and locally uniformly for t ∈ R. By Theorem 1.1 we arriveat Carleson’s smoothness result mentioned earlier.

2) An example of a mapping f which satisfies the pointwise assumptions ofTheorem 1.1 and fails to satisfy the pointwise TWB-condition∫ ∫

|z|<1Im z>0

|µ(z + t)||z|2

dxdy <∞, t ∈ R,

is given in Section 3.

2 (2002), No. 2 On Regularity of Quasiconformal Curves 369

The second problem deals with geometric properties of quasicircles. Recall thata Jordan curve C ⊂ C is called a quasicircle if it is the image of the unit cir-cle T under a quasiconformal self-mapping f of C. In general, f(T) need not berectifiable, see, e.g. [13, p. 104] and [15, p. 249] and its Hausdorff dimension canbe arbitrarily close to 2. We recall also that a Jordan curve C is smooth if ithas a continuously turning tangent, i.e., the tangent angle β(s) is a continuousfunction of the arclength s.

A homeomorphism f is said to be conformally differentiable, or simply conformal,at a point z if f has a nonzero complex derivative at this point.

The following uniform version of the main conformality theorem in [11] pro-vides a sufficient condition for the quasicircle to be smooth and therefore locallyrectifiable.

Theorem 1.4. Let f be a quasiconformal self-mapping of the complex planewith complex dilatation µ and let M⊂ C be a compact set. If there are positiveconstants R and M such that

(5)

∫ ∫|z|<R

|µ(z + η)|2

|z|2dxdy ≤M

holds for each η ∈M and there exists a finite limit

(6) limr→0

∫ ∫r<|z|<R

µ(z + η)

z2dxdy

uniformly for η ∈ M, then the mapping f is conformally differentiable on Mand the complex derivative f ′(η) is continuous on M.

Corollary 1.5. If a quasiconformal mapping f : C→ C satisfies the assumptionsof Theorem 1.4 with M = T, then the quasicircle C = f(T) is smooth.

This result extends the corresponding statements in [2], [3] and [8].

Corollary 1.6 ([11]). If for some M > 0

(7)

∫ ∫|z|<1

|µ(z + η)||z|2

dxdy ≤M

for every η ∈ T, then the quasicircle C = f(T) is smooth.

In [9, p. 291], E. Dynkin asked for a condition on µ which guarantees the smooth-ness of f on the closed unit disk. Now Theorem 1.4 as well as Corollary 1.6provide such a condition.

The following theorem on the local behavior of a quasiconformal mapping maybe of independent interest.


Theorem 1.7. Let f be a quasiconformal self-mapping of the complex planewith complex dilatation µ and let M⊂ C be a compact set. If there are positiveconstants R and M such that (5) holds for each η ∈M and there exists a finitelimit

(8) limr→0

∫ ∫r<|z|<R

Reµ(z + η)

z2dxdy

uniformly for η ∈M, then

limz→η

|f(z)− f(η)||z − η|

= C(η) > 0

uniformly for η ∈M.

The function C(η) is estimated in terms of integrals (5) and (8) in Remark 3.1.

Remark 1.8. The assumption of Theorem 1.4 can be relaxed if

argf(z)− f(η)

z − ηhas a radial limit as z → η ∈ M. For example, if the quasicircle C = f(T) issmooth, then the conformality assertion of Theorem 1.4 holds if (6) is replacedby condition (8)

We apply Remark 1.8 to study boundary properties of conformal mappings inthe last section.

2. Auxiliary results

Let f : C→ C, f(0) = 0, be a homeomorphism. We write

Mf (r) = max|z|=r|f(z)|,

mf (r) = min|z|=r|f(z)|.

A mapping f : C→ C is called weakly conformal at the origin if

(9) limr→0

Mf (r)

mf (r)= 1

and if for an appropriate branch of the argument

(10) limr→0

[arg f(reiθ1)− arg f(reiθ2)− (θ1 − θ2)

]= 0

uniformly in θ1, θ2 ∈ [0, 2π]. The above conditions have the evident geometricinterpretation typical for conformal mappings.

In what follows we shall also use the notion of the uniform weak conformality of amapping f on a set. This means that conditions (9) and (10) hold for the family


of functions fη(z) = f(z + η)− f(η) uniformly with respect to the parameter ηbelonging to the given set.

