On Lipschitz Continuity of Quasiconformal Mappings

Vladimir Gutlyanskiĭ and Toshiyuki Sugawa

Dedicated to Professor Olli Martio on the occasion of his 60 -th birthday

Abstract

We study the local growth of quasiconformal mappings in the plane. Estimates are given in terms of integral means of the pointwise angular dilatations. New sufficient conditions for a quasiconformal mapping f to be either Lipschitz or weakly Lipschitz continuous at a point are given.

2000 Mathematics Subject Classification. AMS (MOS) 30C62.

1. Introduction

Let Ω be a domain in the complex plane ℂ. It is a classical result that every Kquasiconformal mapping f : Ω → ℂ is locally Hölder continuous with exponent 1/K, which is the best possible, see [1], namely,

|f(z) − f(z_{0})| ≤ C|z − z_{0}|^{1/K}. (1.1)

If one allows the exponent in (1.1) to vary with respect to the point z_{0}, one could get more precise information about the local behavior of f at z_{0}. For instance, Belinskiĭ [7, p. 37] showed that if a Beltrami coefficient μ is continuous, then the corresponding quasiconformal mapping f will be weakly Lipschitz continuous, that is, locally Hölder continuous with every exponent less than one. Iwaniec [12] extended this result to the case when μ has vanishing mean oscillation in the sense of Sarason [18]. Moreover, the weak Lipschitz continuity remains to hold for the higher-dimensional quasiconformal mappings if either the dilatation tensors or the matrix dilatations have vanishing mean oscillation [11].

In this paper we study the local behavior of plane quasiconformal mappings by employing the angular and the radial dilatations. One of our main results here is a local growth estimate for a K-quasiconformal mapping, which implies an estimate for the modulus of continuity in terms of the integral means of the angular dilatations. We will observe that the modulus of continuity of f in the form O(|z − z_{0}| log(1/|z − z_{0}|)) at a given point z_{0} does not necessarily follow from the continuity of μ, which implies μ ∈ VMO, for the corresponding quasiconformal mapping f (see Section 4). This observation answers negatively to a problem on the modulus of continuity, which was discussed during the XVIIIth Rolf Nevanlinna Colloquium held in Helsinki, 2000. Moreover, it is shown that the weak Lipschitz continuity of f at z_{0} is the best possible under the assumption of the approximate continuity of μ (see Theorem 4.12).

Some of our estimates in the following are independent of the maximal dilatations K of the quasiconformal mappings. Then, in most cases, by a standard approximation procedure, we can extend the results for those homeomorphisms in the Sobolev class W_{loc}^{1,1}(Ω) for which the relevant quantities are bounded. For simplicity, we will not pursue this direction here.

We recall some basic definitions and notation. A sense-preserving topological embedding f : Ω → ℂ is called quasiconformal if f ∈ W_{loc}^{1,2}(Ω) and if f satisfies the Beltrami equation

fz̄ = μ(z) f_{z} a.e. in Ω

for a complex-valued measurable function μ with ∥μ∥_{∞} < 1, which is called the complex dilatation of f or the Beltrami coefficient of f. More specifically, f is called K-quasiconformal if ∥K_{μ}∥_{∞} ≤ K, where K_{μ}(z) = (1 + |μ(z)|)/(1 − |μ(z)|).

For a given μ in Ω, we define the angular dilatation D_{μ,z_{0}} of μ at z_{0} ∈ ℂ̂ = ℂ ∪ {∞} by

D_{μ,z_{0}}(z) = \frac{|1 − μ(z)\frac{z̄−z̄_{0}}{z−z_{0}} |^{2}}{1 − |μ(z)|^{2}}

in the case when z_{0} is finite, and by D_{μ,∞}(z) = D_{μ,0}(z) in the case when z_{0} = ∞. For brevity, we will write D_{μ,0} = D_{μ}. The angular dilatation is a measurable function in Ω and satisfies the inequality

1/K_{μ}(z) ≤ D_{μ,z_{0}}(z) ≤ K_{μ}(z) (1.2)

a.e. in Ω for each z_{0} ∈ ℂ̂. The name of D_{μ,z_{0}} comes from the following important relation: if f is a quasiconformal mapping in Ω with complex dilatation μ and if we write z = z_{0} + re^{iθ}, then

|\frac{∂f}{∂θ} (z)|^{2} = r^{2}D_{μ,z_{0}}(z)J_{f}(z) (1.3)

holds for almost all z ∈ Ω, where J_{f} stands for the Jacobian of f. The radial dilatation of μ at z_{0} is defined as D_{−μ,z_{0}}(z) and its name comes from the relation

|\frac{∂f}{∂r} (z)|^{2} = D_{−μ,z_{0}}(z)J_{f}(z) a.e. in Ω. (1.4)

The idea of employing the directional dilatations for the study of quasiconformal mappings in the plane is due to Andreian Cazacu [3] and Reich and Walczak [17], and similar ideas are used in [19], [8] and [10].

2. Growth estimates

Given a Beltrami coefficient μ in the unit disk 𝔻 centered at the origin with ∥μ∥_{∞} ≤ (K − 1)/(K + 1), we define Q_{μ}(r, R) by

Q_{μ}(r, R) = \frac{1}{2π log(R/r)} ∫∫_{r<|z|<R} \frac{D_{μ}(z)}{|z|^{2}}dxdy

for 0 < r < R ≤ 1. This quantity can be regarded as the integral average of the angular dilatation D_{μ}(z) over the annulus r < |z| < R with respect to the logarithmic measure. In particular, since K^{−1} ≤ D_{μ}(z) ≤ K, we see that K^{−1} ≤ Q_{μ}(r, R) ≤ K holds.

