Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 19, 1994, 47–57

EXHAUSTIONS OF JOHN DOMAINS

Jussi Väisälä

Matematiikan laitos, PL 4 (Hallituskatu 15) FIN-00014 Helsingin yliopisto, Finland; jvaisala@cc.helsinki.fi

Abstract. We show that John domains in R^{n} can be exhausted by John subdomains. We also give a slightly modified definition for the John domains and prove a topological folk theorem on shortcuts of paths.

1. Introduction

This paper grew out from the following question of R. Hurri-Syrjänen: Can an unbounded c-John domain in R^{n} be expressed as a union of an ascending sequence of bounded c_{1} -John subdomains with c_{1} = c_{1}(c, n)? For terminology, see 3.3. We shall give an affirmative answer in 4.6. The result was applied by Hurri-Syrjänen [Hu, 4.6] to prove that unbounded John domains are Poincaré domains. We also construct the corresponding exhaustion for bounded John domains, but the proof in this case is rather easy and routine.

We also take this opportunity to give some basic analysis on John domains and on paths and arcs in Hausdorff spaces. The definition of a John domain is usually based on joining points by curves with suitable cigar or carrot neighborhoods. The curve can be considered to be either a path or an arc. We find it convenient to use an intermediate concept, called a road. We believe that this formulation is useful also elsewhere. A road is an equivalence class of paths in the relation defined by an increasing change of the parameter. The basic theory of roads is given in Section 3.

Arcs, roads and paths lead to the same concept of a c-John domain. This is due to the fact that a path can be replaced by an injective path by “leaving out some loops”. This is obvious if the path has only a finite number of selfintersections. In the general case, however, the result is somewhat deeper and seems to belong to the folklore. We give a rigorous treatment of this procedure in Section 2.

Notation. We let B(x, r) and S(x, r) denote open balls and spheres in R^{n} with center x and radius r, and we write B(r) = B(0, r), S(r) = S(0, r). For real numbers a, b we write a ∨ b = max(a, b) and a ∧ b = min(a, b).

1991 Mathematics Subject Classification: Primary 30C65.

2. Shortcuts

2.1. Terminology. Let X be a topological space. A path in X is a continuous map f: Δ → X where Δ = [a, b], a ≤ b, is a closed interval of the real line. Observe that we allow the degenerate case a = b. A shortcut of the path f is a triple (g, E, φ) where E is a compact subset of Δ, and φ: E → Δ′, g: Δ′ → X are continuous maps with the following properties:

(1) Δ is an interval, possibly degenerate.

(2) g is injective.

(3) φ is surjective and increasing.

(4) ∂Δ ⊂ E.

(5) f|E = g ∘ φ.

(6) If s, t ∈ E and s < t, then φ(s) = φ(t) if and only if the interval (s, t) is a component of Δ \ E.

Figure 1.

Thus gΔ′ is an arc joining the points f(a) and f(b) in fΔ. Intuitively, g is obtained from f by “leaving out loops”. An injective path f has the trivial shortcut (f, Δ, id). Figure 1 describes the simplest nontrivial shortcut. Here Δ = Δ′ = [0, 1] and E = [0, s] ∪ [t, 1].

We shall prove in 2.5 that every path f in a Hausdorff space has a shortcut. The shortcut is usually not unique in any sense. Our proof is a modification of an idea of J.L. Kelley, who used it to prove that a Peano space is arcwise connected; see [Wh, p. 39]. Observe that the latter result is a corollary of 2.5.

We first prove a lemma, which is a generalization of the construction of the Cantor function from the Cantor set. The proof is somewhat complicated, since the set may have interior points.

2.2. Lemma. Suppose that E is a compact set in I = [0, 1] such that {0, 1} ⊂ E and such that E \ {0, 1} is nonempty and has no isolated points. Then there is a continuous increasing function h: I → I such that for every y ∈ I, the preimage h^{−1}{y} is either a point in E or the closure of a component of I \ E. In particular, hE = I.

