On the Degenerate Beltrami Equation

V. Gutlyanskiĭ, O. Martio, T. Sugawa and M. Vuorinen

November 19, 2001

Abstract

We study the well-known Beltrami equation under the assumption that its measurable complex-valued coefficient μ(z) has the norm ∥μ∥_{∞} = 1. Sufficient conditions for the existence of a homeomorphic solution to the Beltrami equation on the Riemann sphere are given in terms of the directional dilatation coefficients of μ. A uniqueness theorem is also proved when the singular set of μ is contained in a totally disconnected compact set with a certain geometric condition.

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

1. Introduction

The analytic theory of plane quasiconformal mappings f is based on the Beltrami partial differential equation

fz̄ = μ(z) f_{z} a.e. (1.1)

with the complex-valued measurable coefficient μ satisfying the uniform ellipticity assumption ∥μ∥_{∞} < 1. In the case |μ(z)| < 1 a.e. in ℂ and ∥μ∥_{∞} = 1, equation (1.1) is called a degenerate Beltrami equation and the structure of the solutions heavily depends on the degeneration of μ. In this article, unless otherwise stated, by a Beltrami coefficient in a domain Ω we mean a complex-valued measurable function μ in Ω such that |μ| < 1 a.e. in Ω, and by a solution to the Beltrami equation (1.1) in a domain Ω we mean a function f in the Sobolev space W_{loc}^{1,1}(Ω) whose partial derivatives satisfy (1.1) in Ω. Then, f is called μ-conformal in Ω.

The measurable Riemann mapping theorem (cf. [1]) states that given a measurable function μ in the plane ℂ with ∥μ∥_{∞} < 1 there is a quasiconformal homeomorphism f : ℂ → ℂ, f ∈ W_{loc}^{1,2}(ℂ), satisfying (1.1). Contrary to this, the degenerate Beltrami equation need not have a homeomorphic solution and a solution, if it exists, need not be unique. See, for instance, [11]. Therefore, in order to obtain existence or uniqueness results, some extra constraints must be imposed on μ.

The degeneration of μ is usually expressed in terms of the pointwise maximal dilatation function

K(z) = K_{μ}(z) = \frac{1 + |μ(z)|}{1 − |μ(z)|}. (1.2)

This takes into account the absolute value of μ only. In this paper we show that for sharper results the argument of μ(z) should also be considered. For example, consider the Beltrami coefficients μ_{1}(z) = (1 − |z|)z/z̄ and μ_{2}(z) = (|z| − 1)z/z̄ defined in the unit disk 𝔻. It is immediate that ∥μ_{j}∥_{∞} = 1, j = 1, 2, and K_{μ_{1}}(z) = K_{μ_{2}}(z) whenever z ∈ 𝔻 \ {0}. The radial stretching f_{1} : 𝔻 → 𝔻, defined by

f_{1}(z) = \frac{z}{|z|^{2}} e^{2(1−1/|z|)} (1.3)

for z ∈ 𝔻 \ {0}, f_{1}(0) = 0, is μ_{1}-conformal. The second radial stretching

f_{2}(z) = \frac{z}{|z|(2 − |z|)} (1.4)

in the punctured disk 𝔻 \ {0} is μ_{2}-conformal and has the cavitation effect since it maps 𝔻 \ {0} homeomorphically onto the annulus 1/2 < |z| < 1. Actually, the continuous solution to the Beltrami equation with μ = μ_{2} is unique up to the post-composition of analytic functions, cf. Proposition 4.1 below, and hence the cavitation is inevitable in this case. Thus the cavitation problem requires more precise information on μ than merely on |μ|.

To study the aforementioned problem we employ the angular dilatation coefficient D_{μ,z_{0}}, see (2.13) below, to take into account an effect of the argument of μ as well. On one hand, it allows us to prove the existence of a homeomorphic solution f to the Beltrami equation (1.1) for a given Beltrami coefficient μ with ∥μ∥_{∞} = 1 provided that D_{μ,z_{0}} satisfies a local integrability condition for each z_{0}, see Theorem 3.5. We also obtain an estimate for the modulus of continuity of f. On the other hand, we establish a uniqueness theorem for the solution of (1.1) in the case when the singular sets of μ are totally disconnected compacta with certain geometric condition involving D_{μ,z_{0}}, see Theorem 4.3. The modulus estimate for annuli in Lemma 2.19 in terms of the integral means of the angular dilatation coefficients plays a crucial role in the proof of the existence and uniqueness results. A normal family argument is also used, see Propositions 2.1 and 2.3.

The idea of employing μ instead of |μ| in the study of some regularity problems for quasiconformal mappings is due to Andreian Cazacu [2] and Reich and Walczak [19]. O. Lehto [13], [14] was the first who considered the degenerate Beltrami equation from this point of view.

The degeneration of μ in terms of |μ(z)| or K_{μ}(z) has recently been extensively studied. This is due to the close connection of f in (1.1) to the solutions of elliptic partial differential equations. For the earlier studies of μ-homeomorphisms we refer to [4], [5], [18] and [17]. The results of Pesin [18] have been substantially extended by Brakalova and Jenkins [6]. For the recent deep theorems on the existence and uniqueness of μ-homeomorphisms see Iwaniec and Martin [11], who extended the well-known results of David [8] and Tukia [23], and see also [21], [20], [7] and the references therein.

Let us indicate a couple of features of our main results. First, by virtue of adoption of the angular dilatation coefficients, the existence theorem and its primitive, Theorem 2.15, cover many cases when K_{μ} fails to satisfy known integrability conditions, see, for instance, Examples 3.29 and 3.33. Even the case when the singular set of μ consists of finitely many points is of independent interest, see Section 3.

1.5. Theorem. Let μ(z) be a Beltrami coefficient in ℂ̂ such that the set of singularity Sing(μ) consists of finitely many points. Then there exists a homeomorphism f : ℂ̂ → ℂ̂ which is locally quasiconformal in ℂ̂ \ Sing(μ) and whose complex dilatation μ_{f} satisfies |μ_{f}(z)| = |μ(z)| a.e.

Secondly, Theorem 3.5 applied to the classical setting involving K_{μ} only gives the following result.

1.6. Theorem. Suppose that a Beltrami coefficient μ on ℂ satisfies

∫∫_{ℂ} e^{H(K_{μ}(z))} \frac{dxdy}{(1 + |z|^{2})^{2}} < +∞

for a measurable function H : [1, +∞) → ℝ for which there exist an integer n ≥ 1 and numbers c > 0, α ∈ (−∞, 1] such that H(t) ≤ ct/(log t)(log_{2} t) ⋯ (log_{n−1} t)(log_{n} t)^{α} when t is large enough. Then there exists a homeomorphic solution f : ℂ → ℂ to the Beltrami equation with μ such that f ∈ W_{loc}^{1,q}(ℂ) for every 1 ≤ q < 2 and that f^{−1} ∈ W_{loc}^{1,2}(ℂ). Moreover, if α < 1 and if 0 < α in the case n = 1 in addition, f satisfies the inequality

|f(z) − f(z_{0})| ≤ C exp (\frac{−c}{2(1 − α)} (log_{n+1} \frac{1}{|z − z_{0}|})^{1−α}) , |z − z_{0}| < δ_{0},

where the constants C > 0 and δ_{0} > 0 can be taken locally uniformly. In the case when n = 1 and α ≤ 0, the above inequality still holds if the constant c/(2(1−α)) is replaced by any larger number than it.

Here, log_{n} denotes the iterated logarithm, see Section 5. Examples show that this result is close to being optimal. Indeed, for each n, one cannot admit α to be greater than 1 in the integrability condition above. Moreover, the constant c/(2(1 − α)) and the exponent 1 − α in the above estimate for modulus of continuity cannot be replaced by smaller ones.

This paper is organized as follows. In Section 2, we give modulus estimates for ring domains and establish normality theorems for some families of homeomorphisms to prove Theorem 2.15. The existence theorems, modulus of continuity estimates and examples are collected in Section 3. Section 4 is devoted to uniqueness theorems. We provide basic examples for dominating factors and modulus bounds, which are used to formulate our existence and uniqueness theorems, and give a proof of Theorem 1.6 in Section 5.

2. Sequences of Self-Homeomorphisms

The degenerate Beltrami equation need not have a homeomorphic solution nor even a nonconstant solution. A usual approach for the existence of a solution is to consider a sequence f_{n} of quasiconformal homeomorphisms satisfying (1.1) with the Beltrami coefficient μ_{n}, ∥μ_{n}∥_{∞} < 1, such that μ_{n} → μ a.e. and then to use a normal family argument to obtain a limit mapping f. Some conditions must be imposed on μ in order to guarantee the normality.

In the following we introduce a modulus method to study normal families of homeomorphisms. This will be employed to prove the solvability problem of the degenerated Beltrami equation, however, the method is of independent interest.

We introduce some notation. We denote the Euclidean distance and the spherical distance between z and w by d(z, w) = |z − w| and d^{♯}(z, w) = |z − w|/\sqrt{(1 + |z|^{2})(1 + |w|^{2})}, respectively. Also we denote by A(z_{0}, r, R) and by A^{♯}(z_{0}, r, R) the (circular) annuli in the Euclidean and the spherical metric, respectively, i.e.,

A(z_{0}, r, R) = B(z_{0}, R) \ \overline{B}(z_{0}, r) and A^{♯}(z_{0}, r, R) = B^{♯}(z_{0}, R) \ \overline{B}^{♯}(z_{0}, r)

for z_{0} ∈ ℂ̂ and 0 ≤ r < R, where B(z_{0}, r) = {z ∈ ℂ : |z − z_{0}| < r} and B^{♯}(z_{0}, r) = {z ∈ ℂ̂ : d^{♯}(z, z_{0}) < r}. Here, in the case when z_{0} = ∞, we set B(∞, r) = {z ∈ ℂ̂ : |z| > 1/r}, and hence, A(∞, r, R) = A(0, 1/R, 1/r). In the sequel, for subsets E, E_{0}, E_{1} of ℂ̂, diam E and dist(E_{0}, E_{1}) stand for the diameter of E and the distance between E_{0} and E_{1}, respectively, measured in the Euclidean metric d. Similarly, diam^{♯}E and dist^{♯}(E_{0}, E_{1}) stand for those measured in the spherical metric d^{♯}. We also denote by A and A^{♯} the two dimensional Lebesgue measure and the spherical measure, respectively, i.e., A(E) = ∫_{E} ; ∫ dxdy

and A^{♯}(E) = ∫_{E} ; ∫ (1 + |z|^{2})^{−2}dxdy.