2.1. Theorem. Let f : 𝔻 → 𝔻, f(0) = 0, be a quasiconformal embedding with complex dilatation μ. Then

|f(z)| ≤ C (\frac{|z|}{R})^{1/Q_{μ}(|z|,R)} , |z| < R,

for an absolute constant 0 < C ≤ 64.

In particular, because of (1.2), we immediately obtain

|f(z)| ≤ C (\frac{|z|}{R})^{1/P_{μ}(|z|,R)} (2.2)

for all |z| < R. Here

P_{μ}(r, R) = \frac{1}{2π log(R/r)} ∫∫_{r<|z|<R} \frac{K_{μ}(z)}{|z|^{2}}dxdy.

We remark, in particular, that max{1, Q_{μ}(r, R)} ≤ P_{μ}(r, R) ≤ K. Now the aforementioned Hölder continuity estimate

|f(z)| ≤ C|z|^{1/K}, z ∈ 𝔻, (2.3)

is a simple consequence of (2.2).

We do not discuss here the sharp constant C in the case if we allow it to depend on K = (1 + ∥μ∥_{∞})/(1 − ∥μ∥_{∞}). However, we make a few remarks here. One has the expectation that if f : 𝔻 → 𝔻 is a K-quasiconformal homeomorphism, normalized by f(0) = 0, then inequality (2.3) holds with C = 16^{1−1/K}. It is known that 16 is the smallest constant independent of K in this case, see, e.g., [14, p. 68]. Currently it is known that one can choose C = 64^{1−1/K}. More details are given in [21, p. 33].

The proof of Theorem 2.1 is based on the following statement which may be of independent interest.

2.4. Proposition. Let A = {z : r < |z| < R} and f : A → ℂ be a quasiconformal embedding with complex dilatation μ. Then

\frac{1}{Q_{μ}(r, R)} ≤ \frac{mod f(A)}{mod A} ≤ Q_{−μ}(r, R). (2.5)

We remark that the above result is equivalent to the following inequalities:

−\frac{1}{2π} ∫∫_{A} \frac{D_{−μ}(z) − 1}{|z|^{2}}dxdy ≤ mod A − mod f(A)

≤ \frac{mod f(A)}{mod A} · \frac{1}{2π} ∫∫_{A} \frac{D_{μ}(z) − 1}{|z|^{2}}dxdy.

Proof. For simplicity, let us assume that r = 1 and set m = mod A = log R. We first recall the following estimate due to [17]:

mod f(A) ≥ ∫_{1}^{R} \frac{dr}{rψ(r)}, (2.6)

where

ψ(r) = \frac{1}{2π} ∫_{0}^{2π} D_{μ}(re^{iθ})dθ.

A complete proof of (2.6) under somewhat weaker assumptions is also contained in [10]. Here we just sketch the proof of (2.6) for the reader’s convenience. First we may assume that f(A) = {1 < |w| < R′}. Let γ_{r} be the circle |z| = r. Then

2π ≤ ∫_{γ_{r}} |d arg f| ≤ ∫_{0}^{2π} |\frac{f_{θ}(re^{iθ})}{f(re^{iθ})}| dθ

for almost all r ∈ (1, R). Noting (1.3) and making use of Schwarz’ inequality, we get

\frac{2π}{rψ(r)} ≤ ∫_{0}^{2π} J_{f}|f|^{−2}rdθ.

If we integrate both sides of the above inequality with respect to the parameter r over (1, R), we obtain the required estimate.

Next, the observation

\frac{1}{2π} ∫∫_{A} \frac{D_{μ}(z)}{|z|^{2}}dxdy = ∫_{1}^{R} \frac{ψ(r)dr}{r} = ∫_{0}^{m} ψ(e^{t})dt,

and the estimate

m^{2} ≤ ∫_{0}^{m} ψ(e^{t})dt · ∫_{0}^{m} \frac{dt}{ψ(e^{t})},

which is a consequence of Schwarz’ inequality, imply

∫_{0}^{m} \frac{dt}{ψ(e^{t})} ≥ \frac{m^{2}}{\frac{1}{2π}_{∫∫A} \frac{D_{μ}(z)}{|z|^{2}}dxdy} = \frac{m}{Q_{μ}(1, R)}. (2.7)

Combining inequalities (2.6) and (2.7), we obtain the right-hand side in (2.5). The left-hand side of inequality (2.5) can be handled similarly by using (1.4), cf. [17]. □

The following auxiliary lemma will play an important role.

2.8. Lemma ([10]). Let B be a ring domain in ℂ whose complement in ℂ̂ consists of the bounded component E_{0} and the unbounded component E_{1}. If mod B > C_{1}, we have the estimate

diam E_{0} ≤ C_{2} dist(E_{0}, E_{1})e^{−mod B},

where C_{1} and C_{2} are positive absolute constants with C_{1} ≤ 5 log 2 and C_{2} ≤ 64.

6 On Lipschitz Continuity The statement is proved by a direct application of the well-known Teichmüller modulus theorem (see [14, p. 56]).