Proof. The set A = (I \ E) ∪ {0, 1} has a countable number of components A_{0}, A_{1}, . . .. The sequence may be finite or infinite. We choose the numbering so that 0 ∈ A_{0} and 1 ∈ A_{1} . Then A_{j} is an open interval (a_{j}, b_{j}) for j ≥ 2. Moreover, A_{0} is either {0} or an interval [0, b_{0}), and A_{1} is either {1} or an interval (a_{1}, 1].

The function h will have a constant value y_{j} in each A_{j}. We set y_{0} = 0 and y_{1} = 1. Proceeding inductively, assume that p ≥ 2 and that the numbers y_{0}, …, y_{p−1} have been defined in such a way that if A_{j} is the left-hand neighbor of A_{k} in {A_{0}, …, A_{p−1}}, then

(2.3)  y_{k} − y_{j} ≥ d(A_{j}, A_{k}).

To define y_{p}, let A_{j(p)} be the left-hand neighbor of A_{p} in {A_{0}, …, A_{p}}, and let A_{k(p)} be the right-hand neighbor. We set

(2.4)  y_{p} = (y_{j(p)} + a_{p} − b_{j(p)} + y_{k(p)} − a_{k(p)} + b_{p})/2.

Figure 2.

In Figure 2, y_{p} is the midpoint of the interval Δ. The numbers y_{0}, …, y_{p} clearly satisfy (2.3). Setting h|A_{j} = y_{j} we obtain an increasing function h: A → I with h(0) = 0, h(1) = 1.

Suppose that x ∈ E is a left-hand limit point of A. This means that the interval (x − r, x) meets A for all r > 0. We set

h_{1}(x) = sup {h(a) : a ∈ A, a < x}.

Similarly, if x is a right-hand limit point of A, we set

h_{2}(x) = inf {h(a) : a ∈ A : a > x}.

Moreover, we write h_{1}(0) = 0 and h_{2}(1) = 1. We next prove two facts: (i) If F = {u} is a degenerate component of E \ {0, 1}, then h_{1}(u) = h_{2}(u). (ii) If F = [s, t] is a nondegenerate component of E, then h_{1}(s) < h_{2}(t). Suppose that (i) is not true. Then h_{2}(u) − h_{1}(u) = q > 0. For sets P, Q in

the real axis we write P ≤ Q or P < Q if all elements of P and Q satisfy the corresponding inequality. Choose integers j and k such that

(1) A_{j} ≤ u ≤ A_{k},

(2) d(A_{j}, A_{k}) ≤ q/2,

(3) y_{k} − y_{j} ≤ 5q/4,

(4) for m = k ∨ j, the sets A_{j} and A_{k} are neighbors in {A_{0}, …, A_{m}}. Since u is not isolated in E, we can choose the smallest integer p > m for

which A_{j} < A_{p} < A_{k}. Suppose, for example, that u < A_{p}. Setting α = d(A_{j}, A_{p}) and β = d(A_{p}, A_{k}) we have y_{p} = (y_{j} + α + y_{k} − β)/2 by (2.4). Since α ≤ d(A_{j}, A_{k}) ≤ q/2, we obtain

q ≤ y_{p} − y_{j} ≤ (α + y_{k} − y_{j})/2 ≤ (q/2 + 5q/4)/2 < q.

This contradiction proves (i).

To prove (ii), it suffices to show that y_{k} − y_{j} ≥ d(F) whenever A_{j} ≤ F ≤ A_{k}. Write p = j∨k, and let A_{m} and A_{n} be the left-hand and the right-hand neighbors of F, respectively, in {A_{0}, …, A_{p}}. Then (2.3) implies that

d(F) ≤ d(A_{m}, A_{n}) ≤ y_{n} − y_{m} ≤ y_{k} − y_{j},

and (ii) follows.

To complete the proof, we must define h(x) for x ∈ E \ {0, 1}. Let F be the x-component of E. If F = {x}, we use (i) to define h(x) = h_{1}(x) = h_{2}(x). If F = [s, t] is nondegenerate, we define h|F to be the increasing affine map onto [h_{1}(s), h_{2}(t)]. Then h|F is strictly increasing by (ii). Now h is defined on the whole interval I, and it is easy to check that h has the desired properties.