A doubly connected domain is called a ring domain. The modulus m of a ring domain A is the number such that A is conformally equivalent to {1 < |z| < e^{m}} and will be denoted by mod A. When A is conformally equivalent to ℂ^{∗} = ℂ \ {0}, we define mod A = ∞. A non-negative function ρ(z, r, R) in (z, r, R) ∈ ℂ̂ × (0, +∞) × (0, +∞), r < R, will be called a modulus constraint if ρ(z_{0}, r, R) → +∞ as r → 0 for any fixed R ∈ (0, +∞) and z_{0} ∈ ℂ̂.

We denote by H_{ρ} the family of all normalized homeomorphisms f : ℂ̂ → ℂ̂ such that the condition

mod f(A(z_{0}, r, R)) ≥ ρ(z_{0}, r, R)

holds for all z_{0} ∈ ℂ̂ and r, R ∈ (0, +∞) with r < R. Here and hereafter, a homeomorphism f : ℂ̂ → ℂ̂ is said to be normalized if f fixes 0, 1 and ∞. Note that this condition is not Möbius invariant.

Note also that, without changing the family H_{ρ}, we may assume the monotonicity condition ρ(z_{0}, r_{1}, R_{1}) ≤ ρ(z_{0}, r_{2}, R_{2}) for z_{0} ∈ ℂ̂ and r_{2} ≤ r_{1} < R_{1} ≤ R_{2} by replacing ρ by a suitable one.

Similarly, a non-negative function ρ^{♯}(z, r, R) in (z, r, R) ∈ ℂ̂×(0, 1)×(0, 1), r < R, will be called a spherical modulus constraint if the same set of conditions is satisfied for ρ^{♯}. We let H_{ρ^{♯}}^{♯} be the family of all normalized self-homeomorphisms f such that mod f(A^{♯}(z_{0}, r, R)) ≥ ρ^{♯}(z_{0}, r, R) holds for all z_{0}, r, R.

A family of continuous functions is said to be pre-normal or normal if it is relatively compact or compact, respectively, with respect to the topology of local uniform convergence.

Now we are ready to state our main propositions. A similar statement can be found in Lehto’s paper [14] under stronger assumptions (see also Lemma 1 in Section 4 of [6]).

2.1. Proposition. Let ρ^{♯} be a spherical modulus constraint. Then

1) H_{ρ^{♯}}^{♯} is a normal family with respect to the uniform convergence in ℂ̂, and 2) every f ∈ H_{ρ^{♯}}^{♯} satisfies the inequality

d^{♯}(f(z_{1}), f(z_{2})) ≤ Ce^{− \frac{1}{2} ρ^{♯}(z_{0},r_{1},r_{2})}, z_{1}, z_{2} ∈ B^{♯}(z_{0}, r_{1}), (2.2)

for z_{0} ∈ ℂ̂ and 0 < r_{1} < r_{2} < 1/2\sqrt{2}, where C is an absolute constant.

2.3. Proposition. Let ρ be a modulus constraint. Then

1) H_{ρ} is a normal family with respect to the uniform convergence in ℂ̂, and 2) for each R > 0 there is a constant C = C(R, ρ) > 0 depending only on R

and ρ such that every f ∈ H_{ρ} satisfies

|f(z_{1}) − f(z_{2})| ≤ Ce^{−ρ(z_{0},r_{1},r_{2})}, z_{1}, z_{2} ∈ B(z_{0}, r_{1}), (2.4)

for |z_{0}| ≤ R and 0 < r_{1} < r_{2} < R.

For the proof of these propositions we need auxiliary lemmas. The first one is shown by applying the famous Teichmüller’s lemma on his extremal ring domain. See, for example, [10], where the authors assert that we can take C_{0} = π^{−1} log 2(1 + \sqrt{2}) = 0.50118 … below.

2.5. Lemma. There exists a universal constant C_{0} > 0 with the property that for a ring domain B in ℂ with mod B > C_{0} which separates a point z_{0} from ∞ we can choose an annulus A in B of the form A = A(z_{0}, r_{1}, r_{2}), r_{1} < r_{2}, so that mod A ≥ mod B − C_{0} holds.

We need information about the size of components of the complement of a ring domain of sufficiently large modulus. There are several kinds of such estimates, among which the following form due to Lehto-Virtanen [15, Lemma I.6.1] is probably best known. Let B be a ring domain whose complement in ℂ̂ consists of continua E_{0} and E_{1}. Then

min\{diam^{♯}E_{0}, diam^{♯}E_{1}\} ≤ \frac{π}{\sqrt{2mod B}} .

This inequality is very explicit, however, the order in mod B is not best possible. So, we prepare the following result where the order in mod B is the best.

2.6. Lemma. Let B be an arbitrary ring domain in ℂ̂ and let E_{0} and E_{1} be the components of ℂ̂ \ B. Then the inequality

min\{diam^{♯}E_{0}, diam^{♯}E_{1}\} ≤ C_{1}e^{− \frac{1}{2} mod B}

holds, where C_{1} is an absolute constant.

Proof. We may assume that ∞ ∈ E_{1}. Then, we get the desired conclusion by combining Lemma 2.5 with the following elementary but sharp result. In particular, we can use the value C_{1} = 2e^{C_{0}/2} = 2.56957 ….

2.7. Lemma. Let A be an annulus in ℂ̂ whose complement consists of disjoint closed disks E_{0} and E_{1}. Then

min\{diam^{♯}E_{0}, diam^{♯}E_{1}\} ≤ \frac{1}{cosh(\frac{1}{2}mod A)}.

Equality holds if and only if diam^{♯}E_{0} = diam^{♯}E_{1} and if the spherical centers of E_{0} and E_{1} are antipodal.

Proof. If diam^{♯}E_{j} > diam^{♯}E_{1−j} for some j = 0, 1, we can decrease diam^{♯}E_{j} a little while leaving diam^{♯}E_{1−j} and min{diam^{♯}E_{0}, diam^{♯}E_{1}} invariant, then the resulting annulus will have larger modulus. Hence, using this argument, we may assume diam^{♯}E_{0} = diam^{♯}E_{1} without loss of generality. After a suitable isometric Möbius transformation with respect to the spherical metric, we can further assume that E_{0} and E_{1} are symmetric in the imaginary axis and that the center of E_{0} is a positive real number. Let E_{0}∩ℝ = [r, R] and δ be the hyperbolic diameter of E_{0} in the hyperbolic plane ℍ = {z : Re z > 0}. Note that

δ = ∫_{r}^{R} \frac{dx}{2x} = \frac{1}{2} log \frac{R}{r} = log t,

where we set t = \sqrt{R/r} = e^{δ} > 1. Since ℍ \ E_{0} is Möbius equivalent to the annulus A(0, tanh(δ/2), 1), we can compute the modulus of A as follows;

mod A = 2 mod (ℍ \ E_{0}) = 2 log coth \frac{δ}{2} = 2 log \frac{t + 1}{t − 1}.

In particular, we have t = coth(m/2), where we set m = mod A/2. On the other hand,

diam^{♯}E_{0} = d^{♯}(r, R) = \frac{R − r}{\sqrt{1 + R^{2}}\sqrt{1 + r^{2}}} = \frac{(t^{2} − 1)r}{\sqrt{1 + t^{4}r^{2}}\sqrt{1 + r^{2}}}
= \frac{t^{2} − 1}{\sqrt{1 + t^{4} + r^{−2} + t^{4}r^{2}}} ≤ \frac{t^{2} − 1}{\sqrt{1 + t^{4} + 2t^{2}}} = \frac{t^{2} − 1}{t^{2} + 1}
= \frac{coth^{2}(m/2) − 1}{coth^{2}(m/2) + 1} = \frac{1}{cosh m},

where equality holds if and only if rt = 1, equivalently, rR = 1. The last relation means that the spherical center of E_{0} is 1 and vice versa. In this case, the spherical center of E_{1} is −1, which is the antipode of 1. Hence, the last assertion of the lemma follows.

Lemma 2.6 has the best order in mod B, however, if we restrict ourselves to ring domains in the finite plane ℂ, the order is no longer best possible. The following estimate has the best order in mod B in this case, though the extra factor dist(E_{0}, E_{1}) will come into.

2.8. Lemma. 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_{2}, we have the estimate

diam E_{0} ≤ C_{3} dist(E_{0}, E_{1})e^{−mod B},

where C_{2} and C_{3} are positive absolute constants.

Proof. We may assume that dist(E_{0}, E_{1}) = 1, 0 ∈ E_{0} and 1 ∈ E_{1}. Let a be an arbitrary point in E_{0} other than 0. Then, by Teichmüller’s modulus theorem (see [15]), we have

mod B ≤ 2μ (\sqrt{\frac{|a|}{1 + |a|}}) ,

where μ(r) denotes the modulus of the Grötszch ring B(0, 1) \ [0, r]. Using the well-known estimate μ(r) < log(4/r), we obtain

|a| ≤ \frac{16}{e^{mod B} − 16} ≤ 32e^{−mod B}

if mod B > 5 log 2. Hence, we obtain diam E_{0} ≤ 64e^{−mod B} whenever mod B > 5 log 2. In particular, the assertion holds for C_{2} = 5 log 2 and C_{3} = 64.

Proof of Proposition 2.1. Observe first that d^{♯}(0, 1) = d^{♯}(1, ∞) = 1/\sqrt{2} and that d^{♯}(0, ∞) = 1. In particular, when r_{2} < 1/2\sqrt{2}, the disk B = B^{♯}(z_{0}, r_{2}) cannot contain more than one of the three fixed points 0, 1, ∞. Therefore, the component ℂ̂ \ f(B) of the complement of f(A^{♯}(z_{0}, r_{1}, r_{2})) has diameter at least 1/\sqrt{2}. Therefore, if the modulus of f(A^{♯}(z_{0}, r_{1}, r_{2})) is sufficiently large, the image of the disk \overline{B}^{♯}(z_{0}, r_{1}) must be the smaller component of the complement of f(A^{♯}(z_{0}, r_{1}, r_{2})). Now inequality (2.2) follows from Lemma 2.6. This inequality in turn implies the equicontinuity of H_{ρ^{♯}}^{♯}. Since ℂ̂ is compact, then by the Ascoli-Arzela theorem, the family H_{ρ^{♯}}^{♯} is normal.

Let f be the uniform limit of a sequence f_{n} in H_{ρ^{♯}}^{♯}. We show that f is a member of H_{ρ^{♯}}^{♯}. Since f is the uniform limit of homeomorphisms f_{n}, we have deg f = lim deg f_{n} = 1. In particular, f : ℂ̂ → ℂ̂ is a surjective continuous map. We now consider the open set

V = {z ∈ ℂ̂; f is locally constant at z}.