Proof of Theorem 2.1. For a given z with |z| < R, we set m = log(R/|z|) and let A = {w : |z| < |w| < R}. If mod f(A) > 5 log 2, by Lemma 2.8, we obtain (2.9) |f(z)| ≤ 64 dist(E_{0}, E_{1})e^{−mod f(A)} ≤ 64 e^{−mod f(A)}. Otherwise, we see that

|f(z)| < 1 ≤ e^{5 log 2−mod f(A)} = 32 e^{−mod f(A)},

and hence, (2.9) holds. On the other hand, by Proposition 2.4, we have

mod f(A) ≥ \frac{m}{Q_{μ}(|z|, R)}. (2.10)

Combining (2.9) with (2.10), we arrive at the inequality

|f(z)| ≤ 64e^{− log(R/|z|)/Q_{μ}(|z|,R)} = 64 (\frac{|z|}{R})^{1/Q_{μ}(|z|,R)} ,

and thus, we complete the proof. □

Since D_{±μ}(z) ≤ K_{μ}(z) a.e. in 𝔻, we deduce from (2.5) an estimate for the change of moduli under quasiconformal mappings in the plane with pointwise dilatation K_{μ}(z) established by Belinskiĭ [6] (see also [14, p. 222]).

2.11. Corollary. Under the assumption same as in Proposition 2.4

\frac{1}{P_{μ}(r, R)} ≤ \frac{mod f(A)}{mod A} ≤ P_{μ}(r, R). (2.12)

If f is K-quasiconformal mapping, then (2.12) yields

\frac{mod A}{K} ≤ mod f(A) ≤ K mod A

and we recognize the classical Grötzsch inequality for annuli (see, e.g., [14, p. 38]).

2.13. Corollary. Under the assumptions of Proposition 2.4

−\frac{1}{2π} ∫∫_{A} \frac{D_{−μ}(z) − 1}{|z|^{2}}dxdy ≤ mod A − mod f(A)

≤ \frac{1}{2π} ∫∫_{A} \frac{D_{μ}(z) − 1}{|z|^{2}}dxdy.

Proof. Set m = mod A, m′ = mod f(A) and Q = Q_{μ}(r, R). We then have to verify the inequality m − m′ ≤ mQ − m, only. The left-hand side of (2.5) implies m′/m ≥ 1/Q. The trivial inequality (Q − 1)^{2} ≥ 0 yields 1/Q ≥ 2 − Q. Hence, one can get m′/m ≥ 2 − Q, which is nothing but the required inequality. □

In particular, we have the following simple form due to Belinskiĭ [6] (see also [14, p. 222]).

2.14. Corollary. Under the assumptions of Proposition 2.4 the following inequality holds:

|mod A − mod f(A)| ≤ \frac{1}{2π} ∫∫_{A} \frac{K_{μ}(z) − 1}{|z|^{2}}dxdy.

Similar growth estimates for homeomorphic solutions to the degenerate Beltrami equation were recently obtained by Chen [9].

3. Conditions for Lipschitz continuity

We will say that a function f is Lipschitz continuous at z_{0} if |f(z) − f(z_{0})| ≤ C|z − z_{0}| holds in a neighborhood of z_{0} for a constant C. In this section, as applications of the previous results, we give a sufficient condition for a quasiconformal mapping to be Lipschitz continuous at a given point.

Suppose that f(z) is a quasiconformal mapping in 𝔻 normalized by f(0) = 0. It is known, e.g., [14, p. 232], that the condition

∫∫_{|z|<R} \frac{K_{μ}(z) − 1}{|z|^{2}}dxdy < +∞ (3.1)

for some R > 0 implies that f is conformally differentiable at the origin, that is, the limit lim_{z→0} f(z)/z exists and differs from 0 and ∞. The existence of the limit lim_{z→0} |f(z)|/|z| under the hypothesis (3.1) was first proved by Teichmüller [20] and Wittich [22]. The existence of lim_{z→0} arg(f(z)/z) under the assumption of (3.1) is due to Belinskiĭ (see, e.g., [7, p. 54]) and Lehto [13]. So, this result is referred to as the Teichmüller-Wittich-Belinskiĭ-Lehto regularity theorem. Under the weaker assumption than that of (3.1) we have the following statement.

3.2. Theorem. Suppose that f : 𝔻 → 𝔻, f(0) = 0, is a quasiconformal embedding in 𝔻 with complex dilatation μ satisfying the assumption

sup_{0<r<1} ∫∫_{r<|z|<1} \frac{D_{μ}(z) − 1}{|z|^{2}}dxdy < +∞. (3.3)

Then |f(z)| ≤ C|z| (3.4) holds in 𝔻 with some constant C depending only on the value of the above supremum, and therefore, the mapping f is Lipschitz continuous at the origin.

3.5. Remark. 1. If the improper integral converges to a number including −∞, we can state (3.3) simply by

∫∫_{|z|<1} \frac{D_{μ}(z) − 1}{|z|^{2}}dxdy < +∞.

2. The statement of Theorem 3.2 can also be obtained by a careful observation of the proof of Theorem 3.1 in [17].

For the later convenience, we introduce the quantity

T_{μ}(r, R) = \frac{1}{2π} ∫∫_{r<|z|<R} \frac{D_{μ}(z) − 1}{|z|^{2}}dxdy

for 0 < r < R. Note that the obvious relation T_{μ}(r, R) = (Q_{μ}(r, R) − 1) log(R/r) holds. Then condition (3.3) means that lim sup_{r→0} T_{μ}(r, 1) < +∞. This can also be viewed as a Dini type condition. Precisely, (3.3) holds if and only if

lim sup_{r→0} ∫_{r}^{1} \frac{ω_{μ}(t)dt}{t} < +∞, (3.6)

where

ω_{μ}(t) = \frac{1}{πt^{2}} ∫∫_{|z|<t}(D_{μ}(z) − 1)dxdy.

