Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 23, 1998, 205–224

THE QUASIHYPERBOLIC METRIC, GROWTH, AND JOHN DOMAINS

Navah Langmeyer

886 Cecil Avenue, Millersville, MD 21108, U.S.A.; navah@zombie.ncsc.mil

Abstract. A result of Hardy and Littlewood relates Hölder continuity of analytic functions in the unit disk with a bound on the derivative. Gehring and Martio extended this result to the class of uniform domains. We further extend this result to the class of John domains.

1. Introduction

The research in this paper stems from two elements of classical function theory. The first is a criterion due to Hardy and Littlewood for a function to be Hölder continuous in the unit disk B ⊂ C. The second is a class of domains first considered by Fritz John in his studies of plane elasticity and rigidity of local quasi-isometries.

Suppose that f is a function analytic in the unit disk B ⊂ C and that 0 < α ≤ 1. Then the theorem of Hardy and Littlewood mentioned above asserts that

(1.1)  |f′(z)| ≤ m dist(z, ∂B)^{α−1}

for all z ∈ B if and only if

(1.2)  |f(z_{1}) − f(z_{2})| ≤ \frac{M}{α} |z_{1} − z_{2}|^{α}

for all z_{1}, z_{2} ∈ B, where m and M depend only on each other [HL]. By integration along hyperbolic geodesics and by using the Cauchy integeral formula, we obtain extensions of this result to the cases when α = 0 and α < 0 in (1.1) above. If α = 0, then (1.2) becomes

|f(z_{1}) − f(z_{2})| ≤ m log(1 + \frac{2|z_{1} − z_{2}|}{dist(z_{2}, ∂B)}) ≤ 2m log(1 + \frac{|z_{1} − z_{2}|}{dist(z_{2}, ∂B)}).

If α < 0, then (1.2) becomes

|f(z_{1}) − f(z_{2})| ≤ M( min_{j=1,2} dist(z_{j}, ∂B))^{α}.

The goal of this paper is to find geometric criteria for the validity of these extensions of the theorem of Hardy and Littlewood in a simply-connected plane domain. This goal is achieved by the following two main results, proved in Section 4.

1991 Mathematics Subject Classification: Primary 30C65; Secondary 30F45.

Theorem 1.1. If D ⊂ C is simply-connected, then D is a b-John domain if and only if f analytic and satisfying

|f′(z)| ≤ dist(z, ∂D)^{−1}

in D implies

|f(z_{1}) − f(z_{2})| ≤ a log(1 + \frac{λ_{D}(z_{1}, z_{2})}{min_{j=1,2} dist(z_{j}, ∂D)})

for all z_{1}, z_{2} ∈ D where a is a constant which depends only on the constant b.

Theorem 1.2. If D ⊂ C is bounded and simply-connected, then D is a b-John domain if and only if f analytic and satisfying

|f′(z)| ≤ dist(z, ∂D)^{α−1}

in D implies f is in Ord_{α}(D) with

|f(z_{1}) − f(z_{2})| ≤ c( min_{j=1,2} dist(z_{j}, ∂D))^{α}

for all z_{1}, z_{2} ∈ D where α < 0 and c is a constant which depends only on the constants b and α.

A domain D ⊂ R^{n} is a b-John domain if each pair of points x_{1}, x_{2} ∈ D can be joined by an arc γ ⊂ D for which

min_{j=1,2} l(γ(x_{j}, y)) ≤ b dist(y, ∂D)

for all y ∈ γ, where γ(x_{j}, y) is the subarc of γ with endpoints x_{j} and y. A domain is John if it is b-John for some constant b. John domains appear naturally in many areas of analysis, including complex dynamics, approximation theory, and elasticity. (See [NV], [MS].)

I would like to thank the referee for a careful reading and for several helpful comments which greatly improved this paper.

2. Preliminary results

We let dist(A, B) denote the euclidean distance from a set A ⊂ \overline{R}^{n} to a set B ⊂ \overline{R}^{n}. The euclidean distance between two points x, y ∈ R^{n} is denoted by |x − y|. Also, l(α) denotes the euclidean length of a rectifiable path α.

For x ∈ R^{n} and r > 0, B^{n}(x, r) denotes the ball centered at x of radius r. The unit disk in C, {z : |z| < 1}, is denoted by B.

Unless stated otherwise, D will always be a simply-connected domain in R^{n} with at least two boundary points taken with respect to the usual topology in \overline{R}^{n}, and D^{∗} = \overline{R}^{n} \\overline{D} is the exterior of D. A domain D ⊂\overline{C} is a conformal disk if it is conformally equivalent to B; i.e., D is a conformal disk if and only if ∂D is a non-degenerate continuum.

If c is a constant depending only on another constant b, we write “c = c(b)”. For α < 1, we define the α-quasihyperbolic metric k_{D}^{α} in a domain D ⊂ R^{n}

by

(2.1)  k_{D}^{α}(x_{1}, x_{2}) = inf_{γ} ∫_{γ} \frac{ds}{dist(x, ∂D)^{1−α}}

where the infimum is taken over all rectifiable arcs γ joining x_{1} and x_{2} in D. When α = 0, we have k_{D}^{0} = k_{D}, the usual quasihyperbolic metric (see [GO], e.g.).

We define the inner metric λ_{D} in a domain D by

λ_{D}(x_{1}, x_{2}) = inf_{β} l(β)

where β is any path joining x_{1} , x_{2} in D.

We say that D is a b-John domain, 1 ≤ b < ∞, if each pair of points x_{1}, x_{2} ∈ D can be joined by an arc γ ⊂ D for which

(2.2)  min_{j=1,2} l(γ(x_{j}, y)) ≤ b dist(y, ∂D)

for all y ∈ γ. Here γ(x_{j}, y) denotes the part of γ between x_{j} and y. We say that D is John if it is b-John for some b. It follows that a bounded domain D is a John domain if there exists a point x_{0} ∈ D such that each point x ∈ D can be joined to x_{0} by an arc γ ⊂ D for which

(2.3)  l(γ(x, y)) ≤ b dist(y, ∂D)

for all y ∈ γ. The point x_{0} is called a John center; we can take x_{0} such that dist(x_{0}, ∂D) = sup_{x∈D} dist(x, ∂D). Finally when a John domain D is also a conformal disk, we say that D is a John disk. (See [NV].)