First we show the following Claim. If z_{0} ∈ ℂ̂ \ V, then f(z) ≠ f(z_{0}) for any z ∈ ℂ̂ \ {z_{0}}.

By permuting the roles of 0, 1, ∞ if necessary, we may assume that z_{0} ≠ ∞ and that f(z_{0}) ≠ ∞. Pick any point w_{0} other than z_{0}. We will show that f(w_{0}) ≠ f(z_{0}). Choose a small positive number R so that R < min{d^{♯}(z_{0}, w_{0}), d^{♯}(z_{0}, ∞)}. Then inf_{n} mod (f_{n}(A^{♯}(z_{0}, δ, R))) > C_{0} for sufficiently small δ > 0, where C_{0} is the constant in Lemma 2.6. By virtue of Lemma 2.6, we can find an annulus A_{n} of the form A(f_{n}(z_{0}), r_{n}, r′_{n}), r_{n} < r′_{n} in f_{n}(A^{♯}(z_{0}, δ, R)) for n large enough.

Since f is not locally constant at z_{0}, there exists a point z_{1} in the disk d^{♯}(z, z_{0}) < δ with f(z_{0}) ≠ f(z_{1}). The annulus A_{n} separates f_{n}(z_{0}), f_{n}(z_{1}) from f_{n}(w_{0}), ∞, so we obtain |f_{n}(z_{1}) − f_{n}(z_{0})| ≤ r_{n} and r′_{n} ≤ |f_{n}(w_{0}) − f_{n}(z_{0})|. In particular, |f_{n}(z_{1}) − f_{n}(z_{0})| ≤ |f_{n}(w_{0}) − f_{n}(z_{0})| for n large enough. Letting n → ∞, we obtain 0 < |f(z_{1}) − f(z_{0})| ≤ |f(w_{0}) − f(z_{0})|, and hence f(w_{0}) ≠ f(z_{0}).

We next show that V is empty. Suppose that V has a non-empty connected component V_{0}. Then f takes a constant value, say b, in V_{0}. Because V_{0} ≠ ℂ̂, we can take a point z_{0} from ∂V_{0}. By continuity, we have f(z_{0}) = b. On the other hand, from the above Claim, we can deduce that f(z) ≠ f(z_{0}) = b for any point z other than z_{0}, which contradicts the fact that f = b in V_{0}. Thus we conclude that V is empty, namely, f is not locally constant at any point.

By using Claim again, we obtain f(z_{1}) ≠ f(z_{2}) if z_{1} ≠ z_{2}. Thus, the injectivity of f follows. Finally, by the continuity of moduli of ring domains with respect to the Hausdorff convergence (see [15]), we conclude that f ∈ H_{ρ^{♯}}^{♯}.

Proof of Proposition 2.3. It is easy to see that there is a spherical modulus constraint ρ^{♯} such that H_{ρ} ⊂ H_{ρ^{♯}}^{♯}. From Proposition 2.1, the normality of H_{ρ} now follows. We next give an estimate of dist(E_{0}, E_{1}), where E_{0} and E_{1} are the bounded and unbounded components of ℂ̂ \ f(A), A = A(z_{0}, r_{1}, r_{2}), respectively. Since H_{ρ} is normal, S = sup{|g(z)| : g ∈ H_{ρ}, |z| ≤ 2R} < +∞. Now it is clear that dist(E_{0}, E_{1}) ≤ diam f(∂B(z_{0}, r_{2})) ≤ 2S. Set m = mod f(A). If m > C_{2}, we obtain diam E_{0} ≤ 2SC_{3}e^{−m} by Lemma 2.8. Otherwise, we have diam E_{0} ≤ 2S ≤ 2Se^{C_{2}}e^{−m}. Hence, the proof is complete.

Let μ be a Beltrami coefficient defined in ℂ with ∥μ∥_{∞} ≤ 1. In order to apply Proposition 2.3 to the study of the Beltrami equation, we first specify an approximation procedure. This is done in a standard way. For n = 1, 2, ..., we set μ_{n}(z) = μ(z), if |μ(z)| ≤ 1 − 1/n, (2.9)

and μ_{n}(z) = 0 otherwise, and denote by f_{n} the sequence of quasiconformal automorphisms of the extended complex plane preserving 0, 1 and ∞, and having μ_{n} as its complex dilatation. The existence of such f_{n} is guaranteed by the measurable Riemann mapping theorem. We will call such f_{n} the canonical approximating sequence for μ. The topological structure of the set f_{n} with respect to the uniform convergence under some additional assumptions on the generating coefficient μ will be described in Theorem 2.15.

A function H : [0, +∞) → ℝ is called a dominating factor if the following conditions are satisfied;

1. H(x) is continuous and strictly increasing in [x_{0}, +∞) and H(x) = H(x_{0}) for x ∈ [0, x_{0}] for some x_{0} ≥ 0, and

2. the function e^{H(x)} is convex in x ∈ [0, +∞). The convexity of e^{H} implies that H(x) → +∞ as x → +∞. In the sequel, the inverse H^{−1} of H will mean the inverse of the homeomorphism H : [x_{0}, +∞) → [H(0), +∞).

A dominating factor H is said to be of divergence type if

∫_{1}^{+∞} \frac{H(x)dx}{x^{2}} = +∞, (2.10)

and otherwise H is said to be of convergence type. Denote by D the set of all dominating factors of divergence type. Note that H(ηx) and ηH(x) are dominating factors of the same type as H(x) if H(x) is a dominating factor and if η is a positive constant. The following is the most important feature of the divergence condition (2.10).

2.11. Lemma. Let H be a dominating factor. Then H is of divergence type if and only if

∫_{t_{1}}^{+∞} \frac{dt}{H^{−1}(t)} = +∞ (2.12)

for a sufficiently large number t_{1}.

Proof. By the change of variables t = H(x) and integration by parts, we see

∫_{t_{1}}^{t_{2}} \frac{dt}{H^{−1}(t)} = ∫_{x_{1}}^{x_{2}} \frac{dH(x)}{x} = \frac{H(x_{2})}{x_{2}} − \frac{H(x_{1})}{x_{1}} + ∫_{x_{1}}^{x_{2}} \frac{H(x)dx}{x^{2}},

where t_{j} = H(x_{j}) for j = 1, 2. Noting that H(x)/x is positive for x large enough, we easily get to the “only if ” part. In order to show the “if ” part, assume that (2.10) fails. Then, in view of the above identity, (2.12) would imply that H(x)/x → ∞ if x → ∞. In particular, H(x)/x > C > 0 holds for sufficiently large x, where C is a positive constant. Hence, letting x_{1} be large enough, we would see that ∫_{x_{1}}^{x_{2}} H(x)dx/x^{2} > ∫_{x_{1}}^{x_{2}} C dx/x = C log(x_{2}/x_{1}) → ∞ as x → ∞, which is a contradiction. Thus, (2.10) follows.

Let μ ∈ L^{∞}(Ω) be a Beltrami coefficient with |μ| < 1 a.e. in an open subset Ω of ℂ̂. We define the angular dilatation D_{μ,z_{0}} of μ at z_{0} ∈ ℂ̂ by

D_{μ,z_{0}}(z) = \frac{|1 − μ(z)\frac{z̄−z̄_{0}}{z−z_{0}} |^{2}}{1 − |μ(z)|^{2}} (2.13)

in the case when z_{0} is finite, and by D_{μ,∞}(z) = D_{μ,0}(z) in the case when z_{0} = ∞. Then D_{μ,z_{0}} is a measurable function in Ω and satisfies the inequality 1/K_{μ}(z) ≤ D_{μ,z_{0}}(z) ≤ K_{μ}(z) a.e. for each z_{0} ∈ ℂ̂. Note that D_{μ,0}(z) = K_{μ}(z) holds a.e. if and only if μ(z) has the form −ρ(z)z/z̄ for a non-negative measurable function ρ. The name of D_{μ,z_{0}} comes from the following important relation: if f is μ-conformal in Ω 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) (2.14)

holds for almost all z ∈ Ω, where J_{f} is the Jacobian of f. The similar quantity D_{−μ,z_{0}} is called the radial dilatation of μ at z_{0} because the counterpart of the above relation can be obtained:

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

These are called directional dilatations.

2.15. Theorem. Let μ be a Beltrami coefficient in ℂ. Assume that, for each z_{0} ∈ ℂ̂, one of the following conditions holds for some positive constants M = M(z_{0}) and r_{0} = r_{0}(z_{0}) :

1) D_{μ,z_{0}}(z) ≤ M a.e. in B(z_{0}, r_{0});

2) There is a dominating factor H = H_{z_{0}} of divergence type such that

∫B(z_{0},r_{0})e^{H(D_{μ,z_{0}}(z))}dA(z) ≤ M

holds for z_{0} ∈ ℂ, while the above condition is replaced by

∫B(∞,r_{0})e^{H(D_{μ,0}(z))} \frac{dA(z)}{|z|^{4}} ≤ M

if z_{0} = ∞.

Then the canonical approximating sequence f_{n} for μ forms a pre-normal family with respect to the uniform convergence in ℂ̂ and every limit function f of this sequence is a self-homeomorphism of ℂ̂, and admits the following modulus of continuity estimates according to cases 1) or 2) at each point z_{0} with |z_{0}| ≤ R_{0}, where R_{0} is an arbitrary positive number:

|f(z) − f(z_{0})| ≤ C|z − z_{0}|^{1/M} (2.16)

or

|f(z) − f(z_{0})| ≤ C exp \{− ∫_{1+c}^{2m+c} \frac{dt}{2H^{−1}(t)}\} , (2.17)

respectively, for |z − z_{0}| < r_{1} and 0 < r_{1} ≤ min{r_{0}, R_{0}}, where m = log(r_{1}/|z − z_{0}|), c = log(M/πr_{1}^{2}) and C is a constant depending only on μ and R_{0}.

2.18. Remark. This theorem does not tell anything about the regularity of the limit mapping f nor the existence of μ-conformal homeomorphisms. At the moment, the following can be said at least. Let Sing(μ) be the singular set of a Beltrami coefficient μ, i.e.,

Sing(μ) = {z_{0} ∈ ℂ̂ : ∥μ∥_{L^{∞}(B(z_{0},r))} = 1 for any r > 0}.

Note that Sing(μ) is a compact set in ℂ̂. Note also that Sing(μ) can have positive area although |μ| is always assumed to be less than 1 almost everywhere. Let Ω = ℂ̂ \ Sing(μ). Then, by construction, f is locally quasiconformal in Ω and its complex dilatation agrees with the given μ a.e. and f|_{Ω} is unique up to the post-composition by a conformal map. Further discussions on the existence of μconformal homeomorphisms and their regularity will be made in the next section.