The quantity ω_{μ}(t) can be considered as the integral average of D_{μ}(z) − 1 over the disk |z| < t with respect to the plane Lebesgue measure. In particular, 1/K ≤ ω_{μ}(t) + 1 ≤ K if K_{μ}(z) ≤ K a.e. in 𝔻. As above, (3.6) can be expressed by ∫_{0}^{1} ω_{μ}(t)t^{−1}dt < +∞ in the case when the improper integral is convergent. Since the relation

T_{μ}(r, R) = \frac{ω_{μ}(R) − ω_{μ}(r)}{2} + ∫_{r}^{R} \frac{ω_{μ}(t)dt}{t} (3.7)

holds, we see that Dini type condition (3.6) is equivalent to (3.3). The formula in (3.7) can be proved in a more general setting.

3.8. Lemma. For a given Lebesgue integrable function φ in the disk |z| < R, the function η_{φ} defined by

η_{φ}(t) = \frac{1}{πt^{2}} ∫∫_{|z|<t} φ(z)dxdy, 0 < t ≤ R,

satisfies the relation

\frac{1}{2π} ∫∫_{r<|z|<R} \frac{φ(z)}{|z|^{2}}dxdy = \frac{η_{φ}(R) − η_{φ}(r)}{2} + ∫_{r}^{R} \frac{η_{φ}(t)dt}{t}.

Proof. Set α(t) = πt^{2}η_{φ}(t). This function can be written in the form α(t) = ∫_{0}^{t} ψ(r)dr, where

ψ(r) = r ∫_{0}^{2π} φ(re^{iθ})dθ.

Note that ψ ∈ L^{1}([0, R]) by Fubini’s theorem. In particular, the function α(t) is absolutely continuous on [0, R] and α′(t) = ψ(t) for a.a. t. We now compute

∫∫_{r<|z|<R} \frac{φ(z)}{|z|^{2}}dxdy = ∫_{r}^{R} \frac{ψ(t)dt}{t^{2}} = [\frac{α(t)}{t^{2}}]_{r}^{R} + ∫_{r}^{R} \frac{2α(t)}{t^{3}}dt

= π(η_{φ}(R) − η_{φ}(r)) + 2π ∫_{r}^{R} \frac{η_{φ}(t)}{t}dt,

and thus the required identity follows. □

Theorem 3.2 can be derived from the following more general statement.

3.9. Theorem. Let f : 𝔻 → 𝔻, f(0) = 0, be a quasiconformal embedding with complex dilatation μ. Then the function f satisfies the inequality

|f(z)| ≤ \frac{C}{R}|z|e^{T_{μ}(|z|,R)}

for |z| < R ≤ 1, where C ≤ 64 is an absolute constant.

Proof. Fix z with 0 < |z| < R and let m = log(R/|z|) and A = {w : |z| < |w| < R}. By Corollary 2.14, we have mod f(A) ≥ m−T_{μ}(|z|, R). As in the proof of Theorem 2.1, we now obtain |f(z)| ≤ 64 e^{−m+T_{μ}(|z|,R)} = 64 |z| e^{T_{μ}(|z|,R)}/R. □

We now combine (3.7) with the above theorem to obtain the following statement.

3.10. Corollary. Let f : 𝔻 → 𝔻, f(0) = 0, be a K-quasiconformal embedding with complex dilatation μ. Then the function f satisfies the inequality

|f(z)| ≤ C|z| exp (∫_{|z|}^{1} \frac{ω_{μ}(t)dt}{t})

for |z| < 1, where C = C(K) is a constant depending on K only.

The following result produces examples of quasiconformal mappings which satisfy (3.3) but do not necessarily satisfy (3.1).

3.11. Proposition. Let ρ be an arbitrary non-negative real-valued measurable function in a neighborhood U of the origin with ∥ρ∥_{∞} < 1. Then there exists a quasiconformal embedding f : U → ℂ, f(0) = 0, which is Lipschitz continuous at the origin, with complex dilatation μ satisfying

|μ(z)| = ρ(z) and D_{μ}(z) = 1 a.e. in U

Proof. Given ρ(z) we define a Beltrami coefficient μ(z) a.e. in U which takes one of the following two values

(ρ^{2}(z) ± iρ(z)\sqrt{1 − ρ^{2}(z)})) \frac{z}{z̄} (3.12)

at each point z ∈ U in a measurable way. The straightforward computation shows that |μ(z)| = ρ(z) and D_{μ}(z) = 1 for almost all z ∈ U. By the measurable Riemann mapping theorem, there exists a quasiconformal mapping f : U → ℂ, f(0) = 0, with complex dilatation μ. Since (3.3) trivially holds, by Theorem 3.2, the mapping f is Lipschitz continuous at the origin. □

We remark that if ρ(z) = ρ(|z|) for z ∈ 𝔻 and ρ = 0 for z ∈ ℂ \ 𝔻, then the quasiconformal mapping f : ℂ → ℂ, fixing 0 and 1 with complex dilatation μ of the form (3.12) is explicitly given by the formula

f(z) = ze^{iθ(|z|)}, θ(r) = ∫_{r}^{1} \frac{2ρ(t)}{\sqrt{1 − ρ^{2}(t)}} \frac{dt}{t}

and represents a radial rotation. For example, if we set ρ(z) = ε/\sqrt{4 + ε^{2}}, ε ≥ 0, then we obtain the well-known spiral mapping

f(z) = ze^{iε log(1/|z|)},

transforming the rays re^{iθ}, 0 < r < 1, into logarithmic spirals which wind about the origin infinitely many times. Note, in particular, that lim_{z→0} arg(f(z)/z) does not exist, and therefore the mapping f is not conformally differentiable at the origin, nevertheless |f(z)| = |z| holds. This example was observed also by Reich and Walczak [17] in the same context.