The weak conformality conditions (9) and (10) imply the following simple butimportant statements “on unrestricted approach”. If f is uniformly weakly con-formal on a set M and for some τ = τ(η) there exists the uniform radial limit

(11) limr→0

|fη(reiτ(η))|r

= a(η),

then

(12) limz→0

|fη(z)||z|

= a(η),

uniformly with respect to η ∈M. Similarly, the uniform radial limit

(13) limr→0

arg fη(reiθ(η)) = α(η)

implies that

(14) limz→0

arg fη(z) = α(η)

uniformly with respect to η ∈M.

A sufficient condition for f to be uniformly weakly conformal on a set is providedby the following lemma.

Proposition 2.1. Let f be a quasiconformal self-mapping of the complex planewith complex dilatation µ satisfying

(15) limr→0

1

r2

∫ ∫|z|<r

|µ(z + η)|α dxdy = 0

for some α > 0 uniformly in η ∈ M. Then f is uniformly weakly conformalon M.

Proof. By the definition of the uniform weak conformality of the mapping fon the set M, we have to verify conditions (9) and (10) for the family fη =f(z + η)− f(η) at 0 when η ∈M.

Let FQ be the family of all Q-quasiconformal self-mappings f of C normalized bythe conditions f(0) = 0, f(1) = 1. The space FQ equipped with the topology oflocally uniform convergence is sequentially compact, see [13, p. 74]. Suppose nowthat (9) does not hold. Then there exist sequences ηn ∈M, rn = rn(ηn)→ 0 asn→∞ such that

(16) logMfηn (rn)

mfηn (rn)> ε


for some ε > 0 and all n. Let τn be a sequence of complex numbers such that|τn| = rn and mfηn (rn) = |fηn(τn)|. Consider the sequence of K-quasiconformalmappings

(17) Fn(z) =fηn(zτn)

fηn(τn)

of the space FQ with complex dilatations µFn(z) = µ(zτn + ηn). Setting r = Rrnin (15), we can write (15) in terms of µFn(z) as

limn→∞

∫ ∫|z|<R

|µFn(z)|α dxdy = 0.

Hence µFn → 0 in Lαloc(C), α > 0, and since |µ| < 1 a.e., without loss ofgenerality we can assume that simultaneously Fn(z) converges locally uniformlyin C to a quasiconformal mapping F ∈ FQ and that the sequence of their com-plex dilatations µFn(z) converges to 0 almost everywhere in C. Otherwise, onecan work with an appropriate subsequence. By the well-known Bers-BojarskiConvergence Theorem, see [13, p. 187] we conclude that F (z) = z. Let thesequence zn, |zn| = 1, be chosen in such a way that |fηn(znτn)| = Mfηn (rn).Since the unit circle is compact one can assume that zn → z0, |z0| = 1. Hencelog(Mfηn (rn)/mfηn (rn)) = log |Fn(zn)| → log |z0| = 0 as n → ∞ which contra-dicts (16).

Next, suppose that (10) fails. Then for some ε > 0 there exist sequences rn → 0,ηn ∈M, θ1,n and θ2,n such that

(18)∣∣arg fηn(rne

iθ2,n)− arg fηn(rneiθ1,n)− (θ2,n − θ1,n)

∣∣ > ε.

We can additionally assume that eiθj,n → eiθ∗j , j = 1, 2, as n→∞. Substituting

τn = rneiθ1,n in formula (17) and repeating the sequential arguments given above,

we conclude that Fn(z)→ z locally uniformly in C as n→∞. Thus

limn→∞

∣∣arg fηn(rneiθ2,n)− arg fηn(rne

iθ1,n)− (θ2,n − θ1,n)∣∣

= limn→∞

∣∣argFn(ei(θ2,n−θ1,n))− (θ2,n − θ1,n)∣∣

=∣∣argF (ei(θ

∗2−θ∗1))− (θ∗2 − θ∗1)

∣∣ = 0.

This contradicts (18). The lemma follows.

For the sake of completeness, we recall the basic results on the modulus and therotation estimate under quasiconformal mappings given in [11]. We write modEfor the conformal modulus of a doubly connected plane domain E, i.e., modE =2π/E(Γ) where E(Γ) stands for the 2-modulus of the curve family Γ joiningthe boundary components of E in E. In particular, modA(r, R) = log(R/r) ifA(r, R) is the annulus r < |z| < R.