The proof of the theorem is based on Proposition 2.3 and the estimate for the change of moduli of ring domains under homeomorphisms in the Sobolev space W_{loc}^{1,1} stated in the lemmas below.

The first result is essentially due to [2] and [19] and is given now under somewhat weaker assumptions.

2.19. Lemma. Let μ be a Beltrami coefficient on a domain Ω in ℂ and f : Ω → ℂ be a μ-conformal embedding. Suppose that D_{μ,z_{0}}(z) is locally integrable in the annulus A = A(z_{0}, r_{1}, r_{2}) ⊂ Ω. Then

mod f(A) ≥ ∫_{r_{1}}^{r_{2}} \frac{dr}{rψ_{μ}(r, z_{0})}, (2.20)

where

ψ_{μ}(r, z_{0}) = \frac{1}{2π} ∫_{0}^{2π} D_{μ,z_{0}}(z_{0} + re^{iθ})dθ. (2.21)

Proof. We may assume that z_{0} = 0 and Ω = A = A(0, 1, R). Composing a suitable conformal mapping, we may further assume that A′ = f(A) = A(0, 1, R′). Let ν be the positive Borel measure on A defined by ν(E) = A(f(E)). By a usual argument (see, e.g., [1] or [15]), we obtain J_{f}dA ≤ dν on A in the sense of measure, where J_{f} = |f_{z}|^{2} − |fz̄|^{2}. In particular, J_{f} ∈ L_{loc}^{1}(A).

Denote by γ_{r} the circle |z| = r. Then the assumption f ∈ W_{loc}^{1,1}(A) together with the Gehring-Lehto theorem (see [15]) implies that f is absolutely continuous on γ_{r} and totally differentiable at every point in γ_{r} except for a set of linear measure 0 for almost all r ∈ (1, R). By Fubini’s theorem, we observe that D_{μ} and J_{f} are integrable on γ_{r} for almost all r ∈ (1, R). For such an r, we have

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

Taking (2.14) into account, we use Schwarz’s inequality to obtain

(2π)^{2} ≤ r^{2} ∫_{0}^{2π} D_{μ}(re^{iθ})dθ ∫_{0}^{2π} \frac{J_{f}}{|f|^{2}} (re^{iθ})dθ,

and hence

\frac{2π}{rψ_{μ}(r)} ≤ r ∫_{0}^{2π} \frac{J_{f}}{|f|^{2}} (re^{iθ})dθ

for almost all r ∈ (1, R), where ψ_{μ}(r) = ψ_{μ}(r, 0). Integrating both sides with respect to r from 1 to R, we obtain

2π∫_{1}^{R} \frac{dr}{rψ_{μ}(r)} ≤ ∫_{1}^{R}∫_{0}^{2π} \frac{J_{f}}{|f|^{2}} rdθdr = _{∫A} \frac{J_{f}dA}{|f|^{2}}

≤ _{∫A} \frac{dν}{|f|^{2}} = _{∫A′} \frac{dA(w)}{|w|^{2}} = 2π log R′ = 2π mod A′

and thus arrive at the required inequality (2.20).

The following auxiliary result may be of independent interest. The basic idea is due to Brakalova-Jenkins [6], see also [17, p. 51].

2.22. Lemma. Let f be a μ-conformal embedding of A = A(z_{0}, r_{0}e^{−m}, r_{0}) into ℂ. Suppose that a dominating factor H satisfies

\begin{matrix}_{∫A}e^{H(D_{μ,z_{0}}(z))}dA(z) ≤ M, if z_{0} ∈ ℂ, and \\ _{∫A}e^{H(D_{μ,0}(z))} \frac{dA(z)}{|z|^{4}} ≤ M, if z_{0} = ∞. \end{matrix} (2.23)

Then we have

mod f(A(z_{0}, r_{0}e^{−m}, r_{0})) ≥ ∫_{1/2}^{m} \frac{dt}{H^{−1}(2t + log(M/πr_{0}^{2}))} (2.24)

Proof. Let first z_{0} ≠ ∞. Setting

h(r) = \frac{r^{2}}{2π} ∫_{0}^{2π} e^{H(D_{μ,z_{0}}(z_{0}+re^{iθ}))}dθ,

we rewrite inequality (2.23) in the form

2π ∫_{r_{0}e^{−m}}^{r_{0}} h(r)\frac{dr}{r} = 2π ∫_{0}^{m} h(r_{0}e^{−t})dt ≤ M.

By Chebyshev’s inequality, the set T = {t ∈ (0, m) : h(r_{0}e^{−t})) > L} has the length

|T | = _{∫T} dt ≤ \frac{M}{2πL}.

Since e^{H} is a convex function, Jensen’s inequality yields e^{H(ψ(r))} ≤ h(r)/r^{2} where ψ(r) = ψ_{μ}(r, z_{0}). This implies the inequality ψ(r_{0}e^{−t}) ≤ H^{−1}(2t + log(L/r_{0}^{2})) for t ∈ (0, m) \ T. By Lemma 2.19 we get

mod f(A) ≥ ∫_{0}^{m} \frac{dt}{ψ(r_{0}e^{−t})} ≥ _{∫(0,m)\T} \frac{dt}{H^{−1}(2t + log(L/r_{0}^{2}))}
≥ ∫_{|T |}^{m} \frac{dt}{H^{−1}(2t + log(L/r_{0}^{2}))} ≥ ∫_{M/2πL}^{m} \frac{dt}{H^{−1}(2t + log(L/r_{0}^{2}))}.

Finally, letting L = M/π, we obtain (2.24).

The remaining case is that z_{0} = ∞. Let φ(z) = 1/z be the inversion. Let μ̂ be the complex dilatation of the map g = φ ∘ f ∘ φ. Then D_{μ̂,0}(z) = D_{μ,0}(1/z). Now the required inequality immediately follows from the previous one.

Proof of Theorem 2.15. Let f_{n} be the canonical approximating sequence for μ. Suppose that μ satisfies assumption 2) at z_{0} ∈ ℂ. Since

H ∘ D_{μ_{n},z_{0}}(z) ≤ max {H ∘ D_{μ,z_{0}}(z), H(1)} (2.25)

for the dominating factor H, the sequence

M_{n} = _{∫B(z_{0},r_{0})}e^{H(D_{μn,z_{0}}(z))}dA(z) (2.26)

satisfies

lim_{n→∞} M_{n} = _{∫B(z_{0},r_{0})}e^{H(D_{μ,z_{0}}(z))}dA(z) ≤ M = M(z_{0})

by Lebesgue’s dominating convergence theorem. We now see by Lemma 2.22 that

mod f_{n}(A(z_{0}, re^{−m}, r)) ≥ ∫_{1/2}^{m} \frac{dt}{H^{−1}(2t + log(M_{n}/πr^{2}))} =: ρ_{n}(z_{0}, re^{−m}, r)

for 0 < r ≤ r_{0} and m > 1/2. Set ρ̂_{n} = inf{ρ_{k}; k ≥ n}. Then, by virtue of Lemma 2.11, ρ̂_{n}(z_{0}, re^{−m}, r) → ∞ as m → +∞. For z_{0} = ∞ the same conclusion follows.

In the case when μ satisfies assumption 1), the situation is simpler. Since ψ_{μ}(r, z_{0}) ≤ M for a.e. 0 < r < r_{0}, by Lemma 2.19, we obtain mod f_{n}(A(z_{0}, re^{−m}, r)) ≥ m/M for 0 < r ≤ r_{0} and m > 0, so we just set ρ̂_{n}(z_{0}, re^{−m}, r) = m/M in this case. For the other points (z_{0}, r, R), we simply set ρ̂_{n}(z_{0}, r, R) = 0.

The sequence f_{k}, k ≥ n, then belongs to the class Hρ̂_{n} for each n, and hence is normal by Proposition 2.3. Let f be a limit function of the sequence f_{k}. We show the estimates of modulus of continuity of f. Since case 1) is easier to treat, we consider only case 2). Letting n → ∞, we obtain f ∈ H_{ρ_{∞}}, where ρ_{∞} = lim inf_{n→∞} ρ_{n}. By Proposition 2.3, we have

|f(z) − f(z_{0})| ≤ C exp ( − ρ_{∞}(z_{0}, r_{1}e^{−m}, r_{1}))

for |z − z_{0}| ≤ r_{1}e^{−m}, |z_{0}| ≤ R_{0} and r_{1} ≤ min{r_{0}, R_{0}}, where C is a constant depending only on μ and R_{0}. Letting |z − z_{0}| = r_{1}e^{−m}, we obtain (2.17) by

mod f(A(z_{0}, r_{1}e^{−m}, r_{1})) ≥ ρ_{∞}(z_{0}, r_{1}e^{−m}, r_{1}) ≥ ∫_{1+c}^{2m+c} \frac{dt}{H^{−1}(t)},

where c = log(M/πr_{1}^{2}).

3. Existence Theorems

In order to give some applications of Theorem 2.15 to the study of the degenerate Beltrami equation we need the well-known regularity results, see, e.g., [6], Lemmas 4 – 6 and Proposition 10, and also [18], [17].

3.1. Proposition. Let μ be a Beltrami coefficient in ℂ and suppose that a homeomorphism f : ℂ̂ → ℂ̂ is a uniform limit of the canonical approximating sequence f_{n} for μ. If K_{μ} ∈ L_{loc}^{p}(ℂ) for some p > 1, then f ∈ W_{loc}^{1,q}(ℂ), q = 2p/(1+p), and f satisfies the Beltrami equation with μ. Moreover, f^{−1} ∈ W_{loc}^{1,2}(ℂ).

3.2. Remark. It is shown in [6, Lemma 3] that K_{μ} ∈ L_{loc}^{1}(ℂ) implies the ACL property of f.

3.3. Remark. The Sobolev embedding theorem for spheres [16] or Gehring’s oscillation inequality, see, e.g., [11, Lemma 5.2], implies that if f : ℂ → ℂ is a μ-conformal homeomorphism with K_{μ} ∈ L_{loc}^{1}(ℂ), then for every compact set E in ℂ there exist constants C and a such that

|f(z_{1}) − f(z_{2})| ≥ C e^{−a/|z_{1}−z_{2}|^{2}} (3.4)

for z_{1}, z_{2} ∈ E with z_{1} ≠ z_{2}. The same inequality is also obtained in [22, p. 75] as a consequence of the Length-Area principle.

Proposition 2.1 together with Theorem 2.15 yields the following statement.