2.5. Theorem. Every path f: Δ → X in a Hausdorff space X has a shortcut.

Proof. We may assume that Δ = I = [0, 1] and that f(0) ≠ f(1). Let H be the family of all compact sets F ⊂ I such that

(1) {0, 1} ⊂ F,

(2) f(s) = f(t) for every component (s, t) of I \ F.

We may assume that H ≠ ∅, since otherwise f is injective and has the trivial shortcut (f, I, id). Suppose that F_{1} ⊃ F_{2} ⊃ ⋯ is a decreasing sequence in H, and let F be the intersection of all the sets F_{j}. Clearly F satisfies (1). We show that it satisfies also (2). Let (s, t) be a component of I \ F. Choose ε with 0 < ε < (t − s)/2 and then an integer j such that (s + ε, t − ε) ⊂ I \ F_{j}. Let (u, v) be the component of I \ F_{j} containing (s + ε, t − ε). Then s ≤ u ≤ s + ε and t − ε ≤ v ≤ t. Since F_{j} ∈ H, we have f(u) = f(v). Since f is continuous and since X is Hausdorff, this implies that f(s) = f(t). Thus F ∈ H.

We can now apply Brouwer’s reduction theorem [Wh, p. 17] to choose a minimal member E of H. Then {0, 1} ⊂ E, and E \ {0, 1} and has no isolated points. Let h: I → I be the function given by Lemma 2.2 for this E. The restriction φ = h|E: E → I is a continuous increasing surjection satisfying the condition (5) in 2.1. We show that there is a unique map g: I → X with g ∘ φ = f|E. Assume that s, t ∈ E are such that s < t and φ(s) = φ(t). Then (s, t) is a component of I \ E by 2.2. Since E ∈ H, we have f(s) = f(t). This proves the uniqueness of g. Since φ is a continuous surjection of a compact set, φ is an identification map. Hence g is continuous.

To prove that (g, E, φ) is a shortcut of f, it suffices to show that g is injective. Assume that 0 ≤ u < v ≤ 1 with g(u) = g(v). Choose s, t ∈ E with φ(s) = u, φ(t) = v. Since φ is increasing, we have s < t. Moreover, (s, t) is not a component of I \ E, since otherwise φ(s) = φ(t). Thus E_{1} = E \ (s, t) is a proper subset of E. Since f(s) = f(t), E_{1} belongs to H, which contradicts the minimality of E.

2.6. Remark. Theorem 2.5 is not true in arbitrary topological spaces. For example, let X be the two-point space {0, 1} with the trivial topology and let f: I → X be any map with f(0) = 0, f(1) = 1. Then f is a path which has no shortcut.

3. Roads and John domains

3.1. Roads. Let X be a metric space. We say that two rectifiable paths f: Δ → X and f′: Δ′ → X are equivalent if they have the same arc-length parametrization f^{0}: [0, λ] → X where λ = l(f) = l(g) and l denotes the length. The equivalence class [f] of f is called a road in X.

If f = f′h for some increasing homeomorphism h: Δ → Δ′, then f and f′ are equivalent. The converse is true if f and f′ are not constant on any nondegenerate interval. We do not need nonrectifiable paths in this paper, but the equivalence relation could be extended to all paths in X as follows: f and g are equivalent if there is a path f_{0}: Δ_{0} → X and continuous increasing surjections h: Δ → Δ_{0} , h′: Δ′ → Δ_{0} such that f = f_{0}h and f′ = f_{0}h′.

Let E = [f] be a road with f: Δ → X, Δ = [a, b], a ≤ b. The starting point st E = f(a) and the terminal point ter E = f(b) of E are well-defined.

The inverse E^{−1} of E is the road [f ∘ h], where h: Δ → Δ is any decreasing homeomorphism. If F is a road with st F = ter E, the composition EF is welldefined. Moreover, the composition of roads is associative. If Δ_{1} is a subinterval of Δ, the road F = [f|Δ_{1}] is a subroad of the road E. The length of the road E is well-defined and written as l(E).