Lemma 2.2. Let f : A(r, R) → C be a quasiconformal mapping with complexdilatation µ. Then

(19) −I(r, R;−µ) ≤ modA(r, R)−mod f(A(r, R)) ≤ I(r, R;µ)

where

(20) I(r, R;µ) =1

π

∫ ∫A(r,R)

|µ(z)|2 + Re µ(z)zz

1− |µ(z)|2dxdy

|z|2.

Let f , f(0) = 0, be a quasiconformal self-mapping of the complex plane. Thesingle valued function df (z) = arg(f(z)/z), z ∈ C \ {0}, for an assigned branchof the argument is called the angular displacement of the point z under themapping f(z) because of its geometric meaning.

Theorem 2.3. Let f , f(0) = 0, be a Q-quasiconformal mapping and supposethat the image of the annulus A(r, R), 0 < r < R, contains A(r(1+ε), R/(1+ε))and is contained in A(r/(1+ε), R(1+ε)). Then there exists an angle θ0 ∈ [0, 2π]such that the following estimates hold

(21) df (rei(θ0+log(R/r)))− df (Reiθ0) ≤ I(r, R;−iµ) + 2ε(Q+ 1),

(22) df (Reiθ0)− df (rei(θ0−log(R/r))) ≤ I(r, R; iµ) + 2ε(Q+ 1),

where I(r, R;µ) is defined by formula (20).

We complete this section with a remark, which follows from the statements on“unrestricted approach”, cf. [6].

Remark 2.4. Let f be uniformly weakly conformal on a set M. If for someτ(η) and θ(η) there exist radial limits

limr→0

|fη(reiτ )||r|

= C(η) 6= 0

andlimr→0

arg fη(reiθ) = α(η)

uniformly with respect to η ∈ M, then fη(z) is conformal at z = 0 for everyη ∈M and f ′η(0) is continuous on M.

3. Proofs of the main theorems

Proof for Theorem 1.7. We have to show that the mapping f satisfying con-ditions (5) and (8) has a continuous nonvanishing limit

limz→η

|f(z)− f(η)||z − η|

= C(η)


on the compact set M or equivalently, to show that

limz→0

|fη(z)||z|

= C(η)

uniformly in η ∈M. Here fη(z) stands for the family of quasiconformal mappingsfη(z) = f(z + η)− f(η) with the complex dilatations µ(z + η). To this end, wefirst remark that the elementary estimate

1

r2

∫ ∫|z|<r

|µ(z + η)|2 dxdy <∫ ∫|z|<r

|µ(z + η)|2

|z|2dxdy,

which holds for each η ∈ M, together with the convergence of the integral (5)imply that

limr→0

1

r2

∫ ∫|z|<r

|µ(z + η)|2 dxdy = 0

uniformly in η ∈ M. By Proposition 2.1 the mapping f is uniformly weaklyconformal on M. Hence, by the statement on unrestricted approach, it sufficesto show that there exists a finite radial limit limr→0 log(Mfη(r)/r) uniformly withrespect to η ∈M.

On the other hand, conditions (5), (8) and the Q-quasiconformality of f implythat the improper integrals I(0, R;±µη), see formula (20), with µη(z) = µ(z+η)exist in the sense of the principal value and converge uniformly with respect toη ∈M. Indeed, the existence of the improper integrals∫ ∫

|z|<R

|µη(z)|2 ± Re µη(z)z

z

|z|2dxdy

is an immediate consequence of the two conditions (5) and (8). Since |µη(z)| ≤(Q− 1)/(Q+ 1) a.e., we see that∣∣∣∣∣∣∣

∫ ∫|z|<R


z

1− |µη(z)|2dxdy

|z|2−∫ ∫|z|<R


z

|z|2dxdy

∣∣∣∣∣∣∣≤ Q− 1

2

∫ ∫|z|<R

|µη(z)|2

|z|2dxdy.