Let f be a quasiconformal self-mapping of ℂ. Since f has locally integrable partial derivatives, the image Jordan curve of the circle |z| = r under f, which is usually called a quasicircle, is rectifiable for almost all r > 0. However, we cannot say that the specific curve f(∂𝔻) is rectifiable in general, see e.g., [14, p. 104] and [16, p. 249]. The following statement provides a sufficient condition for the quasicircle to be rectifiable.

3.13. Theorem. Let f : ℂ → ℂ be a quasiconformal mapping with complex dilatation μ and Γ be a rectifiable curve in ℂ. If there are positive constants R and M such that

∫∫_{|z−z_{0}|<R} \frac{D_{μ,z_{0}}(z) − 1}{|z − z_{0}|^{2}}dxdy ≤ M

holds for each z_{0} ∈ Γ, then f is Lipschitz continuous on Γ and therefore, the image curve f(Γ) is rectifiable, too.

Proof. Given z_{0} ∈ Γ, we consider the quasiconformal mapping F defined by F (z) = f(z +z_{0})−f(z_{0}). This mapping preserves the origin and has the complex dilatation ν(z) = μ(z + z_{0}). In order to apply Theorem 3.2, we first calculate the angular dilatation D_{ν}. By definition, we see that

D_{ν}(z) = \frac{|1 − μ(z + z_{0})\frac{z̄}{z}|^{2}}{1 − |μ(z + z_{0})|^{2}} = D_{μ,z_{0}}(z + z_{0}),

and therefore, we have

∫∫_{|z|<R} \frac{D_{ν}(z) − 1}{|z|^{2}}dxdy = ∫∫_{|z|<R} \frac{D_{μ,z_{0}}(z + z_{0}) − 1}{|z|^{2}}dxdy
= ∫∫_{|z−z_{0}|<R} \frac{D_{μ,z_{0}}(z) − 1}{|z − z_{0}|^{2}}dxdy ≤ M.

By Theorem 3.2, |F (z)|/|z| = |f(z + z_{0}) − f(z_{0})|/|z| is bounded in |z| ≤ R uniformly with respect to z_{0} ∈ Γ. Hence, the mapping f is Lipschitz continuous on Γ. □

More general conditions for the rectifiability of the image curve can be deduced from Theorem 3.2 by decomposing f into a suitable quasiconformal mapping and an affine mapping. We will illustrate this procedure in the next section. For related results on the rectifiability of quasicircles, see, for instance, [2], [5], [15],

[4] and the references therein.

4. Weak Lipschitz continuity

We will say that a quasiconformal mapping f is weakly Lipschitz continuous at z_{0} if for every 0 < α < 1 there is a constant C > 0 such that

|f(z) − f(z_{0})| ≤ C|z − z_{0}|^{α} (4.1)

holds if |z − z_{0}| < δ, where δ > 0 is a sufficiently small number.

4.2. Theorem. Let f : 𝔻 → 𝔻, f(0) = 0, be a quasiconformal embedding with complex dilatation μ satisfying

lim sup_{t→0} \frac{1}{πt^{2}} ∫∫_{|z|<t}(D_{μ}(z) − 1)dxdy ≤ 0. (4.3)

Then f is weakly Lipschitz continuous at 0.

Proof. It suffices to show that

log^{+} \frac{|f(z)|}{|z|} = o (log \frac{1}{|z|}) , z → 0, (4.4)

where log^{+} x = max{log x, 0}. Condition (4.3) means that lim sup_{t→0} ω_{μ}(t) ≤ 0, and thus,

lim sup_{z→0} \frac{1}{log(1/|z|)} ∫_{|z|}^{1} \frac{ω_{μ}(t)dt}{t} ≤ 0.

Now, from Corollary 3.10, the conclusion follows. □

Note that this result is qualitatively best possible as we will see in Theorem 4.12 below. Corollary 3.10 can be considered to give more precise information about the behavior of f(z)/z at the origin in terms of the local quantity ω_{μ}(t) defined by the complex dilatation.

As might be expected in view of the above proof, if the limit in (4.3) is negative, we may obtain even a stronger consequence.

4.5. Theorem. Let f : 𝔻 → 𝔻, f(0) = 0, be a quasiconformal embedding with complex dilatation μ. Suppose that

lim sup_{t→0} \frac{1}{πt^{2}} ∫∫_{|z|<t} D_{μ}(z)dxdy ≤ \frac{1}{α} (4.6)

holds for a constant α > 0. Then f is weakly Hölder continuous with exponent α, that is, for each 0 < α′ < α, there is a constant C such that

|f(z)| ≤ C|z|^{α′}, z ∈ 𝔻.

Proof. By assumption, there is a non-negative function ε(t) with lim_{t→0} ε(t) = 0 such that ω_{μ}(t) ≤ α^{−1} − 1 + ε(t) holds for t > 0. By (3.7), we have

(Q_{μ}(r, 1) − 1) log(1/r) = ∫_{r}^{1} \frac{ω_{μ}(t)dt}{t} + O(1) ≤ (α^{−1} − 1) log(1/r) + o(log(1/r))

as r → 0. This implies that

Q_{μ}(r, 1) ≤ \frac{1}{α} + o(1)

as r → 0. We can now apply Theorem 2.1 to get the conclusion. □

The following statement produces examples of quasiconformal mappings which satisfy (4.3) but do not necessarily satisfy the stronger assumption

lim_{t→0} \frac{1}{t^{2}} ∫∫_{|z|<t}(K_{μ}(z) − 1)dxdy = 0.