3.5. Theorem. Suppose that μ is a Beltrami coefficient in ℂ such that:

1) K_{μ} ∈ L_{loc}^{p}(ℂ) for some p > 1; and

2) for each point z_{0} ∈ ℂ̂ either D_{μ,z_{0}} ≤ M a.e. in B(z_{0}, r_{0}) or

\begin{matrix}_{∫B(z_{0},r_{0})}e^{H(D_{μ,z_{0}}(z))}dA(z) ≤ M, & if z_{0} ∈ ℂ, and \\ _{∫B(∞,r_{0})}e^{H(D_{μ,0}(z))} \frac{dA(z)}{|z|^{4}} ≤ M, & if z_{0} = ∞ \end{matrix} (3.6)

for some dominating factor H = H_{z_{0}} of divergence type and positive constants M = M(z_{0}) and r_{0} = r_{0}(z_{0}).

Then there exists a normalized homeomorphic solution f : ℂ̂ → ℂ̂ of (1.1) such that f ∈ W_{loc}^{1,q}(ℂ), q = 2p/(1 + p), and f^{−1} ∈ W_{loc}^{1,2}(ℂ). This homeomorphism admits the modulus of continuity estimate |f(z) − f(z_{0})| ≤ C|z − z_{0}|^{1/M} or

|f(z) − f(z_{0})| ≤ C exp \{−\frac{1}{2} ∫_{1+c}^{2m+c} \frac{dt}{H_{z_{0}}^{−1}(t)}\} , (3.7)

respectively, for |z − z_{0}| < r_{1}, |z_{0}| ≤ R_{0} and r_{1} ≤ min{r_{0}(z_{0}), R_{0}}, where m = log(r_{1}/|z − z_{0}|), c = log(M(z_{0})/πr_{1}^{2}), R_{0} is a fixed number and C is a constant depending only on μ and R_{0}.

Proof. Let f_{n} be the canonical approximating sequence corresponding to the Beltrami coefficient μ. From Theorem 2.15 we see that f_{n} forms a pre-compact family with respect to the uniform convergence in ℂ̂ and every limit function f of this family is a self-homeomorphism of ℂ̂. Passing to subsequence, we may assume that f_{n} → f uniformly in ℂ̂ and μ_{n} → μ a.e. as n → ∞. Since K_{μ} ∈ L_{loc}^{p}(ℂ) for some p > 1, we see by Proposition 3.1 that f ∈ W_{loc}^{1,q}(ℂ), q = 2p/(1 + p) and f satisfies (1.1). Moreover, f_{n}^{−1} → f^{−1} uniformly in ℂ̂ as n → ∞ and f^{−1} ∈ W_{loc}^{1,2}(ℂ). The modulus of continuity estimate in (3.7) follows from Theorem 2.15.

3.8. Remark. Assumption 2) in Theorem 3.5 implies D_{μ,z_{0}} ∈ L^{p}(B(z_{0}, r_{0})) for all p > 1. Assumption 1), however, cannot be dropped in order to have the regularity condition f ∈ W_{loc}^{1,1}(ℂ). Indeed, for μ(z) = (1+i|z|^{2})^{−1}z/z̄, |z| < 1 and μ(z) = 0, |z| > 1, the normalized μ-conformal homeomorphism f : ℂ̂ → ℂ̂ has the form f(z) = ze^{i(1/|z|^{2}−1)}, |z| < 1 and f(z) = z, |z| ≥ 1. A simple calculation shows that D_{μ,0}(z) = 1 a.e. in ℂ, and therefore D_{μ,z_{0}} is bounded near z_{0} for each z_{0} ∈ ℂ̂. Hence, assumption 2) is fulfilled, but |fz̄| = 1/|z|^{2} is not locally integrable near the origin, and thus f ∉ W_{loc}^{1,1}(ℂ). Note that K_{μ}(z) ≥ 1/|z|^{4} and hence K_{μ} ∉ L_{loc}^{1}(ℂ).

Note also that assumption 1) can be stated in terms of the radial dilatation of μ. In fact, we can replace it by the condition that D_{−μ,z_{0}} belongs to L^{p}(B(z_{0}, r_{0})) for each finite z_{0}, where p > 1 is a fixed number. To see the equivalence, we have only to note the relation

\frac{K_{μ}}{2} ≤ D_{μ,z_{0}} + D_{−μ,z_{0}} = \frac{1 + |μ|^{2}}{1 − |μ|^{2}} ≤ K_{μ}.

Specifying the dominating factor H(x) we obtain more concrete consequences from Theorem 3.5. Typical examples are ηx and ηx/(1 + log^{+} x) for a positive constant η. The assumptions corresponding to (3.6) can be described by the local exponential integrability condition

∫B(z_{0},r_{0})e^{ηD_{μ,z_{0}}(z)}dA^{♯}(z) < +∞ (3.9)

and by the local subexponential integrability condition

∫B(z_{0},r_{0}) exp \{\frac{ηD_{μ,z_{0}}(z)}{1 + log^{+} D_{μ,z_{0}}(z)}\} dA^{♯}(z) < +∞ (3.10)

for the angular dilatation coefficient D_{μ,z_{0}}(z). More examples will be given in Section 5.

The following result can be viewed as an extended version of the corresponding existence theorems from [6] and [11].

3.11. Theorem. Let H be a dominating factor of divergence type. Suppose that a Beltrami coefficient μ on ℂ with ∥μ∥_{∞} ≤ 1 satisfies

∫ℂe^{H(K_{μ}(z))}dA^{♯}(z) < +∞, (3.12)

where dA^{♯}(z) = (1 + |z|^{2})^{−2}dA(z) denotes the spherical area element. Then there exists a normalized homeomorphic solution f to the Beltrami equation with μ such that f ∈ W_{loc}^{1,q}(ℂ) for every 1 ≤ q < 2 and that f^{−1} ∈ W_{loc}^{1,2}(ℂ).

Proof. Since H is of divergence type, assumption (3.12) implies that K_{μ} ∈ L_{loc}^{p}(ℂ) for every 1 < p < ∞. On the other hand, the inequality D_{μ,z_{0}}(z) ≤ K(z) a.e. and the convergence of the integral (3.12) imply the local assumption (3.6). Hence, by Theorem 3.5, we have the required conclusion.

3.13. Remark. Condition (3.12) is optimal for the solvability of Beltrami equations in the following sense.

Assume that H is a dominating factor of convergence type. We may assume that H is smooth enough and H(1) = 1. Then, by Theorem 3.1 of [11], there exists a μ satisfying (3.12) for which the following holds:

1. μ = 0 off the unit disk B,

2. K_{μ} is locally essentially bounded in ℂ \ {0},

3. there are no W_{loc}^{1,1}-solutions to the Beltrami equation with μ in the unit disk which are continuous at the origin other than constant functions, and

4. there is a solution f to the Beltrami equation in the weak-W^{1,2}(B) Sobolev space, where weak-W^{1,2}(B) = ⋂_{1≤q<2} W^{1,q}(B), which maps the punctured disk B \ {0} homeomorphically onto the annulus A(0, 1, R) for some 1 < R < +∞.

3.14. Corollary. Suppose that μ is a Beltrami coefficient in ℂ such that for some η > 0 ∫ℂe^{ηK_{μ}(z)}dA^{♯}(z) < +∞. (3.15)

Then there exists a μ-conformal homeomorphism f : ℂ̂ → ℂ̂ such that f ∈ W_{loc}^{1,q}(ℂ) for every q < 2 and f^{−1} ∈ W_{loc}^{1,2}(ℂ). Moreover, for every compact set E ⊂ ℂ there are positive constants C, C′ and a such that

C exp (−\frac{a}{|z_{1} − z_{2}|^{2}} ) ≤ |f(z_{1}) − f(z_{2})| ≤ C′ |log \frac{1}{|z_{1} − z_{2}|}|^{−η/2} (3.16)

for z_{1}, z_{2} ∈ E with 0 < |z_{1} − z_{2}| < 1/e. The exponent −η/2 is sharp.

The right hand side of (3.16) can be written in the more precise form

|f(z_{1}) − f(z_{2})| ≤ C dist(E_{0}, E_{1}) (\frac{1 + log(M/πR^{2})}{log 1/|z_{1} − z_{2}|})^{η/2}

for z_{1}, z_{2} ∈ B(0, R/2) with |z_{1} − z_{2}| < 1/e where

M = _{∫B(0,R)}e^{ηK_{μ}}dA,

E_{0} = f(\overline{B}(0, R/2)), E_{1} = f(ℂ̂ \ B(0, R)) and C is a constant depending only on η.

The sharpness can be seen by the following examples. Let f : ℂ → ℂ be the radial stretching defined by

f(z) = \frac{z}{|z|} (1 + \frac{2}{β} log \frac{1}{|z|})^{−β/2} (3.17)

for |z| ≤ 1 and f(z) = z for |z| > 1. (This example was given in [11, §11.1].) Then K(z) = 1+(2/β) log(1/|z|) for |z| < 1. Hence integrability condition (3.15) holds for β > η but not for β ≤ η. Since |f(z)−f(0)| = (1+2β^{−1} log 1/|z|)^{−β/2} ∼ (log 1/|z|)^{−β/2} as z → 0, the exponent η/2 in (3.16) cannot be replaced by any larger number.

3.18. Remark. It is noted in [3] that a necessary and sufficient condition for a measurable function K(z) ≥ 1 to be majorized in Ω ⊂ ℂ by a function M ∈ BMO(ℂ) is that ∫Ωe^{ηK(z)}dA^{♯}(z) < +∞ (3.19)

for some positive number η. Moreover, M can be chosen so that ∥M∥_{BMO} ≤ C/η where C is an absolute constant. For mappings of BMO-bounded distortion David [8] has proved the estimate

|f(z_{1}) − f(z_{2})| ≤ A |log \frac{1}{|z_{1} − z_{2}|}|^{−b/∥M∥_{BMO}} (3.20)

for some positive constant b, which agrees with our modulus estimate (3.16).

3.21. Remark. It is shown in [11] that if μ has a compact support, then there exists a number η_{0} > 1 such that the Beltrami equation with μ satisfying (3.15) with η ≥ η_{0} admits a unique principal solution f with f(z) − z ∈ W^{1,2}(ℂ).

The following consequence is due to [11, Theorem 14.2] except for the modulus of continuity estimate. The almost same result has been obtained by [6] earlier (see the remark below).