Hence convergence of the integral (5) implies the existence of the required im-proper integrals I(0, R;±µη).Thus, for given ε > 0 there exists δ > 0 such that |I(r, ρ;±µη)| < ε/2, for allη ∈M provided that r < ρ < δ. Then, by Lemma 2.2, we have

(23)∣∣∣mod fη(A(r, ρ))− log

ρ

r

∣∣∣ < ε

2


for such radii r and ρ. Applying the monotonicity property of the modulus andagain the uniform weak conformality property (9), we can assume that

(24)

∣∣∣∣mod fη(A(r, ρ))− logMfη(ρ)

Mfη(r)

∣∣∣∣ < ε

2

for all r < ρ < δ and η ∈M. Hence for r < ρ < δ the triangle inequality yields

(25)

∣∣∣∣logMfη(ρ)

ρ− log

Mfη(r)

r

∣∣∣∣ < ε

which is Cauchy’s criterion for the existence of the required uniform finite limitand the proof is complete.

Remark 3.1. The integrals I(0, R;±µη) control the conformal dilatation at thepoint η ∈ T. Using the monotonicity of the modulus, we obtain that

logmfη(R)

Mfη(r)≤ mod fη(A(r, R)) ≤ log

Mfη(R)

mfη(r)

By Lemma 2.2

−I(r, R;−µη) ≤ logR

r−mod fη(A(r, R)) ≤ I(r, R;µη).

Hence

logmfη(r)

r− log

Mfη(R)

R≤ I(r, R;µη),

logMfη(r)

r− log

mfη(R)

R≥ −I(r, R;−µη).

Letting r tend to zero, we obtain the estimates

mfη(R)e−I(0,R;−µη)

R≤ C(η) ≤

Mfη(R)eI(0,R;µη)

R

which hold for every fixed η ∈ T.

Proof for Theorem 1.4. We have to show that the mapping f satisfying con-ditions (5) and (6) admits continuous nonvanishing complex derivative f ′(η) onthe setM. The later assertion is equivalent to the fact that the family of quasi-conformal mappings fη(z) = f(z + η) − f(η) with complex dilatations µ(z + η)is conformal at 0 uniformly with respect to η ∈M.

First we show that there exists

limz→0

|fη(z)||z|

= C(η)

uniformly with respect to η ∈ M. To this end note that condition (6) impliescondition (8) of Theorem 1.7. Hence conditions (5) and (8) hold and thereforethe existence of the required limit follows from Theorem 1.7.


Next we show that limz→0 arg(fη(z)/z) also exists and is uniform with respectto η ∈ M. Now the uniform weak conformality property (10) implies that theexistence of the limit is equivalent to the existence of the uniform radial limit

(26) limt→0

dfη(teiθ)

for a fixed θ. Suppose that (26) does not hold. Then there exist sequences {rn}and {Rn}, n = 1, 2, ..., tending to 0, rn < Rn, and ηn ∈ T such that

(27) limn→∞

[dfηn (rne

iθ)− dfηn (Rneiθ)]

= δ 6= 0.

Because of the weak conformality property, δ does not depend on θ. We nowsuppose that δ > 0. By the first half of the proof |fη(z)|/|z| → C(η) as z → 0uniformly with respect to η ∈ M and we can assume that C = 1 for a fixed η.From (5) and (6) it follows that the improper integrals I(0, R;±iµη) exist inthe sense of the principal value and converge uniformly with respect to η ∈ M.Indeed, the existence of the improper integrals∫ ∫

|z|<R

|µη(z)|2 ± Im µη(z)z

z

|z|2dxdy

is an immediate consequence of conditions (5) and (6). On the other hand, since|µη(z)| ≤ (Q− 1)/(Q+ 1) a.e., we see that∣∣∣∣∣∣∣

∫ ∫|z|<R


z

1− |µη(z)|2dxdy

|z|2−∫ ∫|z|<R


z

|z|2dxdy

∣∣∣∣∣∣∣≤ Q− 1

2

∫ ∫|z|<R

|µη(z)|2

|z|2dxdy.

Hence convergence of the integral (5) implies the existence of the required im-proper integrals I(0, R;±iµη).

Let ε > 0, ε < δ, be a fixed small number. The above remarks together with (27)and the uniform weak conformality property allow us to find a number N = N(ε)such that for n > N the following inequalities hold for all ϕ, θ and η ∈M.

