Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 30, 2005, 159–165

BOUNDARY CORRESPONDENCE UNDER QUASICONFORMAL HARMONIC DIFFEO-MORPHISMS OF A HALF-PLANE

David Kalaj and Miroslav Pavlović

Prirodno-matematički fakultet, Cetinjski put b.b. 8100 Podgorica, Montenegro; davidk@rc.pmf.cg.ac.yu Matematički fakultet, Studentski trg 16 11000 Belgrade, p.p. 550, Serbia; pavlovic@matf.bg.ac.yu

Abstract. It is proved that an orientation-preserving homeomorphism ψ of the real axis can be extended to a quasiconformal harmonic homeomorphism of the upper half-plane if and only if ψ is bi-Lipschitz and the Hilbert transformation of the derivative ψ′ is bounded.

1. Introduction

A homeomorphism f: D ↦ G, where D and G are subdomains of the complex plane C, is said to be quasiconformal if f is absolutely continuous on a.e. horizontal and a.e. vertical line, and there exists a constant K < ∞ such that

|\frac{∂f}{∂x} |^{2} + |\frac{∂f}{∂y} |^{2} ≤ KJ_{f} a.e. on D,

where J_{f} is the Jacobian of f (cf. [1, pp. 23–24]). If D = G = U, where U denotes the upper half-plane,

U = {x + yi ∈ C : y > 0},

then f extends to a homeomorphisms of \overline{U} onto onself, where \overline{U} is the closure of U in C ∪ {∞}.

We denote by QC(U) the group of all quasiconformal homeomorphisms of \overline{U} onto itself fixing the point ∞. By the famous theorem of Beurling and Ahlfors [1], the ‘restriction’ of QC(U) to the real axis R coincides with the class of all quasisymmetric functions, i.e., of those strictly increasing homeomorphisms ψ of R such that

\frac{1}{M} ≤ \frac{ψ(x + t) − ψ(x)}{ψ(x) − ψ(x − t)} ≤ M

2000 Mathematics Subject Classification: Primary Primary 30C55, 30C62.

The second author is partially supported by MNTR grant No. 101863, Serbia.

for some constant M ≥ 1 and for all x ∈ R and t > 0.

In this paper we consider the classes

HQC(U) = {f ∈ QC(U), f harmonic in U}

and

HQS(R) = {f|_{R} : f ∈ HQC(U)},

where f|_{R} denotes the restriction of f to the real axis. The study of the analogous classes of functions on the unit disc D was begun by Martio [6]. Various interesting results and examples concerning that case can be found in Partyka and Sakan [7], [8]. Information on univalent (not necessarily quasiconformal) harmonic mappings can be read in [3].

The classes HQC(D) and HQS(∂D) were characterized in [9]; in particular, f is in HQC(D) if and only if f is bi-Lipschitz. The same holds for the half-plane. In fact we can say somewhat more:

Theorem 1.1 ([5]). Let f be a quasiconformal harmonic mapping of U into U. Then the following assertions are equivalent.

(a) f is in QC(U).

(b) There are positive constants c and M such that v(z) = cy, and 1/M ≤ u_{x} ≤ M and |u_{y}| ≤ M for all z ∈ U.

(c) f is a bi-Lipschitz mapping of U onto U.

It follows that if ψ ∈ HQS(R), then ψ is bi-Lipschitz, i.e., ψ is absolutely continuous and

(1.1)  1/C ≤ ψ′(x) ≤ C (x ∈ R, a.e.),

for some constant C. However this condition is far from being sufficient for ψ to be in HQS(R), as the following theorem shows.

Theorem 1.2. Let ψ be an increasing homeomorphism of R. Then ψ belongs to HQS(R) if and only if it is bi-Lipschitz and the Hilbert transformation of ψ′ belongs to L^{∞}(R).

An analogous result holds in the case of the unit disc (see [9]).

The Hilbert transformation of ϕ ∈ L^{∞}(R) is defined by

Hϕ(x) = lim_{ε→0} H_{ε}ϕ(x),

where

H_{ε}ϕ(x) = \frac{1}{π} ∫_{|x−t|>ε}(\frac{1}{x − t} + \frac{t}{t^{2} + 1})ϕ(t) dt.

It is known that the limit exists almost everywhere, but the function Hϕ need not be in L^{∞} (cf. [4]).

2. Proof of Theorem 1.1.

Lemma 2.1. If f = u + iv: U ↦ U is a quasiconformal mapping of class C^{1} such that v(z) ≡ cy for some constant c > 0, then f is onto and bi-Lipschitz.

