Throughout this section, we assume that Ω is an unbounded simply connected domain in ℂ with 0 ∈ Ω. As in the previous section, σ represents the Riemann mapping of
onto Ω with σ(0) = 0 and σ'(0) > 0. We will also define φ via the Schröder functional equation. Throughout we let
so that δΩ(w) is the Euclidean distance from w to the boundary of Ω.
Theorem 1. Let φ be a univalent function mapping
into
, φ(0) = 0. Suppose that the closure of
intersects
only at finitely many fixed points and is contained in a Stolz angle of opening no greater than απ there.
If |φ'(0)| > 16 tan(απ/ 2) then the following are equivalent
1. C
φ
is compact on
;
2.
;
3. For every n > 0,
.
Remark: It has recently been shown by Smith [6] that compactness of C
φ
on
is equivalent to compactness of C
φ
on
, BMOA and VMOA when φ is univalent and so in the above theorem, the first condition could read: C
φ
is compact on
,
, BMOA and VMOA Before proceeding, we prove the following lemma.
Lemma 1. Under the hypotheses of the theorem, w and γw tend to the same prime end at ∞, and ∂γ Ω ⊂ Ω.
Proof. The first assertion follows from the fact that the closure of
touches
only at fixed points. Suppose now that the second assertion is false and there are distinct prime ends ρ1 and ρ2 with ρ1 = γρ2. Then under the boundary correspondence given by σ there are distinct points η,
with
It follows that
and therefore ζ is a fixed point of φ. Hence, we have the contradiction ρ1 = ρ2. □
Proof. We first prove that 1 is equivalent to 2.
By the results of Madigan and Matheson [2], and Smith [6] cited above C
φ
is compact on
if and only if
However, by Schröder's equation
Since Ω is simply connected, λΩ (w) ≍ 1/δΩ (w) and so C
φ
is compact on
if and only if
Since γ Ω ⊂ Ω, γw → ∂Ω implies that w → ∂Ω. Therefore, (3) holds if and only if
By the Lemma, we see that γw → ∂Ω means w → ∞ and w ∈ γ Ω, and we have shown that 1 and 2 are equivalent.
Suppose that 2 holds and let ε > 0 be given. Then we can find a R > 0 so that δΩ (w) <εδΩ (γw) for all |w| > R, since there are only a finite number of prime ends at ∞. Choose w ∈ Ω arbitrarily with modulus greater than R and let n satisfy |γ|-nR < |w| ≤ |γ |-n -1R.
Then we have that δΩ (w) < εnδΩ (γn w) and hence
Now as w → ∞ in γ Ω, γnw lies in a closed set properly contained in Ω and therefore δΩ (γn w) is bounded below by a constant independent of w. We thus have that
and since ε was arbitrary, the left-hand side of the above inequality must tend to ∞. Hence, we have shown that limw→∞|w| β δΩ (w) = 0 for every β > 0.
Now
may be interpreted geometrically as limw→∂Ωn|w| n-1δΩ (w) = 0 and this follows from the above argument. Therefore, 2 implies 3.
To show that 3 implies 2, we need to show that if
then 2 holds.
To complete the proof, we require the following lemma whose proof we merely sketch.
Lemma 2. Under the hypotheses of the theorem,
Sketch of Proof. First note that
Now if
lies in a non-tangential angle of opening απ at ζ, then a short calculation shows that
and the assertion follows. □
Now with f defined above, we have
for large enough w. Hence,
as w → ∞ and so 2 holds. □
It is of interest to consider the growth of σ since condition 3 would imply that it has very slow growth. The following corollary follows from 3 and the fact that functions in
grow at most of order log 1/(1 - |z|).
Corollary 1. Suppose that φ satisfies the hypotheses of the Theorem and that any of the equivalent conditions holds, then for r = |z|.
We also provide the following restatement of the hypotheses of Theorem 1 to illustrate the main properties of the Königs domain.