3.22. Corollary. Suppose that μ is a Beltrami coefficient in ℂ such that

∫ℂ exp \{\frac{ηK(z)}{1 + log K(z)}\} dA^{♯}(z) < ∞ (3.23)

for some η > 0. Then there exists a homeomorphic solution f in ℂ̂ to (1.1) such that f ∈ W_{loc}^{1,q}(ℂ) for every q < 2 and f^{−1} ∈ W_{loc}^{1,2}(ℂ). Moreover, for every compact set E ⊂ ℂ there are constants C, C′ and a such that

(3.24) C exp \{−\frac{a}{|z_{1} − z_{2}|^{2}} \} ≤ |f(z_{1}) − f(z_{2})| ≤ C′ (log log \frac{1}{|z_{1} − z_{2}|})^{−η/2}

for z_{1}, z_{2} ∈ E with 0 < |z_{1} − z_{2}| < e^{−e}. The exponent −η/2 is sharp.

The modulus of continuity follows from the fact that η^{−1}y log y < H^{−1}(y) for sufficiently large y where H(x) = ηx/(1 + log^{+} x). More precisely, we have an estimate in the following form:

|f(z_{1}) − f(z_{2})| ≤ C dist(E_{0}, E_{1}) (\frac{log log(M/πR^{2})}{log log(1/|z_{1} − z_{2}|)})^{η/2}

for z_{1}, z_{2} ∈ B(0, R/2) with |z_{1} − z_{2}| < e^{−e}, where

M = _{∫B(0,R)} exp \{\frac{ηK(z)}{1 + log K(z)}\} dA(z),

E_{0} = f(\overline{B}(0, R/2)), E_{1} = f(ℂ̂ \ B(0, R)) and C is a constant depending only on η.

Next we show the sharpness by examples. Let f : ℂ → ℂ be the radial stretching defined by

f(z) = \frac{z}{|z|} (1 + \frac{2}{β} log log \frac{e}{|z|})^{−β/2} (3.25)

for |z| ≤ 1 and f(z) = z for |z| > 1. Then K(z) = log \frac{e}{|z|} (1 + \frac{2}{β} log log \frac{e}{|z|}) for |z| < 1. (This example appeared in [11, §12.1], however, there seems to be an error in the formula for K(z).) Hence

\frac{K(z)}{1 + log K(z)} = \frac{2}{β} log \frac{e}{|z|} (1 − \frac{log_{3} e/|z|}{log_{2} e/|z|} + O (\frac{1}{log_{2} e/|z|}))

as z → 0 and, in particular, we can see that exp{ηK(z)/(1 + log K(z))} is integrable in |z| < 1 if and only if β ≥ η. Therefore the exponent η/2 in (3.24) cannot be replaced by any larger number in general.

3.26. Remark. The first proof of the above statement was given in [6] under the assumption of special behavior of K_{f}(z) around the point at infinity:

∫B(0,R)K(z)dA(z) = O(R^{2}).

This assumption says that K is bounded in the sense of the mean, and hence it is different from (3.23). Theorem 14.2 in [11] contains also the quite accurate regularity assertion

∫ℂ \frac{|Ψ(z)|^{2}}{log(e + Ψ(z)) log_{2}(3 + Ψ(z))}dA^{♯}(z) < +∞

where Ψ(z) stands for the spherical derivative \frac{1+|z|^{2}}{1+|f(z)|^{2}} |Df(z)| of f.

Next, we prove Theorem 1.5 given in Introduction, which shows that if Sing(μ) of μ is a finite set then μ can be modified without changing its absolute value and the modified μ admits a “good” solution. For the convenience of the reader, we recall it in the following more concrete form. Claim. Let μ(z) be a Beltrami coefficient in ℂ̂ such that Sing(μ) consists of finitely many points {a_{1}, ..., a_{n}} ⊂ ℂ̂. Then there exists a homeomorphism f : ℂ̂ → ℂ̂ which is locally quasiconformal in ℂ̂ \ {a_{1}, ..., a_{n}} and whose complex dilatation μ_{f} satisfies |μ_{f}(z)| = |μ(z)| a.e. (3.27)

Proof. Without loss of generality we will assume that all a_{k}’s are finite. Let δ be a positive number such that the disks |z − a_{k}| ≤ δ, k = 1, ..., n, are disjoint. Setting

μ′(z) = |μ(z)| · \frac{z − a_{k}}{z̄ − ā_{k}} , k = 1, ..., n,

for |z − a_{k}| ≤ δ and μ′(z) = μ(z) otherwise, we see that D_{μ′,z_{0}}(z) is essentially bounded in a sufficiently small disk B(z_{0}, r_{0}) for each z_{0} ∈ ℂ̂. By Theorem 2.15 there exists a homeomorphism f : ℂ̂ → ℂ̂ which is locally quasiconformal in Ω = ℂ̂\{a_{1}, ..., a_{n}} and f|_{Ω} satisfies the Beltrami equation with μ′. The condition (3.27) is immediate by construction.

3.28. Remark. For every H ∈ D there exists a homeomorphism f : ℂ̂ → ℂ̂ locally quasiconformal in ℂ̂ \ {a_{1}, ..., a_{n}} with complex dilatation μ such that

∫ℂe^{H(K_{μ}(z))}dA^{♯}(z) = +∞

but weaker directional condition (3.6) holds.

In the above examples the singular set of μ consists of isolated points only. In the following examples the singular sets of μ are the whole extended real axis ℝ̂ = ℝ ∪ {∞}.

3.29. Example. Define the Beltrami coefficient μ in ℂ̂ for any α, 1 ≤ α < 2, by

μ(z) = 1 − \frac{2}{α log \frac{1}{|y|}} , z = x + iy,

for |y| ≤ y_{0} and μ(z) = 0 for |y| > y_{0}, where 0 < y_{0} ≤ e^{−1/α} is a constant. Then integrability assumption (3.15) with η = 1 does not hold at any neighborhood of each point on the real axis, whereas assumption (3.9) with η = 1 still holds. Indeed, since K_{μ}(z) = 1 + α log(1/|y|) for |y| < y_{0}, we see that e^{K_{μ}(z)} = e|y|^{−α}. Now the assumption α ≥ 1 implies that e^{K_{μ}} is not locally integrable at any point on the real axis. On the other hand, writing z = re^{iθ}, we compute for any a ∈ ℝ,

(3.30) D_{μ,a}(z) = D_{μ,0}(z) = \frac{1 − μ}{1 + μ} + \frac{4μ}{1 + μ} · \frac{sin^{2} θ}{1 − μ} ≤ 1 + α sin^{2} θ (log \frac{1}{|y|}) ,

where the inequality 4μ/(1 + μ) ≤ 2 has been used. Since the function x log x is bounded in (0, 1), we obtain

(3.31) exp(D_{μ,a}(z)) ≤ e|y|^{−α sin^{2} θ} ≤ er^{−α}e^{− \frac{α}{2} sin^{2} θ log(sin^{2} θ)} ≤ Cr^{−α}

for some constant C. Hence, for sufficiently small r_{0} > 0, we have

∫B(a,r_{0})e^{D_{μ,a}(z)}dA^{♯}(z) ≤ 2πC ∫_{0}^{r_{0}} r^{1−α}dr < +∞ (3.32)

and the claim follows.

3.33. Example. Another example of Beltrami coefficient μ in ℂ, for which Sing(μ) = ℝ̂, is provided by

μ(z) = 1 − \frac{2}{α log \frac{1}{|y|} log log \frac{1}{|y|}} , z = x + iy,

for |y| ≤ y_{0} and μ(z) = 0 for |y| > y_{0}, where α is a constant with 1 < α < 2e/(e+1) = 1.462 … and y_{0} is a positive constant with y_{0} ≤ e^{−e^{1/α}}. An elementary computation shows that integrability assumption (3.23) does not hold at any neighborhood of each point lying on the real axis. On the other hand, assumption (3.10) still holds. Indeed, since K_{μ}(z) = 1 + α log(1/|y|) log_{2}(1/|y|), |y| < y_{0}, we obtain

\frac{K_{μ}(z)}{1 + log K_{μ}(z)} = \frac{α log \frac{1}{|y|} log_{2} \frac{1}{|y|}}{log_{2} \frac{1}{|y|} + log_{3} \frac{1}{|y|} + O(1)} = α log \frac{1}{|y|} [1 + O (\begin{matrix}log & 3 & \frac{1}{|y|} \\ log & 2 & \frac{1}{|y|} \end{matrix} )]

as y → 0. In particular, we see that e^{K_{μ}(z)/(1+log K_{μ}(z))} is not locally integrable at every point on the real axis. On the other hand, as above, we obtain

D_{μ,0}(z) ≤ 1 + α sin^{2} θ log \frac{1}{|y|} log_{2} \frac{1}{|y|}.

Hence, under the assumption that α sin^{2} θ log(1/|y|) log_{2}(1/|y|) is large enough, we have

\frac{D_{μ,0}(z)}{1 + log^{+} D_{μ,0}(z)}

≤ \frac{1 + α sin^{2} θ log \frac{1}{|y|} [log(α sin^{2} θ log \frac{1}{|y|} log_{2} \frac{1}{|y|}) − log(α sin^{2} θ log_{2} \frac{1}{|y|})]}{1 + log(α sin^{2} θ log \frac{1}{|y|} log_{2} \frac{1}{|y|})}

= 1 + α sin^{2} θ log \frac{1}{|y|} − \frac{α sin^{2} θ log \frac{1}{|y|} [log(α sin^{2} θ) + log_{3} \frac{1}{|y|}]}{1 + log(α sin^{2} θ log \frac{1}{|y|} log_{2} \frac{1}{|y|})}
≤ α sin^{2} θ log \frac{1}{|y|} + \frac{αe^{−1} log \frac{1}{|y|}}{1 + log(α sin^{2} θ log \frac{1}{|y|} log_{2} \frac{1}{|y|})}
≤ α(sin^{2} θ + e^{−1}) log \frac{1}{|y|},

where we used the inequality −x log x ≤ e^{−1} for 0 < x. This yields

exp (\frac{D_{μ,0}(z)}{1 + log^{+} D_{μ,0}(z)}) ≤ |y|^{−α(sin^{2} θ+e^{−1})} ≤ Cr^{−α(1+e^{−1})}(sin θ)^{−α/e}

for some positive constant C. The last function in the polar coordinates (r, θ) is integrable over 0 < r < 1, −π/2 < θ < π/2 because α(1 + e^{−1}) − 1 < 1 and α/e < 1.

3.34. Remark. For the above examples, whose Beltrami coefficients μ(z) = μ(x+iy) depend on y only, we can give normalized μ-conformal homeomorphisms f : ℂ → ℂ in the explicit form

f(z) = x + i ∫_{0}^{y} \frac{1 − μ(it)}{1 + μ(it)}dt.

4. Uniqueness

The following remark is a simple consequence of a well-known removability theorem for analytic functions.