4.7. Proposition. Let ρ(z) be an arbitrary non-negative real-valued measurable function in 𝔻 such that ρ(z) ≤ (K − 1)/(K + 1) a.e. in 𝔻. Then the quasiconformal embedding f : 𝔻 → 𝔻, f(0) = 0, with complex dilatation

μ(z) = ρ(z)\frac{z}{z̄}

satisfies (4.6) with α = K, in particular, f is weakly Hölder continuous with exponent K at the origin.

Proof. Given ρ(z), we see that D_{μ}(z) = 1/K_{μ}(z) ≤ 1/K a.e. in 𝔻 and thus the conclusion follows from Theorem 4.5. □

In particular, if we set ρ(z) = (K − 1)/(K + 1), where K ≥ 1 is a constant, then we see that

f(z) = z|z|^{K−1}.

Then μ fails to be continuous at 0 and it does not have vanishing mean oscillation, however, f is Lipschitz continuous at the origin.

By using the composition procedure with affine mappings, we now deduce more general statement from the previous theorem.

4.8. Theorem. Let f : 𝔻 → 𝔻, f(0) = 0, be a K-quasiconformal mapping. Suppose that for some a ∈ 𝔻 the complex dilatation μ of f satisfies

lim sup_{t→0} \frac{1}{πt^{2}} ∫∫_{|w|<t}(D_{μ_{a}}(w) − 1)dudv ≤ 0, (4.9)

where

μ_{a} = \frac{μ − a}{1 − \overline{a}μ} ∘ L^{−1}

and L is the affine mapping defined by L(z) = z + az̄. Then f is weakly Lipschitz continuous at the origin.

Proof. Let us consider the quasiconformal mapping F (w) = f ∘L^{−1}(w) whose complex dilatation agrees with μ_{a}. By Theorem 4.2, assumption (4.3) implies F to be weakly Lipschitz continuous at the origin, which is equivalent to the condition that

log^{+} \frac{|F (w)|}{|w|} = o (log \frac{1}{|w|})

as w → 0. Hence

log^{+} \frac{|f(z)|}{|L(z)|} = o (log \frac{1}{|L(z)|})

as z → 0, which is equivalent to (4.4). Thus, f is weakly Lipschitz continuous at the origin. □

Note that the above number a necessarily satisfies |a| ≤ (K − 1)/(K + 1).

4.10. Corollary. Let f : 𝔻 → 𝔻 be a quasiconformal embedding whose complex dilatation μ is approximately continuous at the origin. Then f is weakly Lipschitz continuous at the origin.

Proof. The approximate continuity of μ at the origin means that there exists a point a ∈ 𝔻 such that

\frac{1}{πt^{2}} ∫∫_{|z|<t} |μ(z) − a|dxdy → 0

as t → 0. The chain of elementary inequalities

\frac{1}{πt^{2}} ∫∫_{|w|<t}(D_{μ_{a}}(w) − 1)dudv ≤ \frac{1}{πt^{2}} ∫∫_{|w|<t}(K_{F} (w) − 1)dudv
≤ \frac{C}{πt^{2}} ∫∫_{|z|<t/(1−|a|)} |μ(z) − a|dxdy

implies (4.9), where C is a constant depending on K only. Thus, the required weak Lipschitz continuity of f at the origin follows from Theorem 4.8. □

4.11. Remark. If the complex dilatation is approximately continuous uniformly on a compact set E, then the mapping f is weakly Lipschitz continuous uniformly on E, that is, for each 0 < α < 1 there exists a constant C = C(E) > 0 such that |f(z_{1}) − f(z_{2})| ≤ C|z_{1} − z_{2}|^{α} holds whenever z_{1}, z_{2} ∈ E.

The example of the quasiconformal radial stretching

f(z) = z (log \frac{1}{|z|})^{2}

defined on a neighborhood of the origin, whose complex dilatation

μ(z) = \frac{1}{1 − log(1/|z|)} \frac{z}{z̄}

is continuous at the origin, tells us that the weak Lipschitz continuity cannot be replaced by the stronger estimate |f(z)| ≤ C|z| log(1/|z|) in Corollary 4.10. Moreover, we cannot say anymore than (4.1) in the following sense.

4.12. Theorem. Let h : (0, δ] → ℝ be a continuous function such that

(i) h(t) = o(log(1/t)) as t → 0, and

(ii) h(t) → ∞ as t → 0.

Then there exists a quasiconformal mapping f : 𝔻 → 𝔻 with f(0) = 0 such that

lim sup_{z→0} \frac{log^{+}(|f(z)|/|z|)}{h(|z|)} > 0,

while the complex dilatation μ is continuous in 𝔻, vanishes at the origin, and therefore, satisfies (4.3).

Before proving the theorem we will introduce standard radial stretchings. Let M : (0, δ) → [1, +∞) be a bounded measurable function. For convenience, we extend M(t) to be identically 1 for t ≥ δ. We consider the two homeomorphisms f_{M} and g_{M} of the complex plane onto itself defined by

(4.13)f_{M}(z) = \frac{z}{|z|} exp (∫_{1}^{|z|} \frac{dt}{tM(t)}) and g_{M}(z) = \frac{z}{|z|} exp (∫_{1}^{|z|} \frac{M(t)dt}{t}) .

Then the complex dilatations of f_{M} and g_{M} are given by μ_{M} and −μ_{M}, respectively, where

μ_{M}(z) = −\frac{M(|z|) − 1}{M(|z|) + 1} · \frac{z}{z̄}.