Corollary 2. Let Ω be an unbounded domain in ℂ with γ Ω ⊂ Ω and 0 ∈ Ω. Suppose that has Ω only finitely many prime ends at ∞ and
In addition, suppose that ∂γ Ω ⊂ Ω. If
, σ(0) = 0, σ'(0) > 0, and φ is defined by Schröder's equation, then the following are equivalent.
1. C
φ
is compact on
;
2.
;
3. For every n > 0,
.
The hypothesis on the boundary of Ω is vital. If we do not assume that ∂γ Ω ⊂ Ω, then we deduce from the proof of the Theorem that
is equivalent to
In this situation, the finite part of the boundary of Ω plays a complicated role in the behaviour of φ. We conclude this section by constructing a domain that displays very bad boundary properties. This answers a question of Madigan and Matheson in [2].
In [2] it was shown that if ∂φ(D) touches
in a cusp, then
. However, it is not sufficient that ∂φ(D) touches
at an angle greater that 0. The question was raised of whether or not it is possible that
can be infinite.
With the hypothesis that ∂γ Ω ⊂ Ω the prime ends at ∞ correspond to points of
that touch
. Therefore,
is at most countable. A natural question to ask is whether or not
can ever be positive, where Λ represents linear measure.
This example is well known in the setting of the unit disk, see [7, Corollary 5.3]. We describe here the construction in terms of the Königs domain.
Theorem 2. There is a univalent function
such that
.
Proof. We construct the domain Ω so that it satisfies (4). Let 0 < γ < 1 be given. We will define a nested sequence
, n = 1, 2, ... so that
where Θ
n
⊂ Θn+1for all n = 1, 2, ....
First let N > 2 be chosen arbitrarily and let Θ1 = {2πk/N : k = 0, ..., N - 1}.
Suppose now that Θ
n
has been defined, then let Θn+1be such that Θ
n
⊂ Θn+1and whenever θ ∈ Θ
n
is isolated, we define a sequence θ
k
∈ Θn+1, k = 1, 2, ..., so that θ
k
→ θ as k → ∞ and for each k there is a j so that θ - θ
k
= θ
j -θ
. Moreover, assume that
In this way, we define the sequence of sets Θ
n
, n = 1, 2, .... We will, furthermore, assume that for each
, there is a sequence θ
n
∈ Θ
n
, n = 1, 2, ...., such that θ
n
→ θ.
We claim that this gives the desired domain Ω with boundary defined by (5).
To see this, let γw ∈ Ω be arbitrary, then by construction, we may find a ζ ∈ ∂Ω so that δΩ (γw) = |ζ - γw|. It is readily seen that for such ζ, there is an n so that ζ ∈ {reiθ: r ≥ γ-n} for some θ ∈ Θ
n
and moreover, θ is isolated in Θ
n
.
If we now consider w, we may find a sequence θ
k
→ θ as k → ∞ so that
for all k hence we may fix a k so that δΩ (w) = |w - n| for
.
By estimating the line segment [w, η] by the arc of
joining w to η, we see that δΩ (w) ≍ |w|| α - θ
k
| where w = reiα.Therefore, we have the estimate δΩ (w) ≤ |w||θk+1- θ
k
|. By a similar argument, we deduce the estimate δΩ (γw) ≍ |γw||θ - θ
k
| and so
by (6) and so the construction is complete.
We claim that if
is defined as usual and φ is given by Schröder's equation, then
.
In fact, if θ ∈ Θ
n
is isolated, then the ray R = {reiθ: r ≥ γ-n-1} is contained in a single prime end of Ω. Therefore, to each such ray, there exists a point
that corresponds to R under σ. Since γR ⊂ ∂Ω, we thus have that ζ corresponds to a prime end p under φ with
.
On the other hand, if θ ∈ Θ
n
is isolated, then R' = {reiθ: γ-n≤ r < γ-n-1} satisfies γR'∩ ∂Ω = ∅, and so there is an arc
such that σ (ρ
θ
) = R' and ρ
θ
has an end-point in
.
Hence, each
is contained in a prime end of
and
The result follows. □