An arc satisfying (2.2) is called a double b-cone arc. An arc satisfying (2.3) is called a b-cone arc. Quasihyperbolic and hyperbolic geodesics in b-John disks are double b_{1} -cone arcs, where b_{1} = b_{1}(b) [GHM], [NV].

A domain D is said to be b-uniform, 1 ≤ b < ∞, if each pair of points x_{1} , x_{2} in D \ ∞ can be joined by a rectifiable arc γ ⊂ D which in addition to (2.2) satisfies

l(γ) ≤ b|x_{1} − x_{2}|.

We say that D is uniform if it is b-uniform for some b. Since uniform domains satisfy (2.2), a uniform domain is a John domain. (See [G2], e.g.)

Given a set A in R^{n}, we let Lip_{α}(A), 0 < α ≤ 1, denote the Lipschitz class of mappings f: A → R^{p} satisfying for some m < ∞

(2.4)  |f(x_{1}) − f(x_{2})| ≤ m|x_{1} − x_{2}|^{α}

for all x_{1} , x_{2} in A. If D is a domain in R^{n}, then f: D → R^{p} belongs to the local Lipschitz class loc Lip_{α}(D) if there exists a constant m < ∞ such that (2.4) holds whenever x_{1} and x_{2} lie in any open ball B which is contained in D.

In Lip_{α}(D) and loc Lip_{α}(D) we shall use the seminorms ∥f∥_{α} and ∥f∥_{α}^{loc} , respectively,

∥f∥_{α} = inf{m : |f(x_{1}) − f(x_{2})| ≤ m|x_{1} − x_{2}|^{α}, x_{1}, x_{2} ∈ D}
∥f∥_{α}^{loc} = inf{m : |f(x_{1}) − f(x_{2})| ≤ m|x_{1} − x_{2}|^{α}, x_{1}, x_{2} ∈ B ⊂ D},

where B ranges over all balls contained in D.

Given a set A in R^{n}, we let Ord_{α}(A), α < 0, denote the class of mappings f: A → R^{p} satisfying for some m < ∞

(2.5)  |f(x_{1}) − f(x_{2})| ≤ m( min_{j=1,2} dist(x_{j}, ∂A))^{α}

for all x_{1} , x_{2} in A. We use this notation to parallel that of Lip_{α}, and because we are examining the order of growth of f(x) as x → ∂D. If D is a domain in R^{n}, then f: D → R^{p} belongs to the class loc Ord_{α}(D) if there exists a constant m < ∞ such that (2.5) holds whenever x_{1} and x_{2} lie in any open ball which is contained in D.

In Ord_{α}(D) and loc Ord_{α}(D) we shall use the seminorms ∥f∥_{α}^{∗} and ∥f∥_{α}^{∗ loc} , respectively,

∥f∥_{α}^{∗} = inf{m : |f(x_{1}) − f(x_{2})| ≤ m( min_{j=1,2} dist(x_{j}, ∂D))^{α}, x_{1}, x_{2} ∈ D}
∥f∥_{α}^{∗ loc} = inf{m : |f(x_{1}) − f(x_{2})| ≤ m( min_{j=1,2} dist(x_{j}, ∂D))^{α}, x_{1}, x_{2} ∈ B ⊂ D},

where B ranges over all balls in D.

3. John domains and the quasihyperbolic metric

Gehring and Osgood essentially showed (up to an additive constant) that a domain D is uniform if and only if it satisfies

k_{D}(x_{1}, x_{2}) ≤ cj_{D}(x_{1}, x_{2})

for all x_{1}, x_{2} ∈ D and some constant c, where

j_{D}(x_{1}, x_{2}) = \frac{1}{2} log(\frac{|x_{1} − x_{2}|}{dist(x_{1}, ∂D)} + 1)(\frac{|x_{1} − x_{2}|}{dist(x_{2}, ∂D)} + 1).

(See [GO], [G2, p. 97].) We define a similar metric j′_{D} by

j′_{D}(x_{1}, x_{2}) = \frac{1}{2} log(\frac{λ_{D}(x_{1}, x_{2})}{dist(x_{1}, ∂D)} + 1)(\frac{λ_{D}(x_{1}, x_{2})}{dist(x_{2}, ∂D)} + 1).

We find that k_{D} and j′_{D} are related in John disks.

Theorem 3.1. A simply-connected proper subdomain D ⊂ C is a b-John disk if and only if there exists a constant c such that

k_{D}(z_{1}, z_{2}) ≤ cj′_{D}(z_{1}, z_{2})

for all z_{1}, z_{2} ∈ D, with c = c(b) and b = b(c).

This result is a planar version of [KL, Theorem 4.1].

4. Domains in the plane

Throughout this section, we will take D ⊂ C.

In the previous section, we discussed analytic functions in the unit disk with various bounds on the derivative. The result of Hardy and Littlewood mentioned at the beginning of the section can be generalized to certain domains in C. Gehring and Martio extended this result to uniform domains [GM1, Corollary 2.2], and called the property that (1.1) implies (1.2) in D for every 0 < α ≤ 1 the Hardy– Littlewood property. This property does not characterize uniform domains; a more general geometric condition than uniformity which implies the Hardy–Littlewood property was introduced in [L]. The Hardy–Littlewood property was further studied by Astala, K. Hag, P. Hag, and Lappalainen [AHHL], in which the relationship between several types of domains with various geometric and extension properties was established. They called the property that (1.1) implies (1.2) in D for some α, 0 < α ≤ 1, the Hardy–Littlewood property of order α.

Gehring and Martio did find that a simply-connected domain D ⊂ C has the Hardy–Littlewood property of order α for some 0 < α ≤ 1 only if D is LLC_{1} (see (5.1.i)).

We will repeatedly use a generalization of a result by Kaufman and Wu [KW]. The proof is essentially identical, so we omit it here. We first introduce an extension of a distance function used by Kaufman and Wu. We define the distance function δ_{D}^{α} on a domain D ⊂\overline{C} for α ≤ 1 by

δ_{D}^{α}(z_{1}, z_{2}) = sup |f(z_{1}) − f(z_{2})|

where the supremum is taken over all analytic functions f on D with

(4.1)  |f′(z)| ≤ dist(z, ∂D)^{α−1}.

We see that δ_{D}^{α} is connected to the metric k_{D}^{α}, introduced in Section 2 and defined in (2.1).