4.1. Proposition. Let μ be a Beltrami coefficient in a domain Ω such that the singular set E = Sing(μ) in Ω is countable. Suppose that a topological embedding f : Ω → ℂ̂ is locally quasiconformal in Ω \ E and satisfies (1.1) with μ there. Then f has the property that for any homeomorphic solution f̂ of (1.1) in Ω \ E there exists a conformal map h in Ω′ = f(Ω) such that f̂ = h ∘ f in Ω \ E. In particular, f̂ extends to an embedding of Ω.

Indeed, let us assume that f̂ is another solution to the Beltrami equation (1.1) with μ in Ω \ E. Then the function h = f̂∘ f^{−1} is an injective holomorphic function in Ω′ \ f(E). Since f(E) is closed and countable in Ω′, it is removable for such a function and we conclude that h can be extended to a conformal map on Ω′.

In order to apply the removability arguments for the uniqueness problem it seems necessary to have information on the singular set E and its image E′ = f(E), simultaneously, see, e.g., [12] and [3].

As Lehto [14] noted, it is reasonable to restrict ourselves to the case when E is totally disconnected. However, this condition does not imply uniqueness in general. We will introduce a geometric condition on totally disconnected compact sets E and show that this condition can be combined with integrability conditions involving the angular dilatation to guarantee the uniqueness.

A positive function m(r) defined on the interval (0, δ) for small δ > 0 is said to be a modulus bound for a dominating factor H of divergence type if it satisfies the condition

(4.2)  lim inf_{r→0} ∫_{0}^{m(r)} \frac{dt}{H^{−1}(2t − 2 log r)} > 0.

Note that m(r) → +∞ as r → 0.

For example, if H(x) = ηx, then m(r) = ε log(1/r) is a modulus bound for H, where ε is an arbitrary positive constant. If H(x) = ηx/(1 + log^{+} x), then m(r) = ε(log(1/r))^{C} is a modulus bound for H, where ε > 0 and C > 1 are arbitrary constants. More examples will be given in Section 5. Each modulus bound m(r) generates a family of annuli A_{m}(z_{0}, r) = A(z_{0}, re^{−m(r)}, r) around every point z_{0} ∈ ℂ̂.

The following notions describe the thinness of the boundary. Let H be a dominating factor. A compact subset E of ℂ̂ is said to be H-coarse at z_{0} ∈ E if there exists a modulus bound m(r) for H such that for any small number δ > 0 there is an r with 0 < r < δ such that A_{m}(z_{0}, r) ⊂ Ω. The set E is said to be H-coarse if E is H-coarse at each point z_{0} ∈ E. We will also say that E is radially coarse at z_{0} ∈ E if a positive constant function can be chosen as m(r) above.

It is clear that z_{0} forms a degenerate boundary component of Ω = ℂ̂ \ E if E is H-coarse at the point.

4.3. Theorem. Let μ be a Beltrami coefficient in ℂ̂ such that E = Sing(μ) is a totally disconnected compact subset of ℂ̂. Assume that one of the following conditions holds for each z_{0} ∈ E :

1) D_{μ,z_{0}}(z) is essentially bounded in a neighborhood of z_{0} and E is radially coarse at z_{0};

2) There is a dominating factor H = H_{z_{0}} of divergence type for which E is H-coarse at z_{0} and ∫V e^{H(D_{μ,z_{0}}(z))}dA^{♯}(z) < +∞ (4.4)

for some open neighborhood V of z_{0}.

Then there exists a homeomorphism f : ℂ̂ → ℂ̂ which is locally quasiconformal in Ω = ℂ̂ \ E and satisfies the Beltrami equation with μ in Ω. If f̂ is another homeomorphic solution to the Beltrami equation in Ω with the same μ, then f̂ = h ∘ f for a Möbius transformation h. In particular, f̂ can be extended to a homeomorphism of ℂ̂.

For the proof of this theorem we need the following auxiliary lemma which is due to Gotoh and Taniguchi. A compact set E is said to be annularly coarse at z_{0} ∈ E if we can find a nesting sequence of disjoint ring domains A_{n} around z_{0} in ℂ̂ \ E with inf_{n} mod A_{n} > 0. Here a sequence A_{n} of disjoint ring domains is said to be nesting if each A_{n} separates A_{n−1} from A_{n+1}. A compact set E is said to be annularly coarse if E is annularly coarse at every point in E.

4.5. Lemma (Gotoh-Taniguchi [9]). If a compact subset E of the Riemann sphere is annularly coarse, then E is removable for conformal mappings off E.

Proof of Theorem 4.3. Let f be a uniform limit of the canonical approximating sequence for μ. Note that the existence of such an f is guaranteed by Theorem 2.15. Set E′ = f(E). We show that E′ is annularly coarse. We may assume that E ⊂ ℂ. Let z_{0} ∈ E. Suppose first that case 2) occurs for z_{0}. Then, by assumption, there is an H ∈ D and a constant M with

∫B(z_{0},r_{0})e^{H(D_{μ,z_{0}}(z))}dA(z) ≤ M

for which E is H-coarse at z_{0} with modulus bound m : (0, δ) → (0, +∞). We then take arbitrarily small r > 0 so that A_{m}(z_{0}, r) ⊂ Ω. By Lemma 2.22, we estimate

mod f(A_{m}(z_{0}, r)) ≥ ∫_{1/2}^{m(r)} \frac{dt}{H^{−1}(2t + log(M/πr^{2}))}.

It is easy to see that

∫_{a}^{b} \frac{dt}{H^{−1}(2t + c − 2 log r)} → 0

as r → 0 for any fixed a, b and c. Hence, by the definition of the modulus bound, we obtain

lim inf_{r→0} ∫_{1/2}^{m(r)} \frac{dt}{H^{−1}(2t + log(M/πr^{2}))} = lim inf_{r→0} ∫_{0}^{m(r)} \frac{dt}{H^{−1}(2t − 2 log r)} > 0.

We can now find a sequence r_{n} with 0 < r_{n+1} < r_{n}e^{−m(r_{n})} such that A_{m}(z_{0}, r_{n}) ⊂ Ω for n = 1, 2, … and that inf_{n} mod f(A_{m}(z_{0}, r_{n})) > 0. Hence, we conclude that E′ is annularly coarse at f(z_{0}).

If case 2) occurs for z_{0}, by assumption, z_{0} is radially coarse in E. Then it is easier to see that E′ is annularly coarse at f(z_{0}) than the above.

Assume that f̂ is another homeomorphic solution to the Beltrami equation (1.1) with μ in Ω. Then the function h = f̂ ∘ f^{−1} is an injective holomorphic function in ℂ̂ \ E′. Since E′ = f(E) is removable for such a function by Lemma 4.5, we conclude that h extends to a Möbius transformation. Thus, the proof is complete.

Specifying H ∈ D, we may obtain some consequences of Theorem 4.3. Let us restrict ourselves to the dominating factors H(x) = ηx and H(x) = ηx/(1 + log^{+} x) corresponding to exponential and subexponential integrability assumptions on K_{μ}(z), respectively.

4.6. Corollary. Let E be a totally disconnected, x-coarse compact subset of ℂ̂. Suppose that μ is a Beltrami coefficient in ℂ with Sing(μ) ⊂ E such that

∫ℂe^{ηK_{μ}(z)}dA^{♯}(z) < ∞ (4.7)

holds for a positive constant η. Then there exists a homeomorphism f : ℂ̂ → ℂ̂ which is locally quasiconformal in Ω = ℂ̂ \ E and satisfies the Beltrami equation with μ in Ω. If f̂: Ω → ℂ̂ is a locally quasiconformal embedding whose Beltrami coefficient agrees with μ a.e. then f̂ = h ∘ f for a Möbius transformation h. In particular, f̂ extends to a homeomorphism of the Riemann sphere.

4.8. Corollary. Let E be a totally disconnected compact subset of ℂ̂ which is x/(1 + log^{+} x)-coarse in E. Suppose that μ is a Beltrami coefficient in ℂ with Sing(μ) ⊂ E such that

∫ℂ exp \{\frac{ηK_{μ}(z)}{1 + log K_{μ}(z)}\} dA^{♯}(z) < ∞. (4.9)

Then there exists a homeomorphism f : ℂ̂ → ℂ̂ which is locally quasiconformal in Ω = ℂ̂ \ E and satisfies the Beltrami equation with μ in Ω. If f̂ : Ω → ℂ̂ is a locally quasiconformal embedding whose Beltrami coefficient agrees with μ a.e. then f̂ = h ∘ f for a Möbius transformation h. In particular, f̂ extends to a homeomorphism of the Riemann sphere.

5. Dominating factors and Modulus bounds

In this section, we provide critical dominating factors H of divergence type and their modulus bounds. The functions presented below are more or less standard. For example, similar examples can be found in [11].

We define the functions log_{n}, exp_{n}, Π_{n,α} and Π_{n} for n = 0, 1, 2, … and for α > 0 by

log_{0} x = x, log_{n} x = log(log_{n−1} x) (n = 1, 2, …),
exp_{0} x = x, exp_{n} = exp(exp_{n−1} x) (n = 1, 2, …),
Π_{n,α}(x) = x(log_{1} x) ⋯ (log_{n−1} x)(log_{n} x)^{α}, and
Π_{n}(x) = Π_{n,1}(x) (n = 0, 1, 2, …).

In particular, log_{1} and exp_{1} coincide with the usual log and exp respectively. Note that log_{n} is the inverse function of exp_{n} for each n. We also define the numbers e_{n} by e_{n} = exp_{n} 0. Then for each n > 0 the functions log_{n}, Π_{n,α} and Π_{n} are defined on (e_{n−1}, +∞), positive on (e_{n}, +∞), and greater than 1 on (e_{n+1}, +∞).

We consider the function H(x) = x^{2}/Π_{n,α}(x) for fixed n ≥ 0 and α > 0. Then, H′(x) = (1 + o(1))H(x)/x and H′′(x) = (1 + o(1))H(x)/ log x when x → +∞. In particular, (e^{H})′′/e^{H} = H′′ + (H′)^{2} = (1 + o(1))(H′)^{2}. Hence, we can choose a sufficiently large number x_{n,α} > e_{n} so that the function H_{n,α} defined by

H_{n,α}(x) = \{ \begin{matrix}H(x) = \frac{x^{2}}{Π_{n,α}(x)} & if x > x_{n,α}, \\ H(x_{n,α}) & if 0 ≤ x ≤ x_{n,α}, \end{matrix}

is a dominating factor. Furthermore, H_{n,α} is of divergence type if and only if α ≤ 1 since the integral

∫_{x_{n,α}}^{∞} \frac{H_{n,α}(x)dx}{x^{2}} = ∫_{x_{n,α}}^{∞} \frac{dx}{Π_{n,α}(x)} = ∫_{x_{n,α}}^{∞} \frac{dx}{x(log x) … (log_{n−1} x)(log_{n} x)^{α}}

is divergent if and only if α ≤ 1. Note that the choice of x_{n,α} does not effect on the integrability condition such as _{∫V} e^{H(D_{μ,z_{0}})}dA^{♯} < +∞. We also write H_{n} = H_{n,1}.