(28)1

1 + ε1

≤ |fη(teiϕ)|

t≤ 1 + ε1


for t = rn, t = Rn, η = ηn and ε1 = ε/8(Q+ 1);∣∣dfηn (rneiθ)− dfηn (rne

iϕ)∣∣ <

ε

2;(29)

δ − ε

2< dfηn (rne

iθ)− dfηn (Rneiθ);(30)

|I(rn, Rn;±iµη)| <ε

4.(31)

Because of the estimates (28) and (30), we can make use of Theorem 2.3 in theannulus A(rn, Rn) for a fixed n > N . Since dfηn (rne

iθ) − dfηn (Rneiθ) > 0 for

each θ, we obtain from the inequality (21) that

dfηn (rnei(θ0+log(Rn/rn)))− dfηn (Rne

iθ0) ≤ I(rn, Rn;−iµ) + 2ε1(Q+ 1)

for some angle θ0 = θ0(n) and therefore, by (31),

dfηn (rnei(θ0+log(Rn/rn)))− dfηn (Rne

iθ0) <ε

2.

The uniform weak conformality property (29), for θ = θ0 and ϕ = θ0+log(Rn/rn),yields

−ε2

+ dfηn (rneiθ0) < dfηn (rne

i(θ0+log(Rn/rn)))

and since

δ − ε

2< dfηn (rne

iθ0)− dfηn (Rneiθ0),

we conclude that

δ − ε

2< dfηn (rne

iθ0)− dfηn (Rneiθ0) < ε.

For sufficiently small ε this provides a contradiction in the case δ > 0.

The case where δ < 0 follows analogously from the second estimate of Theo-rem 2.3. The proof of Theorem 1.4 is complete.

Remark 3.2. 1) Consider the monotone majorant of µ given as

σ(t) = ess sup1−t<|z|<1+t

|µ(z)|, t > 0.

If ∫ 1

0

σ(t)

tdt <∞,

then conditions (5) and (6) evidently hold and hence the quasicircle C = f(T)is smooth. If the mapping f is conformal in D, i.e., µ(z) = 0 for z ∈ D, thenlog f ′(z) is continuous on D, see, [3] and [2].

2) If the Teichmuller-Wittich-Belinski integral∫ ∫|z|<1

|µ(z + η)||z|2

dxdy


converges uniformly for η ∈ T, then the quasicircle C = f(T) is smooth. Ifµ(z) = 0 in the disk D, then log f ′(z) is continuous on D. Indeed, if the TWB-integral is dominated by a positive constant M for all η ∈ T, then∫ ∫

|z|<1

|µ(z + η)|2

|z|2dxdy ≤M ||µ||∞

and the improper integral ∫ ∫|z|<1

µ(z + η)

z2dxdy

converges absolutely and uniformly for η ∈ T. The claim now follows fromTheorem 1.4.

3) The above result can be restated as follows: If the integral

(32)

∫ 1

0

ω(η, t)dt

t

converge uniformly for η ∈ T, where

ω(η, t) =1

πt2

∫ ∫|z−η|<t

|µ(z)| dxdy,

then the quasicircle C = f(T) is smooth. If f is additionally conformal in thedisk D, i.e., µ(z) = 0 for |z| < 1, then log f ′(z) admits a continuous extension tothe closed disk, cf. [9, Theorem 2]. Indeed, the identity

1

2π

∫ ∫r<|z|<R

|µ(z)||z|2

dxdy =ω(η,R)− ω(η, r)

2+

∫ R

r

ω(η, t)dt

t

allows us to reduce the problem to the preceding case.

Proof for Theorem 1.1. By the Reflection Principle we extend f(z) to the

lower half-plane by the formula f(z). The extended mapping f(z) has the com-

plex dilatation µ∗(z) = µ(z) for Im z > 0 and µ∗(z) = µ(z) for Im z < 0. Wefirst have to show that

limz→0

|ft(z)||z|

= C(t) > 0

uniformly with respect to t ∈M, where ft(z) = f(z + t)− f(t). Since

1

r2

∫ ∫|z|<r

|µ∗(z + t)|2 dxdy ≤∫ ∫|z|<r

|µ∗(z + t)|2

|z|2dxdy

= 2

∫ ∫|z|<r

Im z>0

|µ(z + t)|2

|z|2dxdy,


we see that condition (2) implies the uniform limit

limr→0

1

r2

∫ ∫|z|<r

|µ∗(z + t)|2 dxdy = 0.