Lemma 4.1 (See [KW, Theorem 1]). In a conformal disk D in C,

k_{D}^{α}(z_{1}, z_{2}) ≤ b_{α}δ_{D}^{α}(z_{1}, z_{2})

for all z_{1}, z_{2} ∈ D, where b_{α} is a constant depending only on α.

First, we examine the Hardy–Littlewood property, and some analogues, in John disks.

Theorem 4.2 ([GM1, Theorem 2.1]. If D is uniform and if f is defined and satisfies

|∂f(z)| = lim sup_{|h|→0} \frac{|f(z + h) − f(z)|}{|h|} ≤ m dist(z, ∂D)^{α−1}

in D, for some 0 < α ≤ 1, then f is in Lip_{α}(D) with

∥f∥_{α} ≤ \frac{cm}{α}

where c is a constant which depends only on the uniformity constant b.

This implies that a uniform domain has the Hardy–Littlewood property with constant depending only on b. The same is not true for John domains, as the following example shows.

Example 4.3. Let D = B\(−1, 0]. Then D is a John disk, but D does not have the Hardy–Littlewood property. For let

f(z) = z^{1/2},  z_{n} = \frac{1}{4}e^{iπn/(n+1)},  w_{n} = \frac{1}{4}e^{−iπn/(n+1)},

n = 1, 2, …. Then f is analytic and

|f′(z)| = \frac{1}{2}|z^{−1/2}| ≤ \frac{1}{2} dist(z, ∂D)^{(1/2)−1}

in D, since 0 ∈ ∂D, but

lim_{n→∞} |f(z_{n}) − f(w_{n})| = \frac{1}{2} + \frac{1}{2} = 1

while

lim_{n→∞} |z_{n} − w_{n}|^{1/2} = 0.

If we take f(z) = z^{α} for any α ∈ (0, 1), we will have

|f′(z)| = α|z^{α−1}| ≤ α dist(z, ∂D)^{α−1}

in D, but

lim_{n→∞} |f(z_{n}) − f(w_{n})| > 0

while

lim_{n→∞} |z_{n} − w_{n}|^{α} = 0.

An analogue of the Hardy–Littlewood property does hold in John disks, however. If we replace the euclidean metric with the inner metric, we get the following result.

Theorem 4.4. If D is a b-John disk and if f is defined and satisfies

(4.2)  |∂f(z)| ≤ m dist(z, ∂D)^{α−1}

in D for some 0 < α ≤ 1, then

|f(z_{1}) − f(z_{2})| ≤ \frac{cm}{α}λ_{D}(z_{1}, z_{2})^{α},

where c is a constant which depends only on b.

Proof. Fix z_{1}, z_{2} ∈ D and let γ be the hyperbolic geodesic joining z_{1}, z_{2} in D. Next let s denote arclength measured along γ from z_{1} , let l = l(γ), and let z(s) denote the corresponding representation for γ.

Set g(s) = f(z(s)). Then

|∂g(s)| = lim sup_{h→0} \frac{|g(s + h) − g(s)|}{|h|} ≤ |∂f(z(s))|.

Since D is a b-John disk,

min(s, l − s) ≤ b_{1} dist(z(s), ∂D),

b_{1} = b_{1}(b). Thus by (4.2),

|∂g(s)| ≤ m dist(z(s), ∂D)^{α−1} ≤ m(\frac{min(s, l − s)}{b_{1}})^{α−1}

for 0 < s < l, and g is absolutely continuous. The Gehring–Hayman inequality gives a constant c_{0} > 0 such that any curve δ ⊂ D with endpoints z_{1} , z_{2} satisfies

l(γ) ≤ c_{0}l(δ)

[GH, Theorem 2], [Ja]. So we have

\begin{aligned}|f(z_{1}) − f(z_{2})| & ≤ ∫_{0}^{l} |∂g(s)| ds ≤ 2mb_{1}^{1−α} ∫_{0}^{l/2} s^{α−1} ds \\ & ≤ \frac{2b_{1}^{(1−α)}m}{α}(\frac{c_{0} λ_{D}(z_{1}, z_{2})}{2})^{α} ≤ \frac{cm}{α}λ_{D}(z_{1}, z_{2})^{α}, \end{aligned}

c = c(b).

We now have Corollary 4.5. If D is a b-John disk and if f is analytic and satisfies

|f′(z)| ≤ m dist(z, ∂D)^{α−1}

for z in D, then

|f(z_{1}) − f(z_{2})| ≤ \frac{cm}{α}λ_{D}(z_{1}, z_{2})^{α},

for all z_{1} , z_{2} in D, where c is a constant which depends only on b.

Next, we examine the case α = 0.

Theorem 4.6. A conformal disk D ⊂ C is b-uniform if and only if f analytic and satisfying

(4.3)  |f′(z)| ≤ dist(z, ∂D)^{−1}

in D implies

(4.4)  |f(z_{1}) − f(z_{2})| ≤ a log(1 + \frac{|z_{1} − z_{2}|}{min_{j=1,2} dist(z_{j}, ∂D)})

for all z_{1}, z_{2} ∈ D where a is a constant which depends only on the constant b.

Proof. First, suppose D is b-uniform. Then by (3.1),

k_{D}(z_{1}, z_{2}) ≤ c log(1 + \frac{|z_{1} − z_{2}|}{dist(z_{1}, ∂D)})(1 + \frac{|z_{1} − z_{2}|}{dist(z_{2}, ∂D)})

≤ a log(1 + \frac{|z_{1} − z_{2}|}{min_{j=1,2} dist(z_{j}, ∂D)})

for all z_{1}, z_{2} ∈ D, where a depends only on b. If f is analytic and satisfies (4.3) in D, then

|f(z_{1}) − f(z_{2})| ≤ k_{D}(z_{1}, z_{2}) ≤ a log(1 + \frac{|z_{1} − z_{2}|}{min_{j=1,2} dist(z_{j}, ∂D)})

as desired.

Now suppose that every f analytic and satisfying (4.3) in D also satisfies (4.4). By Lemma 4.1,

k_{D}(z_{1}, z_{2}) = k_{D}^{0}(z_{1}, z_{2}) ≤ c_{0}δ_{D}^{0}(z_{1}, z_{2}) ≤ a log(1 + \frac{|z_{1} − z_{2}|}{min_{j=1,2} dist(z_{j}, ∂D)})

for all z_{1}, z_{2} ∈ D. So by (3.1), D is uniform.