5.1. Remark. The dominating factor H_{0} = H_{0,1} is essentially same as x. The dominating factor H_{1} = H_{1,1} is equivalent to H(x) = x/(1 + log^{+} x) in the sense that for any η > 1 we have H(x) < H_{1}(x) < ηH(x) for sufficiently large x.

First we give information on the behavior of the inverse function of H_{n,α}, which is useful to derive a modulus of continuity estimate in connection with the previous theorems.

5.2. Lemma. For each n ≥ 1, α ∈ ℝ and c > 0,

(cH_{n,α})^{−1}(y) = \frac{c}{Π_{n,α}(y)} (1 + (b + o(1))\frac{log_{2} y}{log y} ) (y → +∞)

holds, where b = 1 when n > 1 and b = α^{2} when n = 1. Suppose further that α ≤ 1 and, in addition if n = 1, suppose that 0 < α. Then

∫_{t_{1}}^{t} \frac{dy}{(cH_{n,α})^{−1}(y)} = \{ \begin{matrix}\frac{c}{1 − α}(log_{n} t)^{1−α} + O(1) & when α < 1, \\ c log_{n+1} t + O(1) & when α = 1 \end{matrix}

as t → +∞ for a sufficiently large number t_{1}. When n = 1 and α ≤ 0,

∫_{t_{1}}^{t} \frac{dy}{(cH_{1,α})^{−1}(y)} = (\frac{c}{1 − α} + o(1)) (log t)^{1−α}

as t → +∞ for a sufficiently large number t_{1}.

Proof. We assume that n ≥ 2. (The case n = 1 can be treated in the same way even more easily.) Letting η = η(y) is a positive quantity with log η(y) = O(1) as y → ∞, we observe

log (ηΠ_{n,α}(y)) = log η + log y + log_{2} y + ⋯ + log_{n} y + α log_{n+1} y
= (1 + (1 + o(1))\frac{log_{2} y}{log y} ) log y (y → +∞)

for a positive constant η. We can then see inductively that log_{k}(ηΠ_{n,α}(y)) = log_{k} y · (1 + (1 + o(1)) log_{2} y/ log y log_{2} y ⋯ log_{k} y) for every integer k ≥ 1. Therefore,

(5.2)  cH_{n,α}(ηΠ_{n,α}(y)) = \frac{cηΠ_{n,α}(y)}{log(ηΠ_{n,α}(y)) ⋯ (log_{n}(ηΠ_{n,α}(y)))^{α}}
= cηy (1 − (1 + o(1))\frac{log_{2} y}{log y} )

as y → +∞. We now restrict ourselves to the case when cη = 1 + p log_{2} y/ log y for some constant p. We then see that (1 − log_{2} y/ log y)cη = 1 + (p − 1)(1 + o(1)) log_{2} / log y. Since ηΠ_{n,α}(y) > H_{n,α}^{−1}(y) for sufficiently large y if p > 1 and ηΠ_{n,α}(y) < H_{n,α}^{−1}(y) for sufficiently large y if p < 1, the asymptotic formula for the inverse function follows.

We next show the second assertion. For simplicity, we assume that α < 1. (Actually, the case α = 1 can be absorbed to this case because H_{n,1}(y) = H_{n+1,0}(y) for large y.) By the first assertion, we compute

∫_{t_{1}}^{t} \frac{dy}{(cH_{n,α})^{−1}(y)} = ∫_{t_{1}}^{t} \frac{cdy}{Π_{n,α}(y)} + (−bc + o(1)) ∫_{t_{1}}^{t} \frac{log_{2} ydy}{y(log y)^{2} log_{2} y ⋯ (log_{n} y)^{α}}

if n > 1. (If n = 1, we need a slight modification above.) Since the second integral in the right-hand side is convergent as t → +∞ under the assumptions on α, the required asymptotic formula is obtained.

Proof of Theorem 1.5. We are now ready to prove Theorem 1.5 which is given in Introduction. Without loss of generality, we may assume that α < 1 (see the proof of Lemma 5.2). Under the hypothesis, ∫_{ℂ} e^{cH_{n,α}(K_{μ}(z))}dA^{♯}(z) < ∞ holds. Since cH_{n,α} is a dominated factor of divergence type, we conclude the existence and the regularity of the normalized homeomorphic solution f to the Beltrami equation by Theorem 3.11. We now investigate the modulus of continuity of f. From the assumption, for an arbitrary number R > 1, it follows that

∫_{B(z_{0},1)} e^{cH_{n,α}(K_{μ}(z))}dA(z) ≤ M = ∫_{B(0,R+1)} e^{cH_{n,α}(K_{μ}(z))}dA(z) < ∞

for each point z_{0} with |z_{0}| ≤ R. Setting b = log(M/π), we obtain from Theorem 3.5 that

|f(z) − f(z_{0})| ≤ C exp \{−\frac{1}{2} ∫_{1+b}^{2m+b} \frac{dt}{(cH_{n,α})^{−1}(t)}\} , |z − z_{0}| < 1,

where m = log(1/|z − z_{0}|) and C is a constant depending only on μ, c, n, α and R. Applying Lemma 5.2, in the case when n > 1, we have

\frac{1}{2} ∫_{1+b}^{2m+b} \frac{dt}{(cH_{n,α})^{−1}(t)} = \frac{c}{2(1 − α)} log_{n}(2m+b)+O(1) = \frac{c}{2(1 − α)} log_{n}(m)+O(1)

as m → +∞. The case when n = 1 can be shown similarly by using the latter part of Lemma 5.2. Thus we have shown the estimate for the modulus of continuity of f. To see the sharpness, we construct an example below. As we noted, a more abstract approach for a dominated factor of convergence type can be found in [11, Theorem 3.1].

Let M(t) = η log(1/t) ⋯ log_{n}(1/t)(log_{n+1}(1/t))^{α} = ηΠ_{n,β}(log(1/t)) for constants η > 0, n ≥ 0 and α ∈ ℝ. The radial stretching f defined by

f(z) = \frac{z}{|z|} exp ∫_{δ_{0}}^{|z|} \frac{dt}{tM(t)}

in 0 < |z| < δ_{0}, where δ_{0} is a sufficiently small number. Then, as is easily seen, the pointwise maximal dilatation of f is given by K(z) = M(|z|). Note that the function f continuously extends to z = 0 by setting f(0) = 0 precisely when the integral ∫^{+∞} dt/tM(t) diverges, namely, α ≤ 1. In particular, when α > 1, by Theorem 2.15 (and Proposition 4.1 as well), exp(H ∘ K(z)) is not locally integrable around the origin for every dominating factor H of divergence type. On the other hand, we can see that H ∘ K is exponentially integrable around the origin when we choose H = cH_{n,α}, where c is a constant satisfying c < 2/η. We fix a number ε with cη < ε < 2. From (5.2), it follows that H(K(t)) = cH_{n,α}(ηΠ_{n,α}(log(1/t))) = cη(1 + o(t)) log(1/t) as t → +0. Therefore, there is a number r_{0} > 0 such that H(M(t)) ≤ ε log(1/t) for all 0 < t < r_{0}. We now compute

∫_{B(0,r_{0})} e^{H(K(z))}dA(z) = 2π ∫_{0}^{r_{0}} te^{H(M(t))}dt ≤ 2π ∫_{0}^{r_{0}} t^{1−ε}dt < +∞.

In particular, we find that the exponential integrability condition for H_{n,α} ∘ K_{μ} does not imply the existence of a homeomorphic solution in the case α > 1.

Next, we assume that α < 1. (As we noted, the case α = 1 can be included in the case α = 0.) Then, the function f can be expressed in the form

f(z) = \frac{Cz}{|z|} exp (\frac{−1}{(1 − α)η} (log_{n+1}(1/|z|))^{1−α}) , 0 < |z| < δ_{0}.

Since η can be chosen arbitrarily as long as η < 2/c, the sharpness is obtained. Now the proof of the theorem in Introduction has been completed.

To state results on modulus bounds of the above dominating factors, we prepare some auxiliary functions. For constants C > 1 and β, δ > 0 and a nonnegative integer n, we set

φ_{n,C}(x) = exp_{n} ( log_{n} x + log C), and
ψ_{n,β,δ}(x) = exp_{n} ([( log_{n} x)^{β} + δ]^{1/β})

for x ≥ e_{n}. By definition, φ_{n,C} = ψ_{n,1,log C}. For example, we see that

φ_{0,C}(x) = x + log C,
φ_{1,C}(x) = Cx, and
φ_{2,C}(x) = x^{C}.

5.3. Proposition. For a positive integer n a modulus bound for H_{n} can be given by

m(r) = εφ_{n+1,C}( log 1/r)

for r small enough, where ε and C are arbitrary constants with ε > 0 and C > 1.

Proof. From Lemma 5.2, we see that

lim inf_{r→0} ∫_{0}^{m(r)} \frac{dt}{H_{n}^{−1}(2t − 2 log r)}
= \frac{1}{2} lim inf_{r→0} [log_{n+1} (2m(r) − 2 log r) − log_{n+1}(−2 log r)] .

Noting that log r = o(m(r)) as r → 0, we obtain

log_{n+1} (2m(r) − 2 log r) = log_{n+1}(φ_{n+1,C}(log 1/r)) + o(1)
= log_{n+1}(log 1/r) + log C + o(1)

and log_{n+1}(c(r)) = log_{n+1}(log 1/r) + o(1) as r → 0. Hence, we have confirmed that m(r) is a modulus bound for H_{n}.

In the same way, we can also show the following.

5.4. Proposition. For a positive integer n and α ∈ (0, 1), a modulus bound for H_{n,α} can be given by

m(r) = εψ_{n,1−α,δ}( log 1/r)

for r small enough, where ε and δ are arbitrary positive constants.

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Olli Martio and Matti Vuorinen: Department of Mathematics, P.O. Box 4 (Yliopistonkatu 5), FIN–00014 University of Helsinki FINLAND Fax: +358–9–19123213 E–mail: martio@cc.helsinki.fi, vuorinen@csc.fi

Vladimir Ya. Gutlyanskiĭ: Institute of Applied Mathematics and Mechanics, NAS of Ukraine, ul. Roze Luxemburg 74, 83114, Donetsk UKRAINE Fax: +38-0622-552265 E–mail: gut@iamm.ac.donetsk.ua, gut@mat-92.pc.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