Lemma 2.1 shows that the mapping f is uniformly weakly conformal on theset M. Hence, by the statement on the unrestricted approach, it suffices toshow that there exists a finite uniform limit limr→0 log(Mft(r)/r). Conditions (2)and (3) and the Q-quasiconformality of f imply that the improper integralsI(0, R;±µ∗t ) with µ∗t (z) = µ∗(z + t) exist in the sense of the principal value andconverge uniformly with respect to t ∈ M. Thus for given ε > 0 there existsδ > 0 such that |I(r, ρ;±µ∗t )| < ε/2, for all t ∈ M provided that r < ρ < δ.Then, by Lemma 2.2, we have

(33)∣∣∣mod ft(A(r, ρ))− log

ρ

r

∣∣∣ < ε

2

for such radii r and ρ. Applying the monotonicity property of the modulus andagain the weak conformality property (9), we can assume that

(34)

∣∣∣∣mod ft(A(r, ρ))− logMft(ρ)

Mft(r)

∣∣∣∣ < ε

2

for all r < ρ < δ and x ∈M. Hence for r < ρ < δ the triangle inequality yields

(35)

∣∣∣∣logMft(ρ)

ρ− log

Mft(r)

r

∣∣∣∣ < ε,

which is Cauchy’s criterion for the existence of the required uniform finite limit.

Now we make use of the fact that limr→0 arg f(t + r) = 0 for r > 0 and t ∈ R.Then, by Lemma 2.4, the mapping f(z) is conformally differentiable on the realaxis, i.e.,

limf(z)− f(t)

z − t= C(t), t ∈ R.

Since this limit is locally uniform with respect to t ∈ R, the function C(t) iscontinuous. Thus, if conditions (2) and (3) hold at a fixed point t ∈ R, then theboundary correspondence x 7→ f(x) has a nonvanishing derivative at this point.If the above integrals converge locally uniformly for every t ∈ R, then f ′(x) iscontinuous.

Remark 3.3. 1) Recall that a quasisymmetric homeomorphism h : R → R iscalled asymptotically symmetric if

limt→0

h(x+ t)− h(x)

h(x)− h(x− t)= 1

uniformly with respect to x ∈ R, see [10]. Conditions (2) and (3) imply confor-mality of the mapping f on the real axis, and hence, under these conditions, theboundary correspondence x 7→ f(x) is asymptotically symmetric.


2) Given an arbitrary alternating conditionally convergent series∑

(−1)nan with0 < an+1 < an and

∑an =∞, define a decreasing sequence of real numbers

tn+1 =

(1

e

)exp(a0+a1+···+an)

, t0 =1

e,

such that tn → 0 as n → ∞. For Im z > 0 let a Beltrami coefficient µ(z) bedefined as

µ(z) =

(−1)n(z − z)

4i|z| log 1|z|· zz

for tn+1 < |z| < tn,

0, for |z| > 1/e,

and denote by f(z), f(0) = 0, f(1) = 1, the quasiconformal self-mapping ofthe upper half-plane with this complex dilatation. The existence of such f fol-lows from the measurable Riemann Mapping Theorem, see, e.g. [13, p. 194].An elementary computation shows that µ satisfies the pointwise assumptions ofTheorem 1.1. Indeed, if z = reiϕ, we see that∫ ∫

|z|<1/eIm z>0

|µ(z + t)|2

|z|2dxdy <

1

4

∫ 1/e

0

∫ π

0

sin2 ϕ

|z + t|2 log2(1/|z + t|)r drdϕ <∞

for each real t. Next∫ ∫|z|<1/eIm z>0

|µ(z + t)||z|2

dxdy =1

2

∫ 1/e

0

∫ π

0

sinϕ

|z + t| log(1/|z + t|)drdϕ <∞

for each t 6= 0. Therefore, condition (3) holds for every fixed t 6= 0. If t = 0, then∫ ∫|z|<1/eIm z>0

µ(z)

|z|2dxdy =

∞∑n=0

(−1)nan <∞.

Thus, the boundary correspondence x 7→ f(x) has a derivative f ′(x) at eachpoint x ∈ R. On the other hand, the Carleson condition fails, because∫ 1

0

η(t)

tdt =

∫ 1

0

dt

2t log 1t

=∞.

Since ∫ ∫|z|<1/eIm z>0

|µ(z)||z|2

dxdy =

∫ 1/e

0

dt

t log 1t

=∞,

the TWB-condition also fails.


4. Applications

The main theorems can be used to study the behavior of f ′(z) of a conformalmapping on the boundary T of the unit circle D.