Theorem 4.7. A conformal disk D ⊂ C is b-John if and only if f analytic and satisfying

(4.5)  |f′(z)| ≤ dist(z, ∂D)^{−1}

in D implies

(4.6)  |f(z_{1}) − f(z_{2})| ≤ a log(1 + \frac{λ_{D}(z_{1}, z_{2})}{min_{j=1,2} dist(z_{j}, ∂D)})

for all z_{1}, z_{2} ∈ D where a is a constant which depends only on the constant b.

Proof. First, suppose D is b-John. Then by Theorem 3.1,

k_{D}(z_{1}, z_{2}) ≤ a log(1 + \frac{λ_{D}(z_{1}, z_{2})}{min_{j=1,2} dist(z_{j}, ∂D)})

for all z_{1}, z_{2} ∈ D, where a depends only on b. If f is analytic and satisfies (4.5) in D, then

|f(z_{1}) − f(z_{2})| ≤ k_{D}(z_{1}, z_{2}) ≤ a log(1 + \frac{λ_{D}(z_{1}, z_{2})}{min_{j=1,2} dist(z_{j}, ∂D)})

as desired.

Now suppose that every f analytic and satisfying (4.5) in D also satisfies (4.6). By Lemma 4.1,

k_{D}(z_{1}, z_{2}) = k_{D}^{0}(z_{1}, z_{2}) ≤ c_{0}δ_{D}^{0}(z_{1}, z_{2}) ≤ a log(1 + \frac{λ_{D}(z_{1}, z_{2})}{min_{j=1,2} dist(z_{j}, ∂D)})

for all z_{1}, z_{2} ∈ D. So by Theorem 3.1, D is John.

Finally, we examine the case α < 0.

Theorem 4.8. If D ⊂ C is bounded and simply-connected, then D is a b-John disk if and only if f analytic and satisfying

(4.7)  |f′(z)| ≤ dist(z, ∂D)^{α−1}

in D implies f is in Ord_{α}(D) with

∥f∥_{α}^{∗} ≤ c

where α < 0 and c is a constant which depends only on the constants b and α.

Proof. First suppose D is a b-John disk, and let f be analytic and satisfying (4.7) in D. Then by integration over double b-cone arcs and by the Cauchy integral formula, f is in Ord_{α}(D).

Now suppose that whenever f is a function analytic in D satisfying (4.7), then f is in Ord_{α}(D). Then, by Lemma 4.1,

k_{D}^{α}(z_{1}, z_{2}) ≤ b_{α}δ_{D}^{α}(z_{1}, z_{2}) = b_{α} sup |f(z_{1}) − f(z_{2})| ≤ b_{α}c( min_{j=1,2} dist(z_{j}, ∂D))^{α}

where the supremum is taken over all analytic functions f on D satisfying (4.7). Theorem 6.5 now implies that D is a John domain.

5. Geometric properties and extension properties

Recall that a simply-connected domain D ⊂ \overline{C} is a K -quasidisk if and only if it is uniform with constant b, K = K(b), b = b(K) [MS]. We say an arbitrary set E ⊂ \overline{C} is c-linearly locally connected (c − LLC), c a constant, if for z_{0} ∈ C and 0 < r < ∞,

(5.1)

(i) points in E ∩\overline{B}(z_{0}, r) can be joined in E ∩\overline{B}(z_{0}, cr); and
(ii) points in E \\overline{B}(z_{0}, r) can be joined in E \\overline{B}(z_{0}, r/c).

The set E is c − LLC_{1} if it satisfies (5.1.i), and c − LLC_{2} if it satisfies (5.1.ii). A simply-connected domain D is a K -quasidisk if and only if it is c-linearly locally connected, K = K(c), c = c(K) [G1], and D is a b-John disk if and only if it satisfies (5.1.ii), b = b(c), c = c(b) [NV, 4.6].

Gehring and Martio found that the geometric property (5.1.i) is necessary for the Hardy–Littlewood property of order α, 0 < α < 1, in simply-connected domains D ⊂ C [GM1, Theorem 3.3]. They go on to use the Hardy–Littlewood property to characterize quasidisks with ∞ in the boundary [GM1, Theorem 4.2]. Combining these results with the work in this paper yields the following corollary.

Corollary 5.1. Suppose that D ⊂ C is bounded and simply-connected. Then D is a quasidisk if and only if for some α_{1} < 0, 0 < α_{2} < 1,

(5.2)  |f′(z)| ≤ dist(z, ∂D)^{α_{1}−1}

in D implies f ∈ Ord_{α_{1}}(D) and

(5.3)  |f′(z)| ≤ dist(z, ∂D)^{α_{2}−1}

in D implies f ∈ Lip_{α_{2}}(D).

Proof. First, suppose D is a quasidisk. Then D is a John disk, so by Theorem 4.8 (5.2) implies f ∈ Ord_{α}(D) for all α < 0. Also, D is uniform, and so it has the Hardy–Littlewood property, i.e. (5.3) implies f ∈ Lip_{α}(D) for all 0 < α < 1.

Now suppose that for some α_{1} < 0, 0 < α_{2} < 1, (5.2) implies f ∈ Ord_{α_{1}}(D) in D and (5.3) implies f ∈ Lip_{α_{2}}(D) in D. By Theorem 4.8 D is a John disk, and thus satisfies (5.1.ii). By [GM1, Theorem 3.3], D satisfies (5.1.i). Therefore, D is a quasidisk.

6. Domains in R^{n}

The Hardy–Littlewood property can be extended to higher dimensions by using the concept of loc Lip_{α}, introduced in Section 2. Recall that a function f: D → R^{p} belongs to the local Lipschitz class loc Lip_{α}(D), 0 < α ≤ 1, if there exists a constant m < ∞ such that (2.4) holds whenever x_{1} and x_{2} lie in any open ball B which is contained in D. For functions analytic in a domain D ⊂ C, this is equivalent to the bound on the derivative

|f′(z)| ≤ m dist(z, ∂D)^{α−1}.

This extension to R^{n} was done by Gehring and Martio in the following way. A domain D ⊂ R^{n} is called a Lip_{α} -extension domain if there exists a constant a depending only on D and α such that f ∈ loc Lip_{α}(D) implies f ∈ Lip_{α}(D) with

∥f∥_{α} ≤ a∥f∥_{α}^{loc}

(see [GM2]).