Therefore, we have D_{μ_{M}} (z) = M(|z|), D_{−μ_{M}} (z) = 1/M(|z|) and K_{μ_{M}} (z) = K_{−μ_{M}} (z) = M(|z|) for almost all z.

4.14. Lemma. Let h be as in Theorem 4.12. Then there exists a function g : (0, δ] → ℝ with properties (a), (b), (c) and either (d) or (d’) below:

(a) g is differentiable, strictly decreasing and satisfies lim_{t→0} g(t) = +∞.

(b) g′ is absolutely continuous on the interval [δ_{1}, δ] for each 0 < δ_{1} < δ.

(c) g′′(t) = o(1/t^{2}) as t → 0.

(d) g(t_{n}) = h(t_{n}) for some sequence t_{n} in (0, δ] with t_{n} → 0. (d’) g(t) ≤ h(t) + O(1) as t → 0.

Note that condition (c) implies that g′(t) = o(1/t) and g(t) = o(log 1/t) as t → 0.

Proof of Lemma 4.14. Set H(x) = h(δe^{−x}) − h(δ) + 1 for x ≥ 0. Since h(t) = o(log 1/t), we note that H(x) = o(x) as x → +∞. For a given y > 0, we define H^{∗}(y) = max{x ≥ 0 : H(x) ≤ y}. One can then see that H^{∗}(y) → +∞ while y/H^{∗}(y) → 0 as y → +∞. Set y_{0} = min{H(x) : x ≥ 0} and x_{0} = 0. Let y_{n}, n = 1, 2, …, be a monotone sequence of positive numbers such that y_{1} > y_{0} and y_{n} → ∞ as n → ∞. Also let x_{n} = H^{∗}(y_{n}). It is easy to see that H(x_{n}) = y_{n} for n ≥ 1, H(x) ≥ y_{n} for x ≥ x_{n}, x_{0} < x_{1} < x_{2} < … and x_{n} → +∞ as n → ∞. Choosing a suitable sequence, we can make the following two conditions valid:

(1) x_{n+1} − x_{n} → +∞ if n → ∞,

(2) p_{n} := (y_{n+1} − y_{n})/(x_{n+1} − x_{n}) decreases to 0 as n → ∞.

First, we give a function g satisfying (a), (b), (c) and (d). Define the preliminary function G_{0} by

G_{0}(x) = y_{n} + p_{n}(x − x_{n}) + \frac{p_{n+1} − p_{n}}{2(x_{n+1} − x_{n})}(x − x_{n})^{2}, x ∈ [x_{n}, x_{n+1}), (4.15)

for n = 0, 1, …. Then, G_{0}(x_{n}) = y_{n}, G′_{0}(x_{n}) = p_{n} and G′_{0}(x_{n+1} − 0) = p_{n+1}. This function has the gap

D_{n} := G_{0}(x_{n+1}) − lim_{x→x_{n+1}−} G_{0}(x) = \frac{p_{n} − p_{n+1}}{2} (x_{n+1} − x_{n})

at the point x_{n+1}. So, we will fill it up by adding a suitable term to G_{0}. To this end, we consider an auxiliary function φ : [0, 1] → [0, 1] with the following properties:

(i) φ ∈ C^{2}[0, 1], (ii) φ(0) = 0, φ′(0) = 0, φ(1) = 1, and φ′(1) = 0, and (iii) 0 ≤ φ′(x) ≤ C_{1} and |φ′′(x)| ≤ C_{2} for x ∈ [0, 1], where C_{1} and C_{2} are

constants. For example, we can take φ(x) = sin^{2}(πx/2).

We now define the continuous function G : [0, +∞) → ℝ by

G(x) = G_{0}(x) + D_{n}φ (\frac{x − x_{n}}{x_{n+1} − x_{n}} ) x ∈ [x_{n}, x_{n+1}),

for n = 0, 1, …. Furthermore, G has the locally absolutely continuous derivative

G′(x) = p_{n} \frac{x_{n+1} − x}{x_{n+1} − x_{n}} + p_{n+1} \frac{x − x_{n}}{x_{n+1} − x_{n}} + \frac{p_{n} − p_{n+1}}{2} φ′ (\frac{x − x_{n}}{x_{n+1} − x_{n}} )

on [x_{n}, x_{n+1}]. Note here that G′(x) ≥ 0 and G′(x) → 0 as x → +∞ by (2). From the expression

G′′(x) = −\frac{p_{n+1} − p_{n}}{x_{n+1} − x_{n}} [1 − \frac{1}{2}φ′′ (\frac{x − x_{n}}{x_{n+1} − x_{n}} )] , x ∈ (x_{n}, x_{n+1}),

we obtain G′′(x) → 0 as x → +∞. We now set g(t) = G(log(δ/t)) for 0 < t ≤ δ. Then, g′(t) = −G′(log(δ/t))/t < 0 and g′′(t) = (G′′(log(δ/t)) − G′(log(δ/t)))/t^{2}, and thus, properties (a), (b) and (c) follow. Let t_{n} = δe^{−x_{n}}. Then g(t_{n}) = G(x_{n}) = H(x_{n}) = h(t_{n}), and (d) has been shown.

