Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 27, 2002, 485–490

ON A PAPER OF CARLESON

Melkana A. Brakalova and James A. Jenkins

The Hotchkiss School, Department of Mathematics Lakeville, CT 06039, U.S.A.; mbrakalo@hotchkiss.org Current address: Institute for Mathematical Sciences SUNY Stony Brook, NY 11794-3660, U.S.A.; melkana@math.sunysb.edu Washington University, Department of Mathematics Campus Box 1146, St. Louis, MO 63130-4899, U.S.A.

Abstract. We study the boundary correspondence under sense-preserving homeomorphic self-mapping of the upper half-plane. Under assumptions that involve integrals of the complex dilatation we prove that the natural extension of the mapping is conformal at every point on the boundary. If the integrals involving the complex dilatation are locally uniformly convergent we prove that the derivative of the boundary extension is continuous. The results obtained are closely related to well-known results in a paper by L. Carleson.

1. Introduction

It is well known that a quasiconformal self-mapping of the open upper halfplane admits an extension as a homeomorphism to the closed half-plane. If the points at infinity correspond, this gives a continuous, monotone increasing realvalued function. Beurling and Ahlfors [1] have given a necessary and sufficient condition for such a function to be the extension of a quasiconformal mapping. In particular they deduced that such a function need not be Lebesgue measurable. On the other hand, Carleson [3] showed that if the complex dilatation of a quasiconformal self-mapping of the upper-half plane satisfies in two instances subsidiary conditions in a strip neighborhood of the real axis the extension will be measurable. In one of the cases the extension is proved to have a continuous derivative.

There have been various alternative proofs and extensions of these results. In particular Gutlyanskii and Vuorinen [4] have shown, in the context of quasiconformal self-mappings of the unit disk, that the uniform convergence of certain integrals associated with the points of the circumference implies the existence of a continuously differentiable extension.

In this paper we consider a sense-preserving homeomorphic self-mapping of the upper half-plane (without the assumption of quasiconformality). We assume that this mapping possesses, in a neighborhood of the real axis, a complex dilatation that satisfies pointwise convergence of certain integrals associated with points on the real axis. Under such assumptions the homeomorphism admits a differentiable homeomorphic extension mapping the real axis onto itself. The natural extension of the mapping is conformal at every point on the boundary. If the convergence of the above-mentioned integrals is locally uniform the derivative is continuous.

1991 Mathematics Subject Classification: Primary 30C99; Secondary 30G15, 30C62.

2. Main results

Let f(z) be a sense preserving homeomorphism of the open upper half-plane H onto itself. Let g(w) be the inverse of f(z) in the open upper half-plane H. Suppose that as w → ∞, w ∈ H, then g(w) → ∞ and as z → ∞, z ∈ H, then f(z) → ∞. We assume that f(z) has complex dilatation k(z) = fz̄/f_{z} a.e. in a neighborhood N ∈ H of the real axis R. Denote by x a real number. For a fixed x and 0 < r_{1} < r_{2} , we define the semiannulus Q(x, r_{1}, r_{2}) = {Im z > 0} ∩ {r_{1} < |z − x| < r_{2}} and assume r_{2} sufficiently small so that Q(x, r_{1}, r_{2}) ⊂ N. For every x ∈ R

h_{1}^{∗}(ϕ, x) = ∫_{r_{1}}^{r_{2}} \frac{|1 + e^{−2iϕ}k(re^{iϕ})|^{2}}{1 − |k(re^{iϕ})|^{2}} \frac{dr}{r} , h_{2}^{∗}(r, x) = ∫_{0}^{π} \frac{|1 − e^{−2iϕ}k(re^{iϕ})|^{2}}{1 − |k(re^{iϕ})|^{2}} dϕ,

are to exist for almost all r ∈ [r_{1}, r_{2}] and ϕ ∈ [0, π].

Denote

J(x, r_{1}, r_{2}) = ∫∫_{Q(x,r_{1},r_{2})} \frac{|k(z)|^{2} + | Re e^{−2iϕ}k(z)|}{1 − |k(z)|^{2}} \frac{dA}{|z − x|^{2}} ,

where ϕ = arg(z − x).

Theorem 1. If f(z) satisfies the assumptions outlined above and

(2.1)  J(x, r_{1}, r_{2}) converges for every real x as r_{1} → 0,

then f(z) admits a differentiable increasing homeomorphic extension f(x) from the real axis to itself. In addition for z ∈ N and x ∈ R

(2.2)  f′(x) = lim_{z→x} \frac{f(z) − f(x)}{z − x} > 0.

Theorem 2. If

(2.3)  J(x, r_{1}, r_{2}) converges locally uniformly for every real x as r_{1} → 0,

then f′(x) is continuous.

3. Auxiliary results

Here we restate results from [2] essential to our proof.