Proof. Since f is quasiconformal the following inequality holds:

|∇u(z)|^{2} + |∇v(z)|^{2} ≤ KJ_{f}(z),

where J_{f} is the Jacobi determinant of f and K is a constant independent of z ∈ U. In this case this means that

u_{x}^{2} + u_{y}^{2} + c^{2} ≤ Ku_{x},

where u_{x} and u_{y} are the partial derivatives of u. It follows that

(2.1)  u_{x} ≥ c^{2}/K and u_{x} ≤ K

and hence that |u_{y}| ≤ K. This implies that the function f satisfies a Lipschitz condition.

On the other hand, it follows from (2.1) and the hypothesis v(z) = cy that f is onto. Since the inverse mapping is of the same form as f the above argument shows that f^{−1} satisfies a Lipschitz condition. The result follows.

Proof of Theorem 1.1. Assuming (a) we have that v is a positive harmonic function on U and therefore, by the Riesz–Herglotz theorem (see [2, Theorem 7.20]), v has the form

v(z) = cy + π^{−1} ∫_{−∞}^{+∞} P (z, t) dμ(t),

where c is a non-negative constant, μ is a non-decreasing function on R, and P is the Poisson kernel,

P (z, t) = \frac{y}{|z − t|^{2}} (z = x + iy ∈ U, t ∈ R).

Therefore

(2.2)  \begin{aligned}v(z) & ≥ cy + π^{−1} ∫_{x}^{x+y} P (z, t) dμ(t) \\ & ≥ cy + π^{−1} ∫_{x}^{x+y} \frac{y}{2y^{2}} dμ(t) \\ & = cy + π^{−1} \frac{μ(x + y) − μ(x)}{y} ≥ 0. \end{aligned}

On the other hand, since f is quasiconformal it is continuous up to the boundary and in particular v(x, y) → 0 as y → 0 for any fixed x ∈ R. From this and (2.2) it follows that the right derivative of μ vanishes everywhere. That the left derivative vanishes everywhere can be proved in a similar way. Hence μ is constant, and this proves that v(z) = cy for some c > 0. Now Lemma 2.1 and its proof conclude the proof that (a) implies (b).

That (a) implies (c) follows from Lemma 2.1 and the implication ‘(a) implies (b)’. That (b) implies (a) follows from the definition of quasiconformality. Finally, it is well known that that (c) implies (a). This completes the proof of Theorem 1.1.

3. A representation of HQC(U)

For a harmonic mapping f = u + iv defined on U let

(3.1)  f(i) = b + ic, φ(z) = ∂u(z) := \frac{∂u}{∂z} = \frac{1}{2}(u_{x} − iu_{y}).

Since the function φ is holomorphic and

u(z) − u(i) = 2 Re ∫_{i}^{z} φ(ζ) dζ,

we have the following reformulation of Theorem 1.1.

Theorem 3.1. Each f ∈ HQC(U) has a unique representation of the form

(3.2)  f(z) = 2 Re ∫_{i}^{z} φ(ζ) dζ + b + ic Im(z),

where

(i) b + ic is a point in U, (ii) φ is a holomorphic function on U such that φ(U) is a relatively compact

subset of the right half-plane H.

Conversely, if (i) and (ii) are satisfied, then the function f defined by (3.2) belongs to HCQ(U).

4. Proof of Theorem 1.2

Let U be a real-valued function harmonic in U. Then there exists a unique harmonic function V , called the harmonic conjugate of U, such that V (i) = 0 and that the function U + iV is analytic in U.

Let U be the Poisson integral of ϕ ∈ L^{∞}(R), i.e., the harmonic function on U defined by

(4.1)  U(z) = \frac{1}{π} ∫_{−∞}^{+∞} P (z, t)ϕ(t) dt.

Then the harmonic conjugate of U is given by

(4.2)  V (z) = \frac{1}{π} ∫_{−∞}^{+∞}(Q(z, t) + \frac{t}{t^{2} + 1})ϕ(t) dt,

where

Q(z, t) = \frac{x − t}{|z − t|^{2}}

is the conjugate Poisson kernel. We have F = U + iV , where F is the analytic function defined by

F (z) = \frac{1}{π} ∫_{−∞}^{+∞}(\frac{i}{z − t} + \frac{t}{t^{2} + 1})ϕ(t) dt.

The Hilbert transformation of ϕ and the harmonic conjugate of U are connected by the formulae

(4.3)  lim_{y→0}(H_{y}ϕ(x) − V (x + iy)) = 0

and

(4.4)  V (z) = \frac{1}{π} ∫_{−∞}^{+∞} P (z, t)H(ϕ)(t) dt.