An arc in X is a subset homeomorphic to a closed interval, possibly degenerate. An arc α is directed if one of the endpoints is chosen to be the starting point st α. A directed arc α determines a unique road E with st E = st α, defined by any homeomorphism f: [a, b] → α with f(a) = st α. In the sequel, we shall identify the directed arc α and the road E. If x and y are points in R^{n}, we let [a, b] denote the directed line segment with starting point a and terminal point b.

3.2. Carrots and cigars. Let f: [a, b] → R^{n} be a rectifiable path in R^{n} and let c ≥ 1. The length c-carrot with core f and parameter c is the open set

car (f, c) = ⋃ {B(f(t), l(f|[a, t])/c) : a < t ≤ b}.

If f and g are equivalent paths, then clearly car (f, c) = car (g, c). Hence the carrot car (E, c) with E = [f] is well-defined. Indeed, we can write

car (E, c) = ⋃ {B(ter F, l(F)/c) : F subroad of E, st F = st E}.

Here E can also be a directed arc.

The length c-cigar with core f is the open set

cig (f, c) = ⋃ {B(f(t), l(f|[a, t]) ∧ l(f|[t, b])/c) : a < t < b}.

The cigar cig (E, c) with E = [f] is again well-defined and can be written as

cig (E, c) = ⋃ {B(ter F_{1}, (l(F_{1}) ∧ l(F_{2}))/c) : E = F_{1}F_{2}}.

We remark that in [NV] and in several other papers, we have used the notation car_{l} and cig_{l} instead of car and cig.

3.3. John domains. A domain D ⊂ R^{n} is a c-John domain, c ≥ 1, if each pair of points in D can be joined by an arc with cig (E, c) ⊂ D. For alternative characterizations, see [MS], [NV, Section 2], and Theorem 3.6 below. We first observe that in the definition, the arcs can be replaced by roads or by paths:

3.4. Theorem. Let D be a domain in R^{n} and let c ≥ 1. Then the following conditions are equivalent:

(1) D is a c-John domain.

(2) Each pair of points in D can be joined by a road E with cig (E, c) ⊂ D.

(3) Each pair of points in D can be joined by a path f with cig (f, c) ⊂ D.

Proof. Clearly (1) ⇒ (2) ⇒ (3). Suppose that (3) is true. Let a and b be points in D, and choose a path f: Δ → D joining a and b with cig (f, c) ⊂ D. By 2.5, there is a shortcut (g, F, φ) of f. Let s and t be points of F with s < t. Since φ is increasing, we easily see that the length of the arc g[φ(s), φ(t)] is not larger than the length of the path f|[s, t]. Let E be the image arc of g. Then cig (E, c) ⊂ cig (f, c) ⊂ D. Hence (1) is true.

3.5. The carrot property. Let D be a domain in R^{n} and let c ≥ 1. We say that D has the c-carrot property with center x_{0} ∈ D if each point x ∈ D can be joined to x_{0} by a road E such that car (E, c) ⊂ D. As in 3.4, the roads can be replaced by arcs or by paths. If D has the carrot property, then either D = R^{n} or D is bounded. Indeed, we have D ⊂ B(x_{0}, cd(x_{0}, ∂D)). For bounded domains, the carrot property is quantitatively equivalent to the John property. In fact, the carrot property was originally used as the definition of a John domain [MS]. We recall the precise formulation and the proof:

3.6. Theorem. Suppose that D is a domain in R^{n} and that c ≥ 1.

(1) If D has the c-carrot property, then D is a c-John domain.

(2) If D is a bounded c-John domain, then D has the c_{1} -carrot property with c_{1} = 4c^{2} .

Proof. (1) Let a_{0} and a_{1} be points in D. Assuming that a_{j} ≠ x_{0} we join a_{j} to the center x_{0} by a road E_{j} with car (E_{j}, c) ⊂ D. Then the road E = E_{0}E_{1}^{−1} joins a_{0} to a_{1} with cig (E, c) ⊂ D. If a_{0} ≠ x_{0} = a_{1} , then E = E_{0} has this property.