Next, we construct a function g satisfying (a), (b), (c) and (d’). First, we change the definition of p_{n} by setting p_{n} = (y_{n} − y_{n−1})/(x_{n+1} − x_{n}), and then, in the definition of G_{0} above, we replace y_{n} by y_{n−1}, where we set y_{−1} = y_{0}, and define g in the exactly same way as above. Properties (a), (b) and (c) now follow similarly. Also, in this case, we see that G_{0}(x) ≤ y_{n} ≤ H(x) for x ∈ [x_{n}, x_{n+1}). If D_{n} is bounded, we obtain the estimate G(x) = G_{0}(x) + O(1) ≤ H(x) + O(1), and hence, we confirm property (d’). Otherwise, taking a positive constant C, we set E_{n} = max{D_{n} − C, 0} for each n. Note that E_{n} > 0 for infinitely many n, in this case. Then, we modify the above definition of G by

G(x) = G_{0}(x) + min\{D_{n}, C\} φ (\frac{x − x_{n}}{x_{n+1} − x_{n}} ) − ∑_{k=0}^{n−1} E_{k}, x ∈ [x_{n}, x_{n+1}),

for n = 0, 1, … and set g(t) = G(log(δ/t)). Then, by definition, G(x) ≤ G_{0}(x) + O(1) as x → +∞. Since G(x_{n+1}) − G(x_{n}) ≥ C if E_{n} > 0, we can see that G(x) → +∞ as x → +∞. Now properties (a), (b), (c) and (d’) are easy to see for this function g. □

Proof of Theorem 4.12. Let g be a function satisfying properties (a), (b), (c) and (d) in Lemma 4.14. Since ε(t) = −tg′(t) is non-negative and converges to zero as t → 0, we can choose a small 0 < δ_{1} ≤ δ so that M(t) = 1/(1 − ε(t)) is bounded in 0 < t < δ_{1}. Now we extend M by setting M(t) = M(δ_{1}) for t ≥ δ_{1}. Then the radial stretching f_{M} has continuous complex dilatation μ_{M}. We also note that f_{M} has the form f_{M}(z) = Cz exp(g(|z|)) for |z| < δ_{1}, where C is a constant. Let t_{n} be the sequence in property (d). Then

lim sup_{z→0} \frac{log^{+}(|f(z)|/|z|)}{h(|z|)} ≥ lim sup_{n→∞} \frac{g(t_{n}) + log C}{h(t_{n})} = 1 > 0.

Thus, the assertions of the theorem have been proved. □

Now we show that (3.1) is qualitatively best possible condition for a function f to be Lipschitz continuous at the origin. The same is true for (3.3). Recall that (3.3) is equivalent to Dini type condition (3.6).

4.16. Theorem. Let h : (0, δ] → ℝ be a continuous function satisfying h(t) → +∞ if t → 0. Then there exist a quasiconformal radial stretching f : 𝔻 → 𝔻, f(0) = 0, with complex dilatation μ and a constant C such that D_{μ} = K_{μ} and that

∫_{r}^{1} \frac{ω_{μ}(t)dt}{t} ≤ h(r) + C

for r > 0 while f fails to be Lipschitz continuous at the origin.

Proof. The proof is somewhat similar to that of the previous theorem. First, we can assume that h(t) = o(log(1/t)) as t → 0. Let g be a function satisfying (a), (b), (c) and (d’) in Lemma 4.14. If we set ε(t) = −tg′(t), by (b) we then have ε(t) = o(1) and tε′(t) = −tg′(t) − t^{2}g′′(t) = o(1) as t → 0. In particular, we can choose a sufficiently small δ_{1} > 0 so that ε(t) < 1/2 and |tε′(t)| < 1 for t ∈ (0, δ_{1}). We now define M(t) by M(t) = 1+ε(t)+tε′(t)/2 for t ∈ (0, δ_{1}) and by M(t) = 1 for t ≥ δ_{1}. Then M is a bounded measurable function with 1 ≤ M ≤ 2. Let us consider the function f_{M} given by (4.13), which has complex dilatation μ = μ_{M}. We compute

ω_{μ}(t) = \frac{2}{t^{2}} ∫_{0}^{t}(M(s) − 1)sds = \frac{1}{t^{2}} ∫_{0}^{t}(s^{2}ε(s))′ds = ε(t)

for 0 < t < δ_{1}, and by (d’) we obtain

∫_{r}^{1} \frac{ω_{μ}(t)dt}{t} = ∫_{r}^{δ_{1}} \frac{ε(t)dt}{t} + O(1) = − ∫_{r}^{δ_{1}} g′(t)dt + O(1)
= g(r) + O(1) ≤ h(r) + O(1)

as r → 0. On the other hand, since 1/M(t) ≤ 1 − (M(t) − 1)/2, we see that

∫_{r}^{1} \frac{dt}{tM(t)} ≤ − log r − \frac{1}{2} ∫_{r}^{δ_{1}} \frac{ε(t)dt}{t} − \frac{1}{4} ∫_{r}^{δ_{1}} ε′(t)dt
= − log r + \frac{g(δ_{1}) − g(r)}{2} − \frac{ε(δ_{1}) − ε(r)}{4}

for sufficiently small r > 0, and therefore, log |f_{M}(z)/z| ≥ g(|z|)/2 + O(1) → ∞ as z → 0. □

Acknowledgement. This research was carried out during the authors’ visit to the University of Helsinki. The authors are thankful to the Department of Mathematics for financial support and its hospitality and would like to thank the referee for useful remarks. The first author was also partially supported by FFUY Grant 01.07/00241.

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Vladimir Ya. Gutlyanskiĭ: Institute of Applied Mathematics and Mechanics, NAS of Ukraine, ul. Roze Luxemburg 74, 83114, Donetsk, UKRAINE Fax: +380-622-552265 E–mail: gut@iamm.ac.donetsk.ua, gut@www.math.helsinki.fi

Toshiyuki Sugawa: Department of Mathematics, Kyoto University, 606 – 8502 Kyoto, JAPAN Fax: +81-75-753-3711 E–mail: sugawa@kusm.kyoto-u.ac.jp