Gehring and Martio, in their examination of Lip_{α} -extension domains, first showed that this extension property is equivalent to a geometric condition. They then used this condition to get subsequent necessary and sufficient conditions for D to be a Lip_{α} -extension domain. Our work in this section parallels the work of Gehring and Martio on Lip_{α} -extension domains, 0 < α ≤ 1, in a way analogous to the preceding sections. We find local properties which for functions analytic in a domain D ⊂ C are equivalent to the bound on the derivative

|f′(z)| ≤ m dist(z, ∂D)^{α−1},

for α = 0 and α < 0. We then define and examine extension domains for these properties.

We first examine Bloch-extension domains.

Theorem 6.1. A domain D ⊂ R^{n} is a uniform domain if and only if

(6.1)  |f(x_{1}) − f(x_{2})| ≤ log(1 + \frac{|x_{1} − x_{2}|}{min_{j=1,2} dist(x_{j}, ∂D)})

for all x_{1}, x_{2} ∈ D with |x_{1} − x_{2}| < dist(x_{1}, ∂D) implies

(6.2)  |f(x_{1}) − f(x_{2})| ≤ a log(1 + \frac{|x_{1} − x_{2}|}{min_{j=1,2} dist(x_{j}, ∂D)})

for all x_{1}, x_{2} ∈ D.

Proof. Suppose D is uniform. Then by (3.1),

k_{D}(x_{1}, x_{2}) ≤ a_{1} log(1 + \frac{|x_{1} − x_{2}|}{min_{j=1,2} dist(x_{j}, ∂D)})

for all x_{1}, x_{2} ∈ D. Let f satisfy (6.1). Fix x_{1}, x_{2} ∈ D and let γ ⊂ D be the quasihyperbolic geodesic with endpoints x_{1} and x_{2} . Let γ(s) be the parameterization of γ with respect to arc length measured from x_{1} , l = l(γ). Let y_{1} = x_{1} . We choose positive numbers r_{i} and l_{i}, and points y_{i} ∈ γ as follows:

r_{1} = \frac{1}{2}dist(y_{1}, ∂D),
r_{2} = \frac{1}{2}dist(y_{2}, ∂D),

l_{1} = max\{s : γ(s) ∈\overline{B}^{n}(y_{1}, r_{1})\},_{l_{2} = max\{s : γ(s) ∈\overline{B}^{n}(y_{2}, r_{2})\},} y_{2} = γ(l_{1});_{y_{3} = γ(l_{2});}

and so on. After a finite number of steps, N, say, l_{N} = l and the process stops. Let y_{N+1} = x_{2} . So by [GHM, Lemma 2.6],

|f(x_{1}) − f(x_{2})| ≤ ∑_{i=1}^{N} a_{2} log(1 + \frac{|y_{i} − y_{i+1}|}{dist(y_{j+1}, ∂D)})

≤ a_{2} ∑_{i=1}^{n} k_{D}(γ(y_{i}, y_{i+1})) ≤ a log(1 + \frac{|x_{1} − x_{2}|}{min_{j=1,2} dist(x_{j}, ∂D)})

as desired.

Now suppose (6.1) implies (6.2) in D. Fix x_{0} ∈ D. Let

f(x) = k_{D}(x, x_{0}).

If x_{1}, x_{2} ∈ B ⊂ D, B an open ball, then

|f(x_{1}) − f(x_{2})| ≤ k_{D}(x_{1}, x_{2}).

Let γ ⊂ B be the segment of the circle through x_{1} , x_{2} perpendicular to ∂B with endpoints x_{1} , x_{2} . Then

l(γ) ≤ π|x_{1} − x_{2}|

and

min_{j=1,2} l(γ(x, x_{j})) ≤ π dist(x, ∂B) ≤ π dist(x, ∂D)

for all x ∈ γ. Following the same argument used in the proof of [KL, Theorem 4.1] we get

k_{D}(x_{1}, x_{2}) ≤ ∫_{γ} \frac{ds}{dist(x, ∂D)} ≤ c log(1 + \frac{|x_{1} − x_{2}|}{min_{j=1,2} dist(x_{j}, ∂D)})

where c is independent of x_{0} and B, i.e. (6.1) holds. So

k_{D}(x, x_{0}) ≤ a_{1} log(1 + \frac{|x − x_{0}|}{min\{dist(x, ∂D), dist(x_{0}, ∂D)\}})

for all x ∈ D, where a_{1} is independent of x_{0} . Thus

k_{D}(x_{1}, x_{2}) ≤ a_{1} log(1 + \frac{|x_{1} − x_{2}|}{min_{j=1,2} dist(x_{j}, ∂D)})

for all x_{1}, x_{2} ∈ D, and hence D is uniform by (3.1).

A domain D ⊂ R^{n} is called an Ord_{α} -extension domain, α < 0, if there exists a constant a depending only on D and α such that f ∈ loc Ord_{α}(D) implies f ∈ Ord_{α}(D) with

∥f∥_{α}^{∗} ≤ a∥f∥_{α}^{∗ loc}.

Theorem 6.2. Let α < 0. A domain D ⊂ R^{n} is an Ord_{α} -extension domain if and only if there is a constant M < ∞ such that each x_{1}, x_{2} ∈ D can be joined by a rectifiable curve γ ⊂ D with

(6.3)  ∫_{γ} \frac{ds}{dist(x, ∂D)^{1−α}} ≤ M( min_{j=1,2} dist(x_{j}, ∂D))^{α}.

Proof. First, suppose D is an Ord_{α} -extension domain. So there is a constant c > 0 such that f ∈ loc Ord_{α}(D) implies f ∈ Ord_{α}(D) with

∥f∥_{α}^{∗} ≤ c∥f∥_{α}^{∗ loc}.

Fix x_{0} ∈ D and define a function f by

f(x) = k_{D}^{α}(x_{0}, x) = inf_{γ} ∫_{γ} \frac{ds}{dist(y, ∂D)^{1−α}}

where the infimum is taken over all rectifiable arcs γ joining x_{0} and x in D. By the triangle inequality,

(6.4)  |f(x_{1}) − f(x_{2})| ≤ k_{D}^{α}(x_{1}, x_{2})

for all x_{1}, x_{2} ∈ D.

Assume that x_{1} , x_{2} belong to an open ball B which is contained in D. Let γ be the subarc between x_{1} and x_{2} of the circle through x_{1} , x_{2} perpendicular to ∂B. For x ∈ γ,

min_{j=1,2} l(γ(x_{j}, x)) ≤ \frac{1}{2}π dist(x, ∂B) ≤ \frac{1}{2}π dist(x, ∂D).