(2) Let B(x_{0}, r) be the largest ball contained in D. Fix x ∈ D \ {x_{0}}, and join x to x_{0} by an arc E with cig (E, c) ⊂ D. It suffices to show that car (E, c_{1}) ⊂ D. Let F be a subarc of E with st F = x. Setting y = ter F we show that d(y, ∂D) ≥ l(F)/c_{1} .

Since cig (E, c) contains a ball of radius l(E)/2c, we have l(E) ≤ 2cr. If |y − x_{0}| ≤ r/2, then

d(y, ∂D) ≥ r/2 ≥ l(E)/4c ≥ l(F)/c_{1}.

If |y − x_{0}| ≥ r/2, then

d(y, ∂D) ≥ (l(F) ∧ (r/2))/c ≥ (l(F)/c) ∧ (l(E)/4c^{2}) ≥ l(F)/c_{1}.

4. Exhaustions

In 4.6 we shall give the main result of this paper: Each c-John domain in R^{n} can be exhausted by relatively compact c_{1} -John subdomains, c_{1} = c_{1}(c, n). We first give some preparatory lemmas.

4.1. Lemma. Let E = F_{1}F_{2} be the composition of the roads F_{1}, F_{2} in R^{n}, and let c ≥ 1, c_{1} = c(2c + 1). Then

car (E, c_{1}) ⊂ car (F_{1}, c) ∪ car (F_{2}, c).

Proof. Write z = ter F_{1} = st F_{2} . Let F be a subroad of E with st F = st F_{1} . Set y = ter F. We must show that

(4.2)  B(y, l(F)/c_{1}) ⊂ car (F_{1}, c) ∪ car (F_{2}, c).

Since c_{1} > c, we may assume that F_{1} ⊂ F. We can write F = F_{1}F_{3} where F_{3} ⊂ F_{2} and st F_{3} = z. Setting r = l(F_{1})/c we have B(z, r) ⊂ car (F_{1}, c).

Case 1. l(F_{3}) ≤ r/2. Now |y − z| ≤ r/2, and hence B(y, r/2) ⊂ car (F_{1}, c). Since

l(F) = l(F_{1}) + l(F_{3}) ≤ cr + r/2 ≤ c_{1}r/2,

(4.2) follows.

Case 2. l(F_{3}) ≥ r/2. Now l(F_{1}) = cr ≤ 2cl(F_{3}). Hence

l(F) ≤ (2c + 1)l(F_{3}) = c_{1}l(F_{3})/c,

which implies that

B(y, l(F)/c_{1}) ⊂ B(y, l(F_{3})/c),

and we again obtain (4.2).

4.3. Lemma. Let c ≥ 1 and let Γ be a family of roads in R^{n} with the common terminal point x_{0} . Then the union

D = ⋃ {car (E, c) : E ∈ Γ}

is a domain with the c-carrot property with center x_{0} .

Proof. Since each car (E, c) is a domain containing x_{0} , D is a domain. Let x ∈ D \ {x_{0}}. There is E ∈ Γ with x ∈ car (E, c). We can write E = F_{1}F_{2} such that setting y = ter F_{1} = st F_{2} we have |x − y| < l(F_{1})/c. Now E_{1} = [x, y]F_{2} is a road from x to x_{0} . It is easy to see that car (E_{1}, c) ⊂ D.

4.4. Theorem. Suppose that D is a domain in R^{n} with the c-carrot property with center x_{0} , that Q ⊂ D is compact and that c′ > c. Then there is a domain G such that

(1) Q ⊂ G,

(2) \overline{G} is compact in D,

(3) G has the c′ -carrot property with center x_{0} .

Proof. If D = R^{n}, every ball G = B(x_{0}, r) containing Q has the desired properties. Suppose that D ≠ R^{n} and write 2r = d(Q, ∂D). Choose c_{1} with c < c_{1} < c′. Let G be the union of all carrots car (E, c′) such that

(i) ter E = x_{0} , (ii) d(st E, ∂D) ≥ r, (iii) car (E, c_{1}) ⊂ D. We show that G is the desired domain. The property (3) follows directly from 4.3.