Let w = w(z) be a homeomorphism of the unit disc U_{0} onto itself, w(0) = 0, with a complex dilatation k(z) = wz̄(z)/w_{z}(z) defined a.e. in U_{0} . Assume that

\begin{aligned}h_{1}^{∗}(ϕ) & = ∫_{r_{1}}^{r_{2}} \frac{|1 + e^{−2iϕ}k(x + re^{iϕ})|^{2}}{1 − |k(x + re^{iϕ})|^{2}} \frac{dr}{r} , \\ h_{2}^{∗}(r) & = ∫_{0}^{2π} \frac{|1 − e^{−2iϕ}k(x + re^{iϕ})|^{2}}{1 − |k(x + re^{iϕ})|^{2}} dϕ, \end{aligned}

exist a.e. for r ∈ [r_{1}, r_{2}], 0 < r_{1} < r_{2} < 1 and ϕ ∈ [0, 2π]. In addition we assume that h_{1}^{∗}(ϕ) ∈ L^{1}[0, 2π], h_{2}^{∗}(r) ∈ L^{1}[r_{1}, r_{2}] . We denote by M^{∗}(r_{1}, r_{2}) the module of the image under w(z) of the annulus r_{1} < |z| < r_{2} . Under the above assumptions we have shown the following results.

Lemma 1 ([2, Lemma 3.1]). We have

(3.1) |M^{∗}(r_{1}, r_{2}) − \frac{1}{2π} log \frac{r_{2}}{r_{1}} | < \frac{1}{2π^{2}} ∫∫_{r_{1}<|z|<r_{2}} \frac{|k(z)|^{2} + | Re e^{−2iϕ}k(z)|}{1 − |k(z)|^{2}} \frac{dA}{|z|^{2}} ,

where ϕ = arg z.

Lemma 2 ([2, Corollary 1.1]). If

(3.2)  ∫∫_{U_{0}} \frac{|k(z)|^{2} + | Re e^{−2iϕ}k(z)|}{1 − |k(z)|^{2}} \frac{dA}{|z|^{2}} < ∞

where ϕ = arg z, and if

(3.3)  lim_{r→0} arg w(re^{iθ_{0}}) = a

for a particular value θ_{0} , then w(z) is conformal at z = 0, i.e.

(3.4)  lim_{z→0} \frac{w(z)}{z} = C, C ≠ 0.

A well-known result in the theory of quasiconformal mappings is the Teichmüller–Belinskii theorem according to which (3.4) holds under the restriction that w(z) is K -quasiconformal and:

(3.5)  ∫∫_{U_{0}} \frac{|k(z)|}{1 − |k(z)|} \frac{dA}{|z|^{2}} < ∞.

Since (3.5) is a stronger condition than (3.2) the use of Lemma 2 allows the proof of the differentiability of the boundary mapping under a weaker condition than a condition similar to (3.5), as used by Carleson, without the assumption of K -quasiconformality. We, naturally, take advantage of the fact that when mapping the upper half-plane onto itself (3.3) holds with θ_{0} = 0 and a = 0.

4. Proofs of the main results

Proof of Theorem 1. The semiannulus Q(x, r_{1}, r_{2}) becomes a quadrangle after assigning as a pair of opposite sides the boundary segments on the real axis. Denote by M^{∗}(x, r_{1}, r_{2}) the module of the image Q^{∗}(x, r_{1}, r_{2}) of the quadrangle Q(x, r_{1}, r_{2}) under f(z). In a method similar to the proof of Lemma 1 (or Lemma 3.1 from [2]) one can show that:

|M^{∗}(x, r_{1}, r_{2}) − \frac{1}{π} log \frac{r_{2}}{r_{1}} | < \frac{1}{π^{2}} ∫∫_{Q(x,r_{1},r_{2})} \frac{|k(z)|^{2} + | Re e^{−2iϕ}k(z)|}{1 − |k(z)|^{2}} \frac{dA}{|z − x|^{2}} ,

where where ϕ = arg(z − x). Since (2.1) holds we see that

M^{∗}(x, r_{1}, r_{2}) → ∞, as r_{1} → 0.

This means that the image of the inner semicircle with center x and radius r_{1} tends to a point. We will define this point as f(x). Thus it is obvious that lim_{z→x} f(z) = f(x). Denote by S_{x,d} a semicircle in the upper half plane with center x and diameter d. Let C_{x,d} = [x− \frac{1}{2}d, x+ \frac{1}{2}d]∪S_{x,d}. Consider a sufficiently small interval I of length d centered at f(x_{0}) and take a semicircle S_{x_{0},2r_{0}} such that its image lies in the interior intC_{f(x_{0}),d}. Take any point x in [x_{0}−r_{0}, x_{0}+r_{0}]. The construction of f(x) as outlined above assures that f(x) falls in the interval I. Therefore the image of the interval [x_{0} − r_{0}, x_{0} + r_{0}] falls inside I and f(x) is continuous. Now take two different points x_{1} and x_{2} . We can choose semicircles S_{x_{1},d_{1}} and S_{x_{2},d_{2}} so that intC_{x_{1},d_{1}} ∩ int C_{x_{2},d_{2}} = ∅. Since f(z) is one to one the images of int C_{x_{i},d_{i}}, i = 1, 2 are disjoint. Therefore f(x_{1}) ≠ f(x_{2}), thus f(x) is one to one.