Let f map the unit disk D conformally onto the inner domain bounded by aquasicircle C. Then f admits a quasiconformal extension to the complex planewith some complex dilatation µ. The following statement is an immediate con-sequence of Theorem 1.4.

Proposition 4.1. Let f be a quasiconformal self-mapping of the complex planewith complex dilatation µ which is conformal in the disk D. If conditions (5) and(6) hold uniformly for η ∈ T, then there exists

limz→η

f(z)− f(η)

z − η= f ′(η) 6= 0

uniformly with respect to η ∈ T and therefore log f ′(z) has a continuous extensionto D.

If special quasiconformal extensions are available, then we can obtain more pre-cise integral conditions for the conformality of f up to the boundary. As anexample we consider the case when C is an asymptotically conformal curve, see[15, p. 246] and Becker’s explicit extension [3].

Recall that the Jordan curve C ⊂ C is said to be asymptotically conformal if

(36) maxw∈C(a,b)

|a− w|+ |w − b||a− b|

→ 1 as a, b ∈ J, |a− b| → 0,

where C(a, b) is the smaller arc of C between a and b. The curve C, representedby w(s), has a tangent at the point w(s0) if

arg(w(s)− w(s0))→

{β(s0), as s→ s0 + 0,

β(s0) + π, as s→ s0 − 0.

As we have already noted, a closed Jordan curve C ∈ C is called smooth if it hasa continuously turning tangent at all points. It follows that every smooth curveis asymptotically conformal. On the other hand, an asymptotically conformalcurve may be rather pathological as the example in [15, p. 249] shows. Thefunction f defined by f(0) = 0 and

log f ′(z) =∞∑n=0

bnz2n , z ∈ D,

where bn ∈ C with |bn| < 1/3 and bn → 0 as n → ∞, maps D conformallyonto a quasidisk f(D) bounded by an asymptotically conformal Jordan curve J .Moreover, if additionally

∑|bn|2 =∞, then f is conformal almost nowhere on T

so that J has a tangent only on a set of zero linear measure.


Proposition 4.2. Let f map the unit disk D conformally onto the inner domainof an asymptotically conformal curve C. If for η ∈ T conditions (5) and (6) holdwith

(37) µ(z) = (|ζ|2 − 1)ζf ′′(ζ)

f ′(ζ)· ζζ, ζ =

1

z,

for |z| > 1 and µ(z) = 0 for z ∈ D, then f is conformal at the point η ∈ T. Ifthese conditions hold uniformly with respect to η ∈ T, then there exists

limz→η

f(z)− f(η)

z − η= f ′(η) 6= 0

uniformly with respect to η ∈ T.

Proof. Since f maps the unit disk D conformally onto the inner domain of anasymptotically conformal curve C, we can extend the mapping f to the complexplane quasiconformally by the explicit Becker formula [3]

f(z) = f(1z) +

(z − 1

z

)f ′(1

z).

This extension has the complex dilatation µ(z) given by (37). The applicationof Theorem 1.4 completes the proof.

Corollary 4.3 ([3]). If

ess sup1−t<|z|<1

(1− |z|2)

∣∣∣∣zf ′′(z)

f ′(z)

∣∣∣∣ = σ(t)

and ∫ 1

0

σ(t)

tdt <∞,

then log f ′(z) has continuous extension to the closed disk D.

In the above consideration the Becker extension formula can be replaced by theAhlfors [1] or the Epstein-Pommerenke [14] extension formula.

In the case when the curve C is smooth, the above statement admits the followingweaker version, see Remark 1.8.

Corollary 4.4. Let f map the unit disk D conformally onto the inner domainof a smooth Jordan curve C ⊂ C. If for some positive constants M and R, andfor all η ∈ T conditions (5) and (8) hold with the complex dilatation µ given bythe formula (37), then there exists

limz→η

f(z)− f(η)

z − η= f ′(η) 6= 0

uniformly with respect to η ∈ T.


Recall that the Jordan curve C is Dini-smooth if it is smooth and if the angleβ(s) of the tangent, considered as a function of the arc length s, satisfies

|β(s2)− β(s1)| ≤ ω(s2 − s1), s1 < s2,

where ω(t) is an increasing function for which

(38)

∫ π

0

ω(t)

tdt <∞.

It is easy to show that if C is Dini-smooth, then the conditions of Proposition 4.2hold and we arrive at the well-known Warschawski’s Conformality Theorem, see,e.g. [15, p. 48].