To prove (1), let x ∈ Q and choose a road E_{0} from x to x_{0} such that car (E_{0}, c) ⊂ D. Next pick x_{1} ∈ D such that |x_{1} − x| ≤ r(1 ∧ (c_{1} − c)/c). Then E = [x_{1}, x]E_{0} is a road from x_{1} to x_{0} , and d(x_{1}, ∂D) ≥ r. It suffices to show that car (E, c_{1}) ⊂ D.

Let F be a subroad of E with st F = x_{1} , and set y = ter F. We must show that d(y, ∂D) ≥ l(F)/c_{1} . If F ⊂ [x_{1}, x], then

d(y, ∂D) ≥ r ≥ l(F) > l(F)/c_{1}.

If [x_{1}, x] ⊂ F, we write F = [x_{1}, x]F_{0} . Now F_{0} is a subroad of E_{0} , and hence l(F_{0}) ≤ cd(y, ∂D). If |y − x| ≤ r, then d(y, ∂D) ≥ r, which gives

l(F) = |x_{1} − x| + l(F_{0}) ≤ (c_{1} − c)r/c + cd(y, ∂D) ≤ c_{1}d(y, ∂D).

If |y − x| ≥ r, then l(F_{0}) ≥ r, and hence

l(F) ≤ (c_{1} − c)l(F_{0})/c + l(F_{0}) = c_{1}l(F_{0})/c ≤ c_{1}d(y, ∂D),

and (1) is proved.

To prove (2), let x ∈ D with d(x, ∂D) ≥ r, and let E be a road from x to x_{0} with car (E, c_{1}) ⊂ D. Let z ∈ car (E, c′). We must find a positive lower bound for d(z, ∂D). Choose a subroad F of E from x to a point y such that |z − y| < l(F)/c′. If l(F) ≤ r/3, then

|z − x| ≤ |z − y| + |y − x| ≤ 2l(F) ≤ 2r/3,

and hence d(z, ∂D) ≥ r/3. If l(F) ≥ r/3, then

d(z, ∂D) ≥ d(y, ∂D) − |z − y| ≥ l(F)/c_{1} − l(F)/c′ ≥ r/3c_{1} − r/3c′ > 0.

Hence d(G, ∂D) > 0.

4.5. Theorem. Suppose that D is an unbounded c-John domain in R^{n} and that Q ⊂ D is compact. Then there is a domain G such that

(1) Q ⊂ G,

(2) \overline{G} is compact in D ,

(3) G is a c_{1} -John domain with c_{1} = c_{1}(c, n).

Figure 3.

Proof. Choose R > 0 such that Q is contained in the ball B(R/2). We may assume that B(R) ⊄ D, since otherwise we can choose G = B(R/2). By an elementary packing argument we can choose a finite set A in the sphere S(R) such that the balls B(a, R/4c), a ∈ A, cover S(R) and such that the cardinality of A is a number N = N(c, n) independent of R.

Write A = {a_{1}^{1}, …, a_{N}^{1}}. Since D is a c-John domain, we can choose roads E_{j}^{1} from a_{j}^{1} to a_{j+1}^{1} , 1 ≤ j ≤ N − 1 such that cig (E_{j}^{1}, c) ⊂ D. We bisect the lengths of the roads E_{j}^{1} by points a_{j}^{2} and choose roads E_{j}^{2} from a_{j}^{2} to a_{j+1}^{2} , 1 ≤ j ≤ N − 2, such that cig (E_{j}^{2}, c) ⊂ D. We continue this process until we obtain the road E_{1}^{N−1} and its midpoint a_{1}^{N}. We write x_{0} = a_{1}^{N}. The process is schematically illustrated in Figure 3, where N = 5.

We see that each a ∈ A can be joined to x_{0} by a road E_{a} = F_{1} ⋯ F_{N−1} , where each F_{i} is either the first half of some road E_{j}^{i} or the inverse of the second half. Since cig (E_{j}^{i}, c) ⊂ D, we have car (F_{i}, c) ⊂ D. Applying 4.1 N − 2 times we see that car (E_{a}, c_{1}) ⊂ D for some c_{1} = c_{1}(c, n).