Consider the images under f of C_{x,d} and a closed Jordan curve γ ∈ int C_{x,d}. Since f(z) is sense preserving, the orientation of f(γ) is the same as γ. By the argument principle the orientation of the image of f(C_{x,d}) is the same as the orientation of C_{x,d}. That shows that f is increasing.

Now we are going to show that f is differentiable. We extend f to a function in a neighborhood N_{0} of R, N_{0} = N ∪\overline{N} ∪R by reflection. Here \overline{N} = \{z : z̄ ∈ N\}. Then for z ∈ \overline{N} the extension f(z) is defined as f(z) = f̄(z̄). The so obtained new mapping will be called f. The complex dilatation k(z) of the extended mapping is defined a.e. in N_{0} . Because of (2.1) for all x

∫∫_{|z−x|<r_{2}} \frac{|k(z)|^{2} + | Re e^{−2iϕ}k(z)|}{1 − |k(z)|^{2}} \frac{dA}{|z − x|^{2}} < ∞,

where ϕ = arg(z−x). Since lim_{r→0} arg f(x+r) = 0, by Lemma 2 (or by Corollary 1.1

from [2])

lim_{z→x} \frac{f(z) − f(x)}{z − x} = C, C ≠ 0.

This implies that f(x) has a nonzero derivative that must be positive since f is increasing and (2.2) follows.

Proof of Theorem 2. Let f(z) and k(z) be the extended map and its complex dilatation N_{0} , as constructed in Theorem 1. Because of (2.3) the integral

∫∫_{r_{1}<|z−x|<r_{2}} \frac{|k(z)|^{2} + | Re e^{−2iϕ}k(z)|}{1 − |k(z)|^{2}} \frac{dA}{|z − x|^{2}} ,

where ϕ = arg(z −x), is convergent locally uniformly with respect to x as r_{1} → 0. Now we can consider the module M(x, r_{1}, r_{2}) of the image of the annulus {z : r_{1} < |z − x| < r_{2}} under f(z). Denote by M(x, r_{2}) the reduced module of the image of the circle |z − x| < r_{2} under f(z). For fixed x_{0} and ε > 0 by Lemma 1, (2.3) and the continuity of the reduced module M(x, r_{2}) in x there exists δ > 0 such that for |x − x_{0}| < δ and r_{1} < r_{2} , r_{2} sufficiently small

|M(x, r_{1}, r_{2}) − M(x_{0}, r_{1}, r_{2})| < ε

and

|M(x, r_{2}) − M(x_{0}, r_{2})| < ε.

On the other hand by (2.2) and the definition of reduced module (see [5])

M(x, r_{2}) = lim_{r_{1}→0}(M(x, r_{1}, r_{2}) + \frac{1}{2π} log(r_{1}f′(x))).

For r_{1} sufficiently small

|M(x, r_{2}) − (M(x, r_{1}, r_{2}) + \frac{1}{2π} log(r_{1}f′(x)))| < ε

and

|M(x_{0}, r_{2}) − (M(x_{0}, r_{1}, r_{2}) + \frac{1}{2π} log(r_{1}f′(x_{0})))| < ε.

Therefore

|\frac{1}{2π} log(r_{1}f′(x)) − \frac{1}{2π} log(r_{1}f′(x_{0}))|

≤ |M(x, r_{2}) − (M(x, r_{1}, r_{2}) + \frac{1}{2π} log(r_{1}f′(x)))

− (M(x_{0}, r_{2}) − (M(x_{0}, r_{1}, r_{2}) + \frac{1}{2π} log(r_{1}f′(x_{0}))))

+ (M(x_{0}, r_{2}) − M(x, r_{2})) + (M(x, r_{1}, r_{2}) − M(x_{0}, r_{1}, r_{2}))|

< 4ε

Thus log(f′(x)) is continuous and so is f′(x).

490 Melkana A. Brakalova and James A. Jenkins

References

[1] Beurling, A., and L. Ahlfors: The boundary correspondence under quasiconformal mappings. - Acta Math. 96, 1956, 113–134.

[2] Brakalova, M.A., and J.A. Jenkins: On the local behavior of certain homeomorphisms. - Kōdai Math. J. 17, 1997, 201–213.

[3] Carleson, L.: On mappings conformal at the boundary. - J. Analyse Math. 19, 1967, 1–13.

[4] Gutlyanskii, V.Ya., and M. Vuorinen: On mappings almost conformal at the boundary. - Complex Variables 34, 1997, 445–464.

[5] Teichmüller, O.: Untersuchungen über konforme und quasiconforme Abbildungen. - Deutsche Mathematik 3, 1938, 621–678.

Received 31 January 2002