Theorem 4.5 (Warschawski’s Theorem, [17]). Let C be a Dini-smooth Jordancurve and f(z) map D conformally onto the inner domain of C. Then log f ′(z)has a continuous extension to D.

Proof. Let the curve C be parameterized as w(t) = f(eit), 0 ≤ t ≤ 2π and ω(t)denote the modulus of continuity of the function ϕ(t) = β(t) − t − π/2, whereβ(t) stands for the tangent direction angle at f(eit). Then, by Proposition 3.3and Proposition 3.4 from [14, pp. 45–47] it follows that ω(t) satisfies the Dinicondition (38) and

(39) (1− |z|2)

∣∣∣∣zf ′′(z)

f ′(z)

∣∣∣∣ ≤ 2

πω(1− |z|) + 2π(1− |z|)

∫ π

1−|z|

ω(t)

t2dt

holds whenever z ∈ D. An elementary calculation shows that there is a constantM > 0 such that ∫ ∫

|z|<1/2

|µ(z + η)||z|2

dxdy ≤M

for all η ∈ T with µ(z) of the form (37). The latter implies that the integralsin (5) and (6) converge uniformly for η ∈ T. By Theorem 1.4, there exists

limz→η

f(z)− f(η)

z − η= f ′(η) 6= 0

uniformly for η ∈ T.

References

1. L. Ahlfors, Sufficient conditions for quasiconformal extension, Princeton Annals of Math.Studies 79 (1974), 23–29.

2. J. M. Anderson, J. Becker, and F. D. Lesley, Boundary values of asymptotically conformalmappings, J. London Math. Soc. 38 (1988), 453–462.

3. J. Becker, On asymptotically conformal extension of univalent functions, Complex Vari-ables 9 (1987), 109–120.

4. P. P. Belinskiı, Behavior of a quasiconformal mapping at an isolated singular point, (inRussian), L’vov Gos. Univ. Uchen. Zap., Ser. Meh.-Mat. 6 29 (1954), 58–70.


5. A. Beurling A. and L. Ahlfors, The boundary correspondence under quasiconformal map-pings, Acta Math. 96 (1956), 113–134.

6. M. Brakalova M. and J. A. Jenkins, On a paper of Carleson, Ann. Acad. Sci. Fenn. Math.27 (2002), 485–490.

7. L. Carleson, On mappings conformal at the boundary, J. Analyse Math. 19 (1961), 1–13.8. D. Drasin, On the Teichmuller-Wittich-Belinskii theorem, Results Math. 10 (1986), 54–65.9. E. M. Dyn’kin, Estimates for asymptotically conformal mappings, Ann. Acad. Sci. Fenn.

Math. 22 (1997), 275–304.10. F. P. Gardiner F.P. and D. P. Sullivan, Symmetric structures on a closed curve, Amer. J.

Math. 114 (1992), 683–736.11. V. Gutlyanskii V. and O. Martio, Conformality of a quasiconformal mapping at a point,

J. Analyse Math. (2003), to appear.12. V. Ya. Gutlyanskii and M. Vuorinen, On mappings almost conformal at the boundary,

Complex Variables 34 (1997), 455–464.13. O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, 2nd Ed., Springer-

Verlag, 1973.14. Ch. Pommerenke, On the Epstein univalence criterion, Results Math. 10 (1986), 143–146.15. , Boundary Behaviour of Conformal Maps, Springer-Verlag, 1992.16. O. Teichmuller, Untersuchungen uber konforme und quasikonforme Abbildung, Deutsche

Math. 3 (1938), 621–678.17. S. E. Warschawski, On differentiability at the boundary in conformal mapping, Proc. Amer.

Math. Soc. 12 (1961), 614–620.18. H. Wittich, Zum Beweis eines Satzes uber quasikonforme Abbildungen, Math. Z. 51 (1948),

275–288.

Vladimir Gutlyanskiı E-mail: [email protected]: Institute of Applied Mathematics and Mechanics, NAS of Ukraine, ul. Roze Lux-emburg 74, 83114, Donetsk, Ukraine

Olli Martio E-mail: [email protected]: Department of Mathematics, P.O. Box 4 (Yliopistonkatu 5), FIN–00014 Universityof Helsinki, Finland

on regularity of quasiconformal curves

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