We next show that each point x ∈ D ∩ B(R/2) can be joined to a point z ∈ S(R) by a road F such that car (F, c) ⊂ D. Since D is unbounded, there is a point y ∈ D with |y| ≥ 9cR. Join x to y by a road E_{0} with cig (E_{0}, c) ⊂ D. Let F be the minimal subroad of E_{0} with st F = x, ter F = z ∈ S(R). Write λ = l(F) and choose a subroad F_{0} of F with st F_{0} = x, l(F_{0}) = λ/2. Then d(ter F_{0}, ∂D) ≥ λ/2c. Since B(R) meets ∂D, this implies that λ/2c ≤ 2R. Hence

2λ ≤ 8cR ≤ |y| − |x| ≤ l(E_{0}).

Since cig (E_{0}, c) ⊂ D, this gives car (F, c) ⊂ D.

Let x ∈ D ∩ B(R/2) and choose F and z ∈ S(R) as above. Then B(z, R/2c) ⊂ D. Pick a point a ∈ A with |a − z| < R/4c. Then car ([z, a], 1) = B(a, |z − a|) ⊂ D. Let E_{a} be the road described above from a to x_{0} with car (E_{a}, c_{1}) ⊂ D, and let E be the composition F [z, a]E_{a}. By 4.1, car (E, c_{2}) ⊂ D with some c_{2} = c_{2}(c, n). We have proved that each point of D ∩ B(R/2) can be joined to x_{0} by a road E with car (E, c_{2}) ⊂ D.

Write c_{3} = c_{2} + 1 and 2r = d(Q, ∂D). Let G be the union of all carrots car (E, c_{3}) such that

(i) ter E = x_{0} (ii) d(st E, ∂D) ≥ r, (iii) |st E| ≤ R, (iv) car (E, c_{2}) ⊂ D. We show that G is the desired domain. The property (3) follows from 4.3 and 3.6.

To prove (1), let x ∈ Q. Since Q ⊂ D ∩ B(R/2), there is a road E_{0} from x to x_{0} with car (E_{0}, c_{2}) ⊂ D. As in the proof of 4.4, we can show that if x_{1} is a point sufficiently close to x, then car ([x_{1}, x]E_{0}, c_{3}) ⊂ D. This gives (1).

To prove (2), let E be a road satisfying (i)–(iv), and let z ∈ car (E, c_{3}). As in the proof of 4.4, we obtain the lower bound d(z, ∂D) ≥ r/3c_{2} − r/3c_{3} > 0. Furthermore, since l(E) ≤ d(x_{0}, ∂D), we obtain the upper bound |z| ≤ R + 2c_{2}d(x_{0}, ∂D). Hence \overline{G} is compact in D.

4.6. Theorem. A c-John domain D ⊂ R^{n} can be written as the union of domains D_{1}, D_{2}, … such that

(1) \overline{D}_{j} is compact in D_{j+1} ,

(2) D_{j} is a c_{1} -John domain with c_{1} = c_{1}(c, n).

Proof. This follows directly from 4.4, 3.6 and 4.5. If D is bounded, c_{1} can be chosen to be any number greater than 4c^{2} .

4.7. Remark. It is natural to ask whether 4.6 is true if the John domains are replaced by uniform domains. P. Jones pointed out to the author that the answer is affirmative and can be proved using his ideas in [Jo]. The construction is rather complicated involving pipelines in Whitney cubes.

References

[Hu] Hurri-Syrjänen, R.: Unbounded Poincaré domains. - Ann. Acad. Sci. Fenn. Ser. A I Math. 17, 1992, 409–423.

[Jo] Jones, P.: A geometric localization theorem. - Adv. Math. 46, 1982, 71–79.

[MS] Martio, O., and J. Sarvas: Injectivity theorems in plane and space. - Ann. Acad. Sci. Fenn. Ser. A I Math. 4, 1979, 383–401.

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

[Wh] Whyburn, G.T.: Analytic topology. - Amer. Math. Soc. Colloquium Publications 28, 1942.

Received 27 October 1992