We then parameterize γ with respect to arc-length measured from x_{1} and let d_{j} = dist(x_{j}, ∂D), j = 1, 2. Then

∫_{γ} \frac{ds}{dist(x, ∂D)^{1−α}} ≤ (∫_{0}^{d_{1}/2} + ∫_{l−d_{2}/2}^{l} + ∫_{d_{1}/2}^{l−d_{2}/2})\frac{ds}{dist(γ(s), ∂D)^{1−α}}

≤ \frac{d_{1}^{α}}{2^{α}} + \frac{d_{2}^{α}}{2^{α}} + ∫_{d_{1}/2}^{l−(d_{2}/2)} \frac{(π/2)^{1−α} ds}{(min\{s, l − s\})^{1−α}}

≤ m( min_{j=1,2} dist(x_{j}, ∂D))^{α},

where m depends only on α. Together with (6.4) this gives

|f(x_{1}) − f(x_{2})| ≤ m( min_{j=1,2} dist(x_{j}, ∂D))^{α}

and so f ∈ loc Ord_{α}(D). By the assumption, f ∈ Ord_{α}(D) and ∥f∥_{α}^{∗} has an upper bound which is independent of x_{0} . The definition of f now yields (6.3).

Now suppose that D satisfies the condition (6.3). Let f be a function with

(6.5)  |f(x_{1}) − f(x_{2})| ≤ dist(x_{1}, ∂D)^{α}

whenever x_{1}, x_{2} ∈ D with |x_{1}−x_{2}| ≤ \frac{1}{2} dist(x_{1}, ∂D). Note that if f ∈ loc Ord_{α}(D), then f satisfies (6.5) up to a constant.

Fix x_{1} and x_{2} in D. We can find a rectifiable arc γ ⊂ D joining x_{1} and x_{2} such that

(6.6)  ∫_{γ} \frac{ds}{dist(y, ∂D)^{1−α}} ≤ M( min_{j=1,2} dist(x_{j}, ∂D))^{α}.

Let γ(s) be the parameterization of γ with respect to arc length measured from x_{1} , l = l(γ). Let y_{1} = x_{1} . We choose positive numbers r_{i} and l_{i}, and points y_{i} ∈ γ as follows:

r_{1} = \frac{1}{4}dist(y_{1}, ∂D),
r_{2} = \frac{1}{4}dist(y_{2}, ∂D),

l_{1} = max\{s : γ(s) ∈\overline{B}^{n}(y_{1}, r_{1})\},_{l_{2} = max\{s : γ(s) ∈\overline{B}^{n}(y_{2}, r_{2})\},} y_{2} = γ(l_{1});_{y_{3} = γ(l_{2});}

and so on. After a finite number of steps, N, say, l_{N} = l and the process stops. Let y_{N+1} = x_{2} . We now have

(6.7)  |f(x_{1}) − f(x_{2})| ≤ ∑_{i=1}^{N} |f(y_{i}) − f(y_{i+1})| ≤ c_{α} ∑_{i=1}^{N} dist(y_{i}, ∂D)^{α}.

Let l_{0} = 0 and for each i = 1, 2, …, N − 1, let

A_{i} = \{s ∈ [l_{i−1}, l_{i}] : γ(s) ∈\overline{B}^{n}(y_{i}, r_{i})\}.

Then A_{i} is closed, and

(6.8)  h_{1}(A_{i}) ≥ r_{i} = |y_{i} − y_{i+1}| = \frac{1}{4}dist(y_{i}, ∂D)

where h_{1} is 1-dimensional Hausdorff measure. Moreover, for s ∈ A_{i},

dist(γ(s), ∂D) ≤ |γ(s) − y_{i}| + dist(y_{i}, ∂D) ≤ 5r_{i},

and hence

dist(γ(s), ∂D)^{1−α} ≤ 5^{1−α}r_{i}^{1−α}.

Together with (6.8) this yields

(6.9)

∫_{γ} \frac{ds}{dist(y, ∂D)^{1−α}} ≥ ∑_{i=1}^{N−1} ∫_{A_{i}} \frac{ds}{dist(γ(s), ∂D)^{1−α}}

≥ c_{α} ∑_{i=1}^{N−1} r_{i}^{−(1−α)}h_{1}(A_{i}) = c_{α} ∑_{i=1}^{N−1} dist(y_{i}, ∂D)^{α}.

Also, x_{2} ∈\overline{B}^{n}(y_{N}, \frac{1}{4} dist(y_{N}, ∂D)), and so

(6.10)  dist(x_{2}, ∂D) ≤ \frac{5}{4} dist(y_{N}, ∂D).

Then combining (6.7), (6.9), (6.10), and (6.9) yields

|f(x_{1}) − f(x_{2})| ≤ c_{α} ∫_{γ} \frac{ds}{dist(y, ∂D)^{1−α}} + c_{α}(\frac{5}{4} dist(x_{2}, ∂D))^{α}

≤ c_{α}( min_{j=1,2} dist(x_{j}, ∂D))^{α}.

Thus f ∈ Ord_{α}(D) with ∥f∥_{α}^{∗} depending only on α, and D is an Ord_{α} -extension domain.

We will need a characterization of John domains in R^{n}, the proof of which requires two lemmas.

Lemma 6.3. If there exists an arc γ ⊂ D joining x_{1} and x_{2} in D with

∫_{γ} \frac{ds}{dist(x, ∂D)^{1−α}} ≤ M( min_{j=1,2} dist(x_{j}, ∂D))^{α},

then there is a number a = a(M, α) with

dist(y, ∂D) ≥ a( min_{j=1,2} dist(x_{j}, ∂D)) for all y ∈ γ.

Proof. We may assume that dist(x_{1}, ∂D) ≤ dist(x_{2}, ∂D). Fix y ∈ γ. If x_{1} ∈\overline{B}^{n}(y, dist(y, ∂D)), then

dist(x_{1}, ∂D) ≤ |x_{1} − y| + dist(y, ∂D) ≤ 2 dist(y, ∂D)

and so dist(y, ∂D) ≥ \frac{1}{2} dist(x_{1}, ∂D). Likewise, if x_{2} ∈\overline{B}^{n}(y, dist(y, ∂D)), then

dist(y, ∂D) ≥ \frac{1}{2} dist(x_{2}, ∂D) ≥ \frac{1}{2} dist(x_{1}, ∂D).