It is a simple but important fact that if a function u is harmonic in U, then so is u_{x} := ∂u/∂x, and the harmonic conjugate of u_{x} is equal to u_{y}(i) − u_{y}.

All the above facts can be found in Garnett [4]. ‘Only if’ part of Theorem 1.2 is a consequence of the following two lemmas.

Lemma 4.1. If f ∈ HQC(U), then the restriction ψ of f to the real axis is bi-Lipschitz and we have the relations

(4.5)  u_{x}(z) = \frac{1}{π} ∫_{−∞}^{+∞} P (z, t)ψ′(t) dt

and

(4.6)  u_{y}(z) − u_{y}(i) = −\frac{1}{π} ∫_{−∞}^{+∞}(Q(z, t) + \frac{t}{t^{2} + 1})ψ′(t) dt.

Proof. The function ψ is bi-Lipschitz on R because the mapping f is bi-Lipschitz on U. The function u_{x} is bounded on U and therefore, by Fatou’s theorem, there exists the limit

lim_{y→0} u_{x}(x, y) = ϕ(x)

for almost all x ∈ R. Furthermore, we have

u_{x}(z) = \frac{1}{π} ∫_{−∞}^{+∞} P (z, t)ϕ(t) dt.

Thus in order to prove (4.5) we have to prove that ψ′(x) = ϕ(x) almost everywhere. We start from the relation

u(x, y) − u(0, y) = ∫_{0}^{x} u_{t}(t, y) dt.

Since u_{t}(t, y) is bounded we have

lim_{y→0} ∫_{0}^{x} u_{t}(t, y) dt = ∫_{0}^{x} ϕ(t) dt,

by the dominated convergence theorem. On the other hand,

lim_{y→0}(u(x, y) − u(0, y)) = ψ(x) − ψ(0)

and therefore

ψ(x) − ψ(0) = ∫_{0}^{x} ϕ(t) dt,

which proves (4.5).

The validity of (4.6) now follows from (4.2) and the fact that the function V = −(u_{y} − u_{y}(i)) is equal to the harmonic conjugate of U = u_{x}.

Lemma 4.2. If f = u + iv ∈ HQC(R), and ψ = f|_{R} , then the function H(ψ′) belongs to L^{∞}(R) and the following equality holds:

(4.7)  u_{y}(z) − u_{y}(i) = −\frac{1}{π} ∫_{−∞}^{+∞} P (z, t)H(ψ′)(t) dt.

Proof. By Theorem 1.1 the function V (z) = u_{y}(i) − u_{y}(z) is bounded on U. Since V is the harmonic conjugate of U = u_{x}, we can use (4.5) and (4.3) (ϕ = ψ′) to conclude that H(ψ′) is in L^{∞}. Then formula (4.7) follows from (4.4).

To prove the ‘if’ part of Theorem 1.2, assume that ψ is bi-Lipschitz, |H(ψ′)| ≤ M =const. a.e. on R, and define U by (4.1), where ϕ = ψ′. Let V be the harmonic conjugate of U, let φ(z) = \frac{1}{2}(U(z) + iV (z)) and define f by

f(z) = 2 Re ∫_{i}^{z} φ(ζ) dζ + i Im(z).

From the inequality 1/C ≤ ψ′ ≤ C and (4.1) it follows that 1/C ≤ U(z) ≤ C (z ∈ U). Since |V (z)| ≤ M, by (4.4), we see that the function φ maps U onto a relatively compact subset of H. From Theorem 3.1 it follows that f ∈ HQC(U). Then by Lemma 4.1, the restriction ψ̃ of F to R is a bi-Lipschitz function and, since (Re f)_{x} = U on U, ψ̃′ = ψ′ a.e. on R. Since ψ and ψ̃ are absolutely continuous, ψ = ψ̃ + a for some a ∈ R. Thus defining f̃ = f + a we see that ψ = f̃|_{R} ∈ HQC(R), which completes the proof.

Boundary correspondence 165

5. A question

The set QS(R) is a group with respect to composition. Is this true for HQS(R)?

Acknowledgment. We would like to express our gratitude to the referee for many useful suggestions.

References

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[4] Garnett, J. B.: Bounded Analytic Functions. - Academic Press, 1981.

[5] Kalaj, D.: Harmonic functions and harmonic quasiconformal mappings between convex domains. - Thesis, Beograd, 2002.

[6] Martio, O.: On harmonic quasiconformal mappings. - Ann. Acad. Sci. Fenn. Math. 1968, 3–10.

[7] Partyka, D., and K. Sakan: A note on non-quasiconformal harmonic extensions. - Bull. Soc. Sci. Lettres Lodź 47, 1997, 51–63. Série: Recherches sur les déformations 23.

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Received 5 April 2004