Now assume x_{1}, x_{2} ∉\overline{B}^{n}(y, dist(y, ∂D)). Let

a = min\{\frac{1}{2}, (\frac{2^{1+α−2}}{αM} )^{−1/α}\}

so that

\frac{−2}{α}(1 − 2^{α})a^{α} ≥ M.

Then if dist(y, ∂D) < a dist(x_{1}, ∂D), we get

∫_{γ} \frac{ds}{dist(x, ∂D)^{1−α}} ≥ 2 ∫_{0}^{dist(y,∂D)} \frac{dt}{(t + dist(y, ∂D))^{1−α}} > Mdist(x_{1}, ∂D)^{α},

a contradiction. Therefore, dist(y, ∂D) ≥ a dist(x_{1}, ∂D) for all y ∈ γ.

Lemma 6.4. Let x_{1} , x_{2} , and γ all be as in Lemma 6.3 above. Then, for δ ∈ (0, ∞),

h_{1}{y ∈ γ : dist(y, ∂D) ≤ δ} ≤ δ^{1−α}M( min_{j=1,2} dist(x_{j}, ∂D))^{α}

where h_{1} is 1-dimensional Hausdorff measure.

Proof. Fix a δ, and let γ′ = {y ∈ γ : dist(y, ∂D) ≤ δ}. Then

∫_{γ′} \frac{ds}{dist(x, ∂D)^{1−α}} ≥ \frac{h_{1}(γ′)}{δ^{1−α}},

and

∫_{γ′} \frac{ds}{dist(x, ∂D)^{1−α}} ≤ ∫_{γ} \frac{ds}{dist(x, ∂D)^{1−α}} .

Thus, by our hypothesis,

\frac{h_{1}(γ′)}{δ^{1−α}} ≤ M( min_{j=1,2} dist(x_{j}, ∂D))^{α}

which gives the desired result.

Theorem 6.5. A bounded domain D is a b-John domain if and only if there is a constant M < ∞ and α < 0 such that each x_{1}, x_{2} ∈ D can be joined by a rectifiable curve γ ⊂ D with

(6.11)  ∫_{γ} \frac{ds}{dist(x, ∂D)^{1−α}} ≤ M( min_{j=1,2} dist(x_{j}, ∂D))^{α}.

Proof. First, suppose D is a b-John domain. So for any pair of points x_{1}, x_{2} ∈ D, we can find an arc γ ⊂ D joining x_{1} and x_{2} with

min_{j=1,2} l(γ(y, x_{j})) ≤ b dist(y, ∂D)

for all y ∈ γ. We may assume dist(x_{1}, ∂D) ≤ dist(x_{2}, ∂D). Let γ(s) be the parameterization of γ with respect to arc-length measured from x_{1} and let d_{j} = dist(x_{j}, ∂D), j = 1, 2.

If l(γ) = l ≥ dist(x_{1}, ∂D), we have the following:

k_{D}^{α}(x_{1}, x_{2}) ≤ ∫_{0}^{l} \frac{ds}{dist(γ(s), ∂D)^{1−α}}

≤ \frac{d_{1}^{α}}{2^{α}} + \frac{d_{2}^{α}}{2^{α}} + ∫_{d_{1}/2}^{l−(d_{2}/2)} \frac{b^{1−α}ds}{(min\{s, l − s\})^{1−α}} ≤ c_{1}( min_{j=1,2} dist(x_{j}, ∂D))^{α}

where c_{1} depends only on α and b.

If l < dist(x_{1}, ∂D), then x_{2} ∈ B^{n}(x_{1}, dist(x_{1}, ∂D)) and there is a universal constant c such that dist(y, ∂D) ≥ c dist(x_{1}, ∂D) for all y in [x_{1}, x_{2}]. So

k_{D}^{α}(x_{1}, x_{2}) ≤ ∫_{[x_{1},x_{2}]} \frac{ds}{dist(y, ∂D)^{1−α}} ≤ c^{1−α} dist(x_{1}, ∂D)^{α}.

Let M = sup{c_{1}, c^{1−α}}.

For the converse, fix x_{0} ∈ D with

dist(x_{0}, ∂D) = max_{x∈γ} dist(x, ∂D) = d_{0}.

Fix x_{1} ∈ D and let γ_{1} be a rectifiable curve joining x_{1} and x_{0} in D satisfying

(6.12)  ∫_{γ_{1}} \frac{ds}{dist(y, ∂D)^{1−α}} ≤ M dist(x_{1}, ∂D)^{α}.

We will construct a rectifiable curve γ ⊂ D joining x_{1} and x_{0} satisfying

l(γ(y, x_{1})) ≤ b dist(y, ∂D)

for all y ∈ γ, where b depends only on α. This will imply that D is John. Let y_{1} = x_{1} .

If 2 dist(x_{1}, ∂D) ≥ d_{0} , then
dist(y, ∂D) ≤ d_{0} ≤ 2 dist(x_{1}, ∂D)

for all y ∈ γ_{1} , and so by Lemma 6.4

l(γ_{1}) = h_{1}{y ∈ γ_{1} : dist(y, ∂D) ≤ 2 dist(x_{1}, ∂D)} ≤ 2^{1−α}M dist(x_{1}, ∂D)

and by Lemma 6.3

dist(y, ∂D) ≥ a dist(x_{1}, ∂D)

for all y ∈ γ_{1} . Then we let y_{2} = x_{0} , and we stop.

If 2 dist(x_{1}, ∂D) < d_{0} , then by Lemma 6.4,

h_{1}{y ∈ γ_{1} : dist(y, ∂D) ≤ 2 dist(x_{1}, ∂D)} ≤ (2 dist(x_{1}, ∂D))^{1−α}M dist(x_{1}, ∂D)^{α}
= 2^{1−α}M dist(x_{1}, ∂D).

So we can find y_{2} ∈ γ_{1} with dist(y_{2}, ∂D) = 2 dist(x_{1}, ∂D) and l(γ_{1}(y_{1}, y_{2})) ≤ 2^{1−α}M dist(x_{1}, ∂D). By Lemma 6.3,

dist(y, ∂D) ≥ a dist(x_{1}, ∂D)

for all y ∈ γ_{1}(y_{1}, y_{2}). Let γ_{2} be a rectifiable curve joining y_{2} and x_{0} in D with

∫_{γ_{2}} \frac{ds}{dist(y, ∂D)^{1−α}} ≤ M dist(y_{2}, ∂D)^{α} = M2^{α} dist(x_{1}, ∂D)^{α}.

Once again, if 4 dist(x_{1}, ∂D) ≥ d_{0} , then

dist(y, ∂D) ≤ d_{0} ≤ 4 dist(x_{1}, ∂D)  for all y ∈ γ_{1}(y_{2}, x_{0}),
l(γ_{2}) ≤ 2^{2−α}M dist(x_{1}, ∂D),
dist(y, ∂D) ≥ 2a dist(x_{1}, ∂D)  for all y ∈ γ_{2}.

Then we let y_{3} = x_{0} , and we stop. Otherwise, we find y_{3} ∈ γ_{2} with dist(y_{3}, ∂D) = 4 dist(x_{1}, ∂D) and l(γ_{2}(y_{2}, y_{3})) ≤ 2^{2−α}M dist(x_{1}, ∂D), as above. By Lemma 6.3,

dist(y, ∂D) ≥ 2a dist(x_{1}, ∂D)

for all y ∈ γ_{2}(y_{2}, y_{3}).

Continue this process to get points y_{j} ∈ B and curves γ_{j}(y_{j}, y_{j+1}) ⊂ D, j = 1, …, m such that letting y_{m+1} = x_{0} yields

(6.13) dist(y_{j}, ∂D) = 2^{j−1} dist(x_{1}, ∂D), l(γ_{j}(y_{j}, y_{j+1})) ≤ 2^{j−α}M dist(x_{1}, ∂D),

and if y ∈ γ_{j}(y_{j}, y_{j+1}), we have

(6.14)  dist(y, ∂D) ≥ 2^{j−1}a dist(x_{1}, ∂D)

The quasihyperbolic metric, growth, and John domains 223

for j = 1, …, m. Let

γ = ⋃_{j=1}^{m} γ_{j}(y_{j}, y_{j+1}).

For any y ∈ γ, there is an N such that y ∈ γ_{N}[y_{N}, y_{N+1}), and so by (6.13)

(6.15) l(γ(x_{1}, y)) ≤ ∑_{j=1}^{N} 2^{j−α}M dist(x_{1}, ∂D) = (2^{N} − 1)2^{1−α}M dist(x_{1}, ∂D)

and (6.14) implies

(6.16)  dist(y, ∂D) ≥ 2^{N−1}a dist(x_{1}, ∂D).

Combining (6.15) and (6.16), we get

\frac{2^{2−α}M}{a} dist(y, ∂D) ≥ \frac{(2^{N} − 1)2^{1−α}M}{2^{N−1}a} dist(y, ∂D) ≥ l(γ(x_{1}, y))

for all y ∈ γ. Thus D is a John domain.

Combining Theorem 6.5 and Theorem 6.2 yields our main result, Theorem 6.6. Theorem 6.6. A bounded domain D is a b-John domain if and only if D

is an Ord_{α} -extension domain for some α < 0.

Remark. Note that in the proof of Theorem 6.5, we have in fact shown that in a bounded b-John domain,

k_{D}^{α}(x_{1}, x_{2}) ≤ M( min_{j=1,2} dist(x_{j}, ∂D))^{α}

in D for every α < 0 (here M varies with α). So such a John domain is an Ord_{α} - extension domain for every α < 0. In addition, we get the following corollary.

Corollary 6.8. A bounded domain D is an Ord_{α} -extension domain for some α < 0 if and only if D is an Ord_{α} -extension domain for all α < 0.

References

[AHHL] Astala, K., K. Hag, P. Hag, and V. Lappalainen: Lipschitz classes and the Hardy– Littlewood property. - Monats. Math. 115, 1993, 267–279.

[G1] Gehring, F. W.: Characteristic Properties of Quasidisks. - Les Presses de l’Université de Montréal, 1982.

[G2] Gehring, F. W.: Uniform domains and the ubiquitous quasidisk. - Jahresber. Deutsch. Math.-Verein. 89, 1987, 88–103.

[GH] Gehring, F. W., and W. K. Hayman: An inequality in the theory of conformal mapping. - J. Math. Pures Appl. 41, 1962, 353–361.

224 Navah Langmeyer

[GHM] Gehring, F. W., K. Hag, and O. Martio: Quasihyperbolic geodesics in John domains. - Math. Scand. 65, 1989, 75–92.

[GM1] Gehring, F. W., and O. Martio: Quasidisks and the Hardy–Littlewood property. - Complex Variables Theory Appl. 2, 1983, 67–78.

[GM2] Gehring, F. W., and O. Martio: Lipschitz classes and quasiconformal mappings. - Ann. Acad. Sci. Fenn. Ser. A I Math. 10, 1985, 203–219.

[GO] Gehring, F. W., and B. G. Osgood: Uniform domains and the quasi-hyperbolic metric. - J. Analyse Math. 36, 1979, 50–74.

[GP] Gehring, F. W., and B. P. Palka: Quasiconformally homogeneous domains. - J. Analyse Math. 30, 1976, 172–199.

[HL] Hardy, G. H., and J. E. Littlewood: Some properties of fractional integrals. II. - Math. Z. 34, 1932, 403–439.

[HR] Heinonen, J., and S. Rohde: The Gehring–Hayman inequality for quasihyperbolic geodesics. - Math. Proc. Cambridge Philos. Soc. 114, 1993, 393–405.

[Ja] Jaenisch, S.: Length distortion of curves under conformal mapping. - Michigan Math. J. 15, 1968, 121–128.

[KL] Kim, K., and N. Langmeyer: Harmonic measure and hyperbolic distance in John disks. - Math. Scand. (to appear).

[KW] Kaufman, R., and J. M. Wu: Distances and the Hardy–Littlewood property. - Complex Variables Theory Appl. 4, 1984, 1–5.

[L] Lappalainen, V.: Lip_{h} -extension domains. - Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertations 56, 1985.

[M] Martin, G. J.: Quasiconformal and bi-Lipschitz homeomorphisms, uniform domains and the quasihyperbolic metric. - Trans. Amer. Math. Soc. 292, 1985, 169–191.

[MS] Martio, O., and J. Sarvas: Injectivity theorems in plane and space. - Ann. Acad. Sci. Fenn. Ser. A I Math. 4, 1978/79, 383–401.

[NV] Näkki, R., and J. Väisälä: John disks. - Exposition. Math. 9, 1991, 3–43.

Received